(v), from values of R/(^') ^i^d V/L), while for CO2, using only Regnault^s coefficient of expansion and Joule and his own cooling effects, he found — 273°*9. Now that CO2 is seen to be in harmony with the other two more perfect gases, the number 27o may be accepted definitely as the absolute temperature of melting ice. The equation therefore applies accurately at high volumes, a fact which we can prove by another test, seeing that Amagat carried out a special research ( Compt.liend. xciii.) to determine the ratios olpv top'v' at different temperatures and up to values of y about 8 atmospheres, v being double v', Q2 220 Mr. William Sutherland on the Table YI. (Yaliies o^ pv/p'v' at high volumes for CO2.) (pzzzh'l metres of mercury : v = 2v\) Temperature 0 50°. 100°. 200°. 1-0040 1-0026 300°. Arnaaat 10145 1-0156 1-0087 1-0092 10020 1-0000 Calculated As the experiments are not free from liability to an error of 1 in 1000, the agreement is again close enough to prove the applicability of the equation at high volumes. In the sequel it will be shown that this equation applies to the great majority of compounds, but meanwhile the only other experimental determinations similar to those ah-eady discussed for ethyl oxide and carbonic dioxide are Amagat^s and C2H, ; Rothes for SO^ and NH3 (Wied. forH2,02,N2,CH, Ann. xi.) ; Janssen^s for N2O (Wied. Beihl. ii.) ; and Ramsay and Young's on methyl and ethyl alcohol (Phil. Mag. Aug. 1887). Our form of characteristic equation applies to SO2, NH3, and ^2^ successfully, but not to H2, Oo, No, and CH4, which require a still simpler type, the alcohols other hand requiring a less simple type. These values of k and I for SOo, NH^,, and NoO : — '2? on are the the 25 S02. 2-08 2740 NH3. 4-8 22040 2-3 3420 with which the following pressures have been calculated for comparison with the experimental data, the latter being taken direct from air or nitrogen manometer without correction for departure from Boyle^s law. Table YII. SO2 at 99° -6. INH3 at 99^ -6. N,0 at 25°-l. i ^'- pexp. p caJc. V. p exp. p ealc. V. 7-0 ^exp. 39-1 p calc. 41-9 7-9 8-0 172 7-6 7-6 37-2 17-6 17-1 170 85-7 14-8 14-7 5-36 42 6 42-4 13-6 21-2 20-8 65-4 18-8 18-9 3-78 43-9 46-7 99 26-0 26-2 40-0 28-6 291 At 183° At 183° At 43°-8. 18-2 21-7 220 160 10-3 10-2 5-83 49-4 47-9 10-0 37-3 36-9 81-6 198 19-6 4-62 55-6 54-4 4-58 69-2 67-5 36-1 40-9 42-1 3-47 61-4 61-0 2-75 98-2 93-5 ! 16-2 82-6 85-4 2-82 641 64-3 Laws of Molecular Force. 221 This comparison has been made only to show that the form is apphcable to other bodies as well as to ethyl oxide and car- bonic dioxide ; full confirmation of the form will come later on, in the study of many of its applications. 2. Estahlishment of the Characteristic Equation for the Gaseous Elements, with proof of continuity during liquefaction. — The simplest plan in the case of the gaseous elements will be to take nitrogen as typical and tabulate for it E,i/(r) and v'^(f)(v) from Amagat's experiments up to 320 metres of mercury. Table VIII.- —Nitrogen. V. :Rvf(v). v-'

{v) is the tendency of ^vf {v) at low A^olumes to double its perfect gas-value. In the case of Hg and O2 the two functions run a similar course to that for N2, but it is a more unexpected fact that they also do the same for the compound methane, CH4, as is shown in Table IX. Table IX.- -Methane. V. Jivfiv). v'^iv). V. nvf{v). v^,p{v). Perfect gas. 3-908 1211 5-24 6900 32-3 4-16 i 1009 5-43 6400 28-2 4-30 5600 ! 807 5-95 6500 l'4-2 4-39 ()2()0 7-27 6-47 7000 20 2 4-73 6900 6-46 680 6800 161 4-73 6200 605 6-73 ()200 It is evident that we have here to do with v'cfy^v) as a constant, 222 Mr. William Sutlierland on the that is with an internal virial varying inversely as the volume down to near the critical volume, and ^vfiv) tending some- where near that point to about double its value in the perfect gas state. The course of ^vf(y) in these four gases is repre- sented by the simple form which attains the value 2R when v = ^. teristic equation down to v = kh I - i?i; = ET 1+ — \ V — Hence the charac- a form which I had already adopted for air (Phil. Mag. Aug. 1887). The following are the values for h and I : — H. N,. 0^. CH,. Air. h . . 12-0 2-64 1-78 5-51 2-47[2^11] I . . 41700 1175 851 6460 1110[910] The values given in brackets for air are those previously found by me from Amagat's data (Compt. Rend, xcix.), but as these data are not carried to such high pressures as those for ^2 ^nd O2, I have calculated values for air by adding to four fifths of the values for E^2 oi^e fiftl^ ^f the values for O2. This equation is almost identical with that of Yan der Waals, but it is a little simpler. It gives the following pres- sures for comparison with Amagat's experimental results : — Table X. Hydrogen. At 17-7° 0. V. pexp. 166-9 100-1 60-1 46-7 57-5 99 176 238 56' 99 176 241 At 100° C. 66-9 74-2 00-1 129 601 230 46-7 311 73-3 128 229 315 Nitrogen. At 17-7° 0. 13-83 6-91 461 3 69 3-23 p exp. 46 92 145 194 223 At 100 C. p calc. 13-83 60 691 125 4-61 200 3 69 270 3-23 320 45-7 91 142 188 9:?(\ 60-5 124 199 266 323 Oxygen. At 14-7° C. 5-73 3-58 2-58 2-43 j> exp. 141 201 216 'p calc At 100° C. 5-73 123 3-58 204 2-58 301) 2-43 322 89-7 142 203 219 124 204 301 327 Laics of Molecular Force. 223 The experimental numbers for oxygen are taken from Amagat's data in the Comptes Rendus, xci. The most delicate test we can apply to our form at high volumes is, in the case of air, to compare the calculated with the experimental Thomson and Joule cooling effect. When I did this with the previous equation for air (^ = 2*11 and Z=910), I assumed the difference '7° to exist between the melting-point of ice on the thermodynamic and gas ther- mometers ; but, as already pointed out, Sir W. Thomson having proved this difference not to exist, there must have been a compensation of errors in the application of the previous equation. Thomson's expression for the cooling effect, applied to our equation for air, becomes Yipdhidp = nimi-kl2, which gives the following calculated values : — Cooling effects of air escaping through a porous plug into the atmosphere under a pressure excess of 2'54: metres of mercury. Temperature C 7°' 1. Experiment . . . '88 Calculation . . . '84 The agreement is the closest to be looked for and proves the accuracy of our equation for air at high volumes. At low volumes we can test the form for all the elementary gases and CH^^ by applying it to the calculation of the critical volume, pressure, and temperature in each case. To do this at the present stage we must assume that our form can be trusted to hold not only to the critical volume but also a little past into the liquid region, a legitimate assumption for the elements, where we have seen the internal virial varying inversely as the volume, and so giving a guarantee of con- tinuity, but not legitimate for the compounds where dis- continuity occurs. Then, applying James Thomson's idea of the passage from the gaseous to the liquid state, as pre- cisionized by Maxwell and Clausius, we have the critical point determined by the conditions '^pI'^v^O^ '^^p/'^v'^ = {J. Along with the characteristic equation these lead to the fol- lowing values; — critical volume Vc = ok/2 ; critical temperature Tc=l()//27R^; critical pressure pc = 4//27/cV"-to compare with the experimental values found by Olszewski for O2 and N2 (Compt. Rend, c), by Wroblewski for air, and by Dewar for CH^ (Phil. Mag. 1884, xviii.). 17°. 39°-5. 92°-8. '86 •75 •51 •80 •71 •55 224 Mr. ^Yilliam Sutherland on the Table XL H^. N,. 0, CH,. Air. Critical f exper. . . . Volume. [calc Critical fexper. ... Temperature. (^ calc Critical fexper. ... Pressure. \ calc —229 '"19 3-4 3-96 -146 -165 27 25 2-5 2-67 -119 -127 38 40 -99-5 -95 37 32 -140 -149 30 27 The agreement is all that can be looked for in view of the difficulties of measuring these low critical temperatures and their associated pressures. With regard to hydrogen all we know is that Olszewski [Compt. Rend, ci.) has submitted it to a temperature estimated by him as —220° without a sign of liquefaction. If our equation is to be trusted, it would indi- cate that he would need to go some 10 degrees lower before the only unliquefied gas is conquered. Wroblewski has pub- lished data on the compressibihty of llg up to pressures of 70 atmospheres at temperatures of —103° and —182° (Journ, Chem. Soc. 1889), and with these our equation is in accord, but there is hardly need of tabulated proof. 3. Brief discussion of exceptional Compounds such as the Alcohols and Ethylene. — To complete our survey of the ex- perimental materia] on bodies above the critical region we have to consider Ramsny and Young's observations on methyl and ethyl alcohol, and Amagat^s on ethylene. Ramsay and Young point out that at low volumes the values of B/^/bT for the alcohols are not so reliable as for ethyl oxide, being deter- mined from a smaller temperature range ; hence our values of Iiv/(t;) and v^(p[v) are not so reliable as before, but they suffice to show the exceptional nature of these bodies. Table XII.— Methyl Alcohol. V. nvAv). ■v^-

{v) by the form vl/{v + a) ; so that the characteristic
equation for ethylene is
^ \ vj t; + a
with ^ = 4-15, a=:l-64, and Z = 6270.
The form for ethylene is intermediate in simplicity between
that for the simple gases and that for compounds, except that
it has an extra constant,
forms
k // k
2
It is also worth noting- that the
(v— o), k/Vj and 2k/{v-\-k)
are special cases of a general form
nk/{v + {n — '[)k},
with n = i, 1, and 2.
n
Laics of Molecular Force. '227
4. Estahlishment of Characteristic Equation below the
region of the Critical Volume. — Now that we have practically
exhausted the available data of the gaseous state, we see that
by themselves they do not give much scope for generaUzation ;
but if we can secure an equation applicable from the critical
volume down to the volumes of liquids in the ordinary state,
then, with two equations covering almost the whole range of
fluidity, we shall have a much larger experimental area laid
under contribution for information on the characters of mole-
cules.
Already we have secured one important fact towards the
acquisition of such an equation, namely that below the critical
volume the internal virial term varies inversely as the volume ;
and in the case of ethyl oxide we know its actual amount l/2v
with Z = 5514. We have therefore only to add to //2y Ramsay
and Young^s values oi pv at different temperatures for different
volumes below the critical, and we obtain the values of the
kinetic-energy term in the desired equation ; we can then
proceed to study how this quantity depends on temperature
and volume_, and express the resulting conclusions in a
formula.
As to the form we have this clue, that it must join on con-
tinuously with the previous one where that ceases to be appli-
cable. Now the first fact to notice is that our form for
compounds above the critical region cannot, like that for the
elements, give a critical point by itself at all ; for given p
and T it is not a cubic but a quadratic in v, and hence cannot
give us the three equal roots which are adopted as charac-
teristic of the critical point when we apply the conditions
This emphasizes the discontinuity in compounds as contrasted
with elements. However, we know as an experimental fact
that at the critical point 'dpl'dv = 0, which with the charac-
teristic equation gives us only two relations between the
critical temperature, pressure, and volume. As a third rehition
that would perfectly define these three quantities I was led to
believe that the critical volume is proportional to k, and found
critical volume v^ = 7k/6
is the relation which, with the two others, gives successfully
the numerics of the critical state in agreement with ex-
periment. As this will be proved subsequently (Section 10)
for a large number of substances, I will not delay at present to
give examples, except for those compounds for which we have
already found k and /.
228
Mr. William Sutherland on the
Table ^IN.
Critical temperature, T^=120?/409E^ ; critical pressure,
(C,H,),0.
CO,
SO,
NH3.
1
N2O.
Critical f exper. ...
Temperature. \ calc
Critical f exper. . . .
Pressure. \ calc
194
199
27-1
29-3
32
52
59
78-6
155
125
60
56
130
96
87
84
35
36
57
57
The want of accuracy in the agreement in parts of this
table is to be ascribed partly to inaccuracy in the ordinary
determinations of the critical point, as I have already pointed
out that capillary action must sometimes largely affect the
numerics of the critical state when these are determined in
capillary tubes (Phil. Mag. August 1887). Regnault, in
his account of his experiments on the saturation-pressures of
CO2, expressly declares that he had liquid COg at 42°^ which
is 10° above the apparent critical temperature in capillary
tubes ; and Cailletet and Colardeau {Compt. Rend, cviii.)
have shown that although the meniscus between gas and
liquid CO2 disappears to the eye about 31° or 32°, yet cha-
racteristic diff'erences between liquid and gas can be proved
to exist several degrees higher than this. Hence an error of
at least 10° is possible in ordinary determinations of critical
temperatures. On the other hand, an error of 5 per cent, in
the value of an absolute temperature of about 400° as given
by our equation would amount to 20°. Table XIY. is to be
taken in the light of these facts.
We have now ascertained a second property that our
equation for volumes below the critical is to possess : it must
begin to apply when v=7k/6, as the other form cannot apply
below this volume at the critical temperature. At this volume
the kinetic-energy term in our form above the critical region
becomes
RT(H- 12/13), or 25RT/13;
so that 25E/13 is the lower limit of the term which in the new
equation is to take the same place as 'Rvf{v) hitherto. Hence
for this term the form
25E(l + F(zO)/13
naturally suggests itself, and as F(i') is to vanish when
v = 7k/6, we get (7k/6 — v)/'\lr(v') as a suggestion for its form ;
and it only 1 emains from the data obtained, as 1 have said, by
Laws of Molecular Force. 229
adding Ij^v to Ramsay and Young^s values oi pv for volumes
of ethyl oxide below k, to determine the form of the function
-^{v). This was found, after a rather tedious search, to come
out in the simple form
VT(»-/3)/B ;
SO that finally we have the following as the equation for ethyl
oxide below the volume k :
^.RT(l+^-.t3>
l_
\ 15 V — P,
with the following values for the constants :
R'=25Il/13, k' = 7k/6, B = 63-l, /5=1-11,
R, k, and I as before.
I propose to call this the infracritical equation. It is to be
noticed that we have introduced only two additional constants ;
so that, as regards number of constants, we could hardly
look for a simpler form.
Above the volume 7k/6 the appropriate form was proved
to be
^ \ v-{-k/ v + k
which I propose to call the supracritical equation.
Between k and lk/6 v/e have the circacritical form
pv
This, then, gives the complete representation of ethyl oxide
in the fluid state if we establish the sufficiency of the infra-
critical form, as we now proceed to do. In the next table are
compared the pressures found by Ramsay and Young and
those given by the equation.
Table XY.— Liquid Ethyl Oxide.
Volume
3-7.
2-75.
2-25.
2.
1-9.
IQqO n f Pressure, exper. .
■ 1 „ calc. ...
28
32
29
26
43
45
i^Ko n f Pressure, exper. .
^'^ ^-l „ calc. ...
19
14
43
48
1 nt\o i-i f Pressure, exper. .
^^^ ^' [ „ calc. ...
19-5
20
230
Mr. William Sutherland on the
For the proper appreciation of this table it must be borne
in mind that as soon as we enter the liquid region the pv term
of the characteristic equation becomes the small difference of
tvvo terms, a small percentage error in either of which becomes
a large one in pv. The fact that the above table brings out
is that from 150° to 195° the relation between volume and
temperature given by the equation is so accurate as to make
only the small errors in pressure in the above table. But to
show this more directly, we will now compare the volumes of
the liquid under a pressure of 9 metres of mercury between
0^ and 100°, as determined by Grim^aldi (Wied. Beihl. x.) and
as given by the equation. The specific gravity of ethyl oxide
at 0° and under one atmosphere is taken as '7366.
Table XVI. (p= 19*5 metre.;
0°.
50°.
100°.
150°.
Volume, experiment
1-355
1-354
1-469
1-467
1630
1633
1-9
1-9
This, taken in conjunction with Table XY., shows that the
equation represents with a high degree of accuracy the ex-
pansion of liquid ethyl oxide right up to the critical volume.
It is now to be tested as to its power to give compressibilities
correctly. The next Table contains the calculated compressi-
bilities of liquid ethyl oxide, and also the experimental as
given by Amagat (Ann. de CJiim. et de Pliys. 5 ser. t. xi.),
Avenarius (Wied. Beihl. ii.), and Grimaldi (Wied. Beibl.x.).
Amagat's values had to be interpolated for comparison with
the others.
Table XYII.
Compressibilities with metre of mercury as pressure-unit.
Temperature 0°.
40°.
60°.
100°.
Amagat i "000200
Avenarius -000178
G-rimaldi 1 -000207
•000309
•000317
•000316
•000300
•000380
•000403
-000407
-000392
•000730
•000654
•000632
•000710
Equation '000183
Laivs of Molecular Force. 231
The agreement here is again satisfactory, and we have now
seen that our form, with only two constants in addition to
those characteristic of the gaseous state, can give both the
expansion and compression of the liquid at low pressures ;
but Amagat has measured these also at high pressures up to
2000 and 3000 atmos. {Compt. Mend. ciii. and cv.), and the
following Table compares first his values of the mean co-
efficient of expansion between 0° and 50"" at pressures from
76 up to 2280 metres with those given by the equation, and,
second, his values of the mean compressibility at 17°*4 and at
pressures up to 1500 metres with those given by the equation.
If Vi and V2 are the volumes at pi and p2, then the mean com-
pressibility is taken as (^i— t'2)/^i(f>2— i^i)- The apparent
compressibilities given by Amagat are converted to true values
by adding '000002, which he has since given as the com-
pressibility of glass.
Table XYIII.
Mean Coefficient of Expansion at high pressures.
p in metres ... 76.
Amagat -00170
Equation -00170
Mean Compressibilities at high pressures*
join metres ... 76 to 114 to 366 to 654 to 933 to 1218 to 1500
I i I I I I
Amagat '000208 -000143 -000112 -000086 -000070 "000062
Equation -000197 "000128 -000085 -000060 -000046 -000037
As regards expansion the equation goes fairly near to the
truth ; except at the lowest pressures, it gives coefficients
somewhat smaller than the experimental, but it parallels
closely the main phenomenon of the rapid diminution of the
coefficient with rising pressure. But in the compressibilities
there is an increasing divergence between experiment and
equation with increasing pressure, although again the equation
is true to the main fact of the rapid diminution of compres-
sibility with increasing pressure. We may conclude from the
last table that our equation holds within the limits of experi-
mental accuracy up to 760 metres ; beyond that it begins to
fail. A simple empirical modification would adapt the form
to the whole of Amagat's range, but as it stands it will be
fomid good enough for our applications.
We will now consider briefly how this form apphes to car-
bonic dioxide below the critical volume ; and the comparison
380.
760.
1140.
1520.
1900.
2280.
00112
■00091
•00077
-00070
•00063
-00056
00101
•00076
•00066
•00056
•00050
-00047
232
Mr. William Sutherland on tJi
le
is interesting, as it relates to temperatures both above and
below the critical. The values of the constants are B = 54,
/3 = -692.
Table XIX.
Carbonic Dioxide below critical volume.
Volume
70° C / ^^^s^^^'^j ®^P- •
■ \ „ calc.
oKO pt r Pressure, exp..
' \ „ calc.
iQo n I Pressure, exp..
■ 1 ,, calc.
1-526.
1-203.
1115.
1-027.
150
152
274
266
69
69
126
120
187
184
320
320
99
97
200
208
The agreement is within the limit of experimental error at
the high pressures. Cailletet and Mathias have determined
{Compt. Rend, cii.) the density of liquid CO2 at various tem-
peratures under the pressure of saturation. Here is a com-
parison with a couple of their results : —
Temperature . . . —34°. 0°.
Volume— Cailletet and Mathias . -946 1'087
„ Equation '943 1-086
As far as compound gases are concerned, the applicability
of the form for volumes below the critical has now been de-
monstrated in two typical cases. The elementary gases have
now to be considered as to their behaviour below the critical
volume. The data are again those furnished by Amagat
(Compt. Mend, cvii. and Phil. Mag. Dec. 1888) on the com-
pressibility of these gases between 760 and 2280 metres of
mercury. Our study of these bodies above the critical volume
has given us the knowdedge that the internal virial term below
k must be l/v, and the kinetic-energy term at the critical
volume is 3RT/2, and wdth these guides the complete form
required is soon found from the experimental numbers. It is
/8
pv = ^nT(i+h''^
with the following values for the
additional constants
and h : —
k.
^. b.
Hydrogen ... 12
4-3 -480
Nitrogen . . . 2 '64
•81 -420
Oxygen . . . 1-78
•604 -4415
Methane . . . [o'Sl]
[1-59] [-447]
Laws of Molecular Force,
23B
The approximate equality of the values of the constant h is
worth noting. I have also reproduced here the values of k at
the side of those for /3, in order to point out that yS is nearly
kl?> in each case. Amagat has not published data for methane
at volumes below the critical region, but the numbers given
in brackets for methane were obtained indirectly as explained
below. These relations of ^ and h give our equation such a
degree of simplicity as largely to establish the soundness of its
form. The next Table shows the degree of accuracy with which
it represents the experimental facts.
Table XX.-
—Oxygen at hi
gh pressures.
Yolume
1-277.
1-097.
1-008.
•949.
-905.
if-ort / Pressure, exp
^^ ^-l „ calc. ...
760
739
1140
1141
1520
1512
1900
1893
2280
2296
The agreement is quite as good for hydrogen and nitrogen.
By means of this equation we can calculate the volumes of
a gramme of liquid nitrogen and oxygen at their boiling-
points under a pressure of '76 metre, for comparison with
Wroblewski and Olszewski's determinations of the same
{Compt. Bend, cii.; Wied. Beibl.^.; and ^Nature,' April
1887).
Volume
at -184^
{Wroblewski
Olszewski
Equation
Oxygen.
•85
•89
•90
at
Nitrogen.
1 1-20
1-26
The equation is seen to give the volumes of these two bodies
at these low temperatures within the present limits of experi-
mental accuracy, and accordingly it covers a range of 2000-
metres pressure and almost the whole experimental range of
temperature. In the case of methane, if we take Olszewski's
value 2*41 for its volume at — 164°_, and assume b is the mean
of b for H2, N2, and O2, then we can calculate the value of jS
which is tabulated above.
To ethylene above the critical region we had to assign a
special form intermediate between that for ordinary compounds
and that for elements ; so that Ave had better do likewise for
its infracritical equation, which I have cast in the form
riiil, Mag. S. 5. Vol. 35. Ko. 214. March 1893.
R
f 34 Mr. William Sutherland on the
with B = 56*5, /3 = 1'53. For the elements we had t7^=3^/2,
for ordinary compounds v^=7k/6 ; so that to make ethylene
intermediate v^ is taken as 5k/ 4:, all these being of the general
form (1 + 2n) /27i, with n=l, 2, 3.
With the values of the constants B and /B given above as
derived from Amagat''s results at high pressures^ we can
determine the density of liquid ethylene; at —21° under
saturation-pressure the density is '414, identical with the
experimental value of Cailletet and Mathias [Compt. Rend.
cii.).
It will be as well at this stage to extract clear from among
the argumentative detail the most important results so far
obtained.
First, in the elements the internal virial varies inversely as
the volume over the whole experimental range.
Second, in compounds there is mathematical discontinuity
in the value of the internal virial at volume k ; from volume k
downwards the internal virial varies inversely as the volume :
from the volume k upwards it tends towards variation inversely
as the volume as the limiting law, the limiting constant being
double that which holds below the volume h ; between the two
limiting cases the internal virial of compounds varies inversely
as (v + k).
Third, a fact of the highest importance in connexion with
the kinetic-energy or temperature term in the equation arrests
our attention, namely, that the coefficient of T in it, or the
apparent rate of variation of the translatory kinetic energy
with temperature at constant volume, attains near the critical
volume double its value in the gaseous state, and below the
critical region increases rapidly with diminishing volume (see
column B^vf{v) in Table I.), becoming at ordinary liquid
volumes as much as ten times as large (see coefficient of T in
infracritical equation). Now the specific heat of liquids at
constant volume, which is the rate of variation of the total
energy with temperature, is rarely much more than twice that
for their vapours. Hence we must seriously consider the
interpretation to be put on the different terms of our equations.
5. A short digression on the general interpretation of Clausius^s
Equation of the Virial. — Returning to Olausius's theorem of
the virial,
^pv^^^mT'-'l.^tt'^r,
we see that strictly the kinetic-energy term includes not only
the energy of the motion of the molecules as wholes, but also
that of the motion of their parts, and at the same time the
I
Laios of Molecular Force* 235
internal virial includes the actions between the parts of the
molecules as well as those between the molecules. Calling
these actions the chemic force, we can write the theorem
thus :
^jpx) = the total kinetic energy— chemic virial— virial of
molecular forces.
Now in the usual treatment of the equation it is assumed
that the chemic virial is equal to that part of the total kinetic
energy which is due to the motion of the parts of the mole-
cules relatively to their centres of mass, and neutrahzes it in
our equation, reducing it to
^pv =■ translatory kinetic energy of molecules as wholes
—virial of molecular forces.
But if we retain the fall equation, and assume that the virial
term we have been finding for various bodies is the true virial
of the molecular forces, and includes none of the chemic virial,
then the term usually regarded as the translatory kinetic
energy of the molecules as wholes is really the total kinetic
energy minus the chemic virial.
Let E be the total kinetic energy of unit mass, Y the virial
of the chemic forces, and P their potential energy ; then, above
the critical volume.
E-V=iRT(l + ^-f-^)
and
2k
|,(E-Y)=JE(l4-^-f-,)
which in the limiting gaseous state becomes 3R/2.
A1.0
^ (B-P) = K„,
the specific heat at constant volume.
Below the critical volume,
and, again,
^,(E-P) = K„.
R2
236 Mr. William Sutherland on the
Now we can calculate K^. from tlie experimental values of
K by the relation
Let us then make a comparison in the case of ethyl oxide,
using E. Wiedemann's value '3725 for K for the vapour at
0°, and Eegnault''s value •529 for the liquid at 0° ; then, con-
verting to ergs per degree C, we get
Vapoiu' at 0°. Liquid at 0°.
A(E-V) = 1-68x10^ 15-3 xlO«,
K, or^(E-P) = 14-4 x 10^ 17-9xlO«.
Thus we see that while in the liquid ^(E— Y)/BT is nearly
equal to ^(E — P)/'^T, there is a great difference in the
vapour :
^(y-P)/^T= 12-7 for the vapour and only 2*6 for the liquid.
Or, while K^. is nearly the same in the two states, B (E — Y)/BT
has in the hquid state increased to nine times its value in the
vaporous. We have here, therefore,"an interesting opening
into the regions of chemic force ; but meanwhile we must
restrict ourselves to the question of molecular force at present
in hand, calhng attention, however, to the fact that our energy
term in its two forms for elements and its two forms for com-
pounds is well worthy of the closest study. It summarizes a
lot of information about the internal dynamics of molecules —
perhaps about the relations of matter and aether ; but these
would need to be extracted by a special research on the term
and its relation to our experimental knowledge of specific heat.
It is worth mentioning here that Clausius^s equation of the
virial, as usually applied to molecular physics, takes no accotmt
of the mutual action of matter and aether — an action which we
know must exist, from the radiation of heat by gases as well as
by hquids and solids. According to ordinary -sdews of the
sether this may be neglected, on account of the smallness of
the mass and of the specific heat of the aether ; but it is well
to remember that we are neDjlectino- it.
6. Consideration of Van der Waals^s generalization. — We
are now in a position to consider how far Yan der Waals^s
generahzation holds, namely : — If the volume, pressure, and
temperature are measured for each substance in terms of the
critical values as units, then one and the same aw holds for
all substances.
i
Laws of Molecular Force. 237
In the first place, we see from what has gone before that
the same law cannot apply to both elements and compounds,
nor can the alcohols and water follow the same law as regular
compounds.
In the case of the elements and methane we have the critical
volume, pressure, and temperature given in terms of R, Z, and
k by three relations (see end of Section 2),
_QZ./0 -^L T_l^ ^
Whence, in the supracritical equation replacing R, /, and k by
their values in terms of v^, p^, T^, we get
which shows that when the critical values are made the units
in the measurements of the variables, one and the same law
holds for the elements above the critical volume.
Below the critical volume we have
Pc% 16T„r' « /S
We have seen that h is nearly the same for these bodies and
that ^jv^ is approximately constant, so that below the critical
volume the elements and methane all follow approximately
the same law.
In the case of compounds, we have (see Section 4, at the
beginning)
^0-^^/^; Po- ^09 k^' ^~409RF
with which, eliminating R, k, and / from the supracritical
equation, we get
?;u___20T/ 2 \ 409 I
.-."7 T^X l-^^i) 42 7_v__^-
pv
pT. - - _ , ^
6 t; ' "^ 6 V
Hence, above the critical volume the compounds follow the
same law among themselves.
In the same way, below the critical volume we get for
compounds: —
238
Mr. William Sutherland on the
pv
20 25 T
7 'IST
409 ^e
■ 98 V
One and the same law holds for compounds below the
critical volume only if B varies as the square root of the
critical temperature, and if ^ is proportional to the critical
volume : in the elements we have found the latter condition
to hold approximately^ and so we are prepared to find it do so
for compounds. The following Table compares B with s/T^
and /3 with k, which is Qvjl for the five compounds for which
we have as yet found k. The values of B and P for NH3 and
N2O were obtained from Andreeff's data for the expansion of
these bodies as Hquids {Ann. Chem. Pharm, ex.) and for SO2
from Jouk's (Wied. BeihL vi.).
Table XXI.
k. /3.
kl^. ! B.
B/VTc.
2-9
30
3-6
3-6
31
(^O.H-VO
4066 1 Ml
1-76 -69
2-08 ! -55
3-7 63
4-0 ; 54
3-8 71
3-9 70
3-5 55
CO,
SO,
NH3
4-8
2-3
1-22
•66
N.O
In these bodies we find a fair approximation to propor-
tionality between /8 and k on the one hand, and between
B and VT^ on the other ; to the same degree of approxi-
mation Yan der Waals's generalization can be applied to
compounds below the critical volume (excluding of course
such exceptional bodies as the alcohols and water).
The accurate statement of the generalization ought then to
be as follows : — When the variables are expressed in terms of
their critical values as units, then down to the critical point
compound bodies with certain exceptions have all one and the
same characteristic equation, but below the critical point they
have closely similar but not identical equations,
It is a remarkable fact that Yan der Waals should have
been led to his valuable generalization by means of a form of
equation which completely fails to apply to the substances
which are the subject of the generalization. As a point in the
history of this branch of molecular physics, it calls for mention
thatWaterston,in the Phil. Mag. vol.xxxv. (1868), had prac-
Laios of Molecular Force, 239
tically discovered the generalization, and expressed it in its
most striking aspects by means of several diagrams for a
number of bodies ; but tbe verbal expression of his results was
so unsystematic, and withal so crabbed, that his work has been
overlooked.
There is one typical application of the generalization which
is of special importance — to the relation between pressure and
temperature of saturation. If with the aid of our equations
we trace the complete isothermals for temperatures below the
critical, we shall get curves with the James Thomson double-
bend as shown in Ramsay and Young's isothermals for ether
(PhiLMag. May 1887).^
According to Maxwell's thermodynamical deduction, the
pressure of saturation at a given temperature is that cor-
responding to the line of constant pressure which cuts off
equal areas in the two bends, a result which Ramsay and
Young verified by actual measurement on their curves.
Let ^3 and v^ be the volumes of saturated vapour and liquid
at pressure P and temperature T ; then Maxwell's principle
gives us that the pressure of saturation is defined by the three
equations : —
F{vs-Vi)= pdv,
'••-='(i+.^)-.-;tj.
^3 is given in terms of P and T by a quadratic, and can
therefore be eliminated from the integral when evaluated ;
Vi is given by a cubic, but as Fvi can for practical purposes
be put 0, a very close approximation to Vi can be obtained also
from a quadratic. The resulting relation between P and T,
which is the law of saturation, involves the constants R, k, /,
B, and /5.
The actual evaluation of the integral would of course
proceed in three stages, corresponding to the supra-, circa-,
and infracritical equations. The law of the integral in the
first stage from v^ to 7k/ 6, with critical values of the variables
as units, would be the same for all compounds ; and we have
tseen that the integral in the other two stages will follow
approximately the same law in all cases. Hence if saturation
pressures and temperatures are expressed in terms of the
critical values, the law of their relation will be approximately
bhe same for all regular compounds.
24:0
Mr. William Sutherland on the
of testing it would be to take Eegnault's formula, with one
exponential term,
\ogp = a-{-hu,
and by a simple recalculation from his values for a, h, and a,
to cast it in the form
proving that the constants e, d, and 7 are approximately the
same for all compounds. But the objection to this plan soon
becomes obvious on trial, as the formula owes its empirical
convenience to tte power of adjustment amongst the con-
stants ; and the same difficulty would be experienced with
any purely empirical equation.
Accordingly, to test this matter, I have thought it best to
compare the pressures of a number of bodies at temperatures
which are constant fractions of their critical temperatures,
such as '6 T^, •? T^, and so on. The ratio of the pressure of
any substance to the corresponding pressure of ethyl oxide
ought to be approximately the same for that substance at all
values of the fraction. Great uncertainty attends the mea-
surement of critical pressures : an error of 20° in the critical
temperature is not a large fraction of its value measured from
absolute zero, but it makes a large difference in a saturation-
pressure, and the critical pressure is the limiting saturation-
pressure. In the subjoined Table the critical-pressure ratios
are given for what they are worth in the column T^.
Tablij XXII.
Katios of Saturation -Pressures at constant fractions of the
critical temperature to the Saturation-Pressures of Ethyl
Oxide at the same fractions of its critical temperature.
Tc.
Fractions of Tc
•6.
•65.
•7.
•75.
•8.
•9.
ro.
Acetous
506
404
428
404
373
305
308
548
556
537
404
450
509
560
•87
1-7
1-7
2-6
31
1-3
1-6
1-7
1-j
Id
1-3
•90
1-5
1-7
2-6
3-6
2-7
1-3
15
1-5
1-4
1=5
1^3
•92
1-4
1-7
2-5
3-2
2'3
1-3
1-5
1-4
1-3
16
1-2
•94
1-4
1^7
2-5
2-8
2'l
1-2
r4
1-4
1 2
15
1-2
•95
1-7
2-5
2-6
1-8
24
1-3
1-2
1-4
11
2-4
1-7
rs
1-4:
2-2
3-2
2-6
21
21
21
ra
1^6
2^1
1-5
Methyl oxide
SO2...
NHg
H.S
cc^ :::::::::::::::
Np
CS2
0CI4
CH013
CH3O1
C.H.Cl
C,H>
Benzene, OgHg ...
Laics of Molecular Force. 241
This table makes it clear enough that, in applying Van der
Waals^s generalization below the critical volume, we have to
do with a first approximation only. The curves for all these
diverse bodies excepting CS2, while not identical, would form
a compact bundle about a mean curve from which each body
would have its own characteristic departure ; and this is just
what our study of B and ^ in Table XXI. should lead us to
expect.
7. Five Metliods of finding the Virial Constant. — The first
method is that which we have already exhausted, namely, by
means of extended enough observations of the compression
and expansion of bodies in the gaseous state.
Second method : to obtain the virial constant I from one
measure of the compressibility and of the expansibility at the
same temperature of the body as a liquid.
Writing our infracritical equation thus,
_R'TA , v/T ^'-tA_ I
§£_E/ 3RVT l-v^?> J__B/,3£
BT~ V 2 B ' v-p 2'2?;2T 2?; 2 T*
But at ordinary low pressures the term p/T is a negligible part
of this expression, and we can write
B^_3 I _E/
Now
BT 4 * v^T 2v
■dp_
BT-
__'bv /'dv _
^p/ 'dp'
V
1 -dv
Vo'dT
' Idv ~
V dp
V
a
where a and /i, are the coefficients of expansion
and the com-
pressibility at T
as usually defined.
3 I
• • 4 „2T
2v
Vo a ,
V fJb '
= J(„„^ + .R,),T = |(„o^H-fp).T.
In addition to giving us a value of /, this last equation gives
a test as to whether the equation applies to a body or not, as
the expression on the right-hand side is to be constant at all
temperatures if //, is measured at low pressures. But on
242
Mr. William Sutherland on the
account of the experimental difficulties hitherto met with in
the measurement of /jl, the equation gives no very delicate
test,, although it might with the improvements in accuracy
made within the last few years. In addition to ethyl oxide,
the two substances for which we have measurements of both a
and fM over the widest range of temperature are ethyl chloride,
studied as to expansion by Drion and as to compression by
Amagat, who has corrected Drion^s coefficients of expansion
for change of pressure, and pentane, studied by Amagat and
Grrimaldi. The following are the values of I calculated from
the data at different temperatures for these two substances,
with the megadyne taken as unit of force.
i
! Temperature C, ..
.1 0°.
j
IP.
13°.
20°.
40^.
60°.
80°.
100°.
1 0.5 Hi2 (Grimaldi)..
CgHjo (Amagat) ..
' OgH-Cl
' 8530
5820
...
9200
7430
7230
8000
5450
8300
9250
9250
5270
Thi3 comparison has been made to show how, from mea-
surements of Hquid compressibility at present available, we
can get only a fair idea of the value of /, but not an accurate
measurement of it.
Amagat has determined the compressibility of several other
liquids at different temperatures [Ann. de Cliim. et de Physique,
ser. 5,t. xi.) ; so have Pagliani and Palazzo (TVied. Beihl. ix.j
and de Heen (Wied. Beihl. ix.), but their discussion would
bring out nothing more than the above comparison has. De
Heen's results would appear to make I diminish with rising
temperature in^ every case ; but he measured his com-
pressibilities in comparison with that of water at the same
temperature, and to calculate their values used Pagliani and
Vincentini^s values for water (yCmdi. Beihl. viii.), which make
the compressibility of water much more variable with tem-
perature than Grassi's. If we used Grrassi's values in
de Heen^s experiments, I would remain nearly constant.
We will accordingly use the compressibility-method of cal-
culating / subsequently, only to illustrate the general agree-
ment of values derived from purely mechanical experiments
with those found by the more accurate methods to which we
now proceed. The values found by this second method are
tabulated later on in Table XXIY.
Third method of finding the virial constant I : from the
latent heat. If we differentiate with respect to T our equation
Laws of Molecular Force. 243
for saturation-pressures (see Section 6),
we get
^T '^'""'^ = J,. BT
Now from thermodynamics we have the relation
J\=(v3-Vi)TcZP/(^T,
where X is the latent heat,
which of course could have been written down immediately,
for if we write it in the form
-m--')
we remind ourselves that the latent heat of evaporation of a
liquid is the heat supplied to neutralize a Thomson and
Joule cooling effect.
We must evaluate the integral in three stages,
using in each integral the equation that holds between its
limits.
In the first,
BT~ vv'^ v+kr
-dT ^^v^k-
In the second,
244 Mr. William Sutherland on the
In the third,
■^BT~2W^^2W '2ir
Hence
3/ /I 1\ E'T, A'
We must now evaluate the only integral that has been left
unevaluated,
These last expressions are the only ones that introduce B and
/3 into the value of JX, and it is desirable to remove these
two constants. At low pressures we can neglect the 'pv term
in the infracritical equation, and we get
R^T VT_/ I Tp,rrVi-/3
B V2i'i ^^)y-v^'>
removing B by means of this we get
Laws of Molecular Force, 245
We can now remove /3 and greatly simplify this equation, if
we apply it to latent heats near the ordinary boiling-point
T5. The last term taken in its entirety has a small numerical
value compared to the rest, so that in it we can make approxi-
mations without any sacrifice of accuracy worth considering :
we have seen that ^ is approximately proportional to k (see
Table XXI.) and v,, the volume of the liquid at the boiling-
point, may be assumed to be approximately proportional to k!
the volume at the critical temperature, and k'=:7k/6: hence
the coefficient of {l/2vi — 'R'T) in the last term is approxi-
mately the same for all bodies and we can evaluate it for
ethyl oxide ; call it 0. Again, in ^(v^^Vi) neglect v^ and
assume the gaseous law Pv3 = RTj. And further, k is small
compared to Vg, so that 2v2/{vg-\-k) is nearly 2, and its value
for ethyl oxide can be applied to all bodies. E,' = 25R/13.
Hence multiplying by M the molecular weight we can write
MR is the same for all bodies, and T^ is the absolute boiling-
point. This equation still involves k as well as / ; when k is
not known we must eliminate it by means of our previous
assumption, namely, that k is proportional to Vi, which we
know to be approximately true ; in so far as it is inexact it
will introduce inexactness into our calculation of /. Accord-
ingly in symbols kjvi = r, where r is the same for all bodies,
and can be found for ethyl oxide. Making the numerical
reductions we get
M//?;i=66-5MX-101T,
as the equation which gives I in terms of the megadyne as
unit of force, when X is the latent heat of a gramme in
calories and Vi its volume in cubic centimetres at the absolute
boiling-point T^. This equation will be abundantly verified
afterwards in Table XXIY. ; but meanwhile, if to test it we
apply it to calculate the latent heat of ethyl oxide, we find
\ = 83'4, whereas several experimenters have agreed in an
estimate of about 90 ; but, on the other hand, Ramsay and
Young (Phil. Trans. 1887) have made a special study of the
terms in the thermodynamic relation J\ = {r-^—vi)TdF/dT,
and have so calculated values of \ almost up to the critical
temperatures, their value at the boiling-point is 84'4, and
there is the same amount of discrepancy between their values
at higher temperatures and Regnault's experimental deter-
minations. Yet Perot, who has made an elaborate study
(Ann, de Ch. et de Ph. ser. 6, t. xiii.) both of X experimentally
246 Mr. William Sutherland on the
and of the quantities involved in its calculation by the ther-
modynamic relation, has found the most perfect harmony
between the results of the two methods. Now at 30° C.
Perot gives as the saturation- volume of the vapour 400*4, and
the saturation-pressure '635 metre, while Ramsay and
Young's values are 374 and '648 metre; but if ethyl oxide
were a perfect gas, under Perot's pressure of '635 metre it
would have a volume of 400*8, almost identical with his value:
yet we cannot imagine that ethyl oxide under this pressure
and at this temperature is so nearly a perfect gas as this would
implj^, unless there is some remarkable discontinuity in its
behaviour at high volumes. Accordingly, in spite of the
thoroughness of the researches of both Perot and Ramsay and
Young, we are on the horns of a triple dilemma, from which
only some experimental repetition can dehver us, and de-
monstrate where the cause of these discrepancies lies. Wiillner
and Grrotrian (Wied. Ann. xi.) have put on record the results
of experiments which indicate the cause; they find the pressure
of condensation measurably different from the ordinarily
measured saturation-pressure, — a fact explaining the difficulty
of measuring v^ accurately, and showing also that the values
of dVldT are not so reliable as usually supposed.
Our last equation is verifi.ed by, and shows us the cause of,
an interesting relation that has been independently discovered
and expressed in different forms, between the molecular latent
heat and the boiling-point, by Pictet {Ann. de Ch. et de Ph.
ser. 5, t. ix. 1876), l>outon (Phil. Mag. xviii. 1884), and
Ramsay and Young (Phil. Mag. xx. 1885), namely, that the
molecular latent heats of fluids are nearly proportional to
their absolute boiling-points. Now we have seen that
T^=120//409R^ (Section 4), and I have noticed that a large
number of substances have their ordinary absolute boiHng-
points nearly equal to 2Ty3, and k is nearly proportional to
vi, say is equal to 2*8 v^ as it is for ethyl oxide. Hence we
have
3^ ^ 120/
409 R 2-8 ri'
/. M?=14-3MRt'il\ = 1190t^iTj,
when the megadyne is the unit of force; hence from our
equation for Ml in terms of X we have
1190T,=66-5M\-101T5;
.-. 1291 Tj- 66-5 M\ or M\ = 19-4T^,
Laws of Molecular Force.
247
or the molecular latent heat is proportional to the absolute
boiling-point.
(It is to be noted that Wl—ll^OviT^ gives a rough means
of obtaining I from the boiling-points and the volumes at the
boiling-points of liquids, which might be convenient when
better data are wanting.)
Robert SchifF {Ann- der Chem. ccxxxiv.) has made the
most accurate determinations to test this relation between
molecular latent heat and boiling-point. For 29 compounds
of the form C^^Hg^Og, from ethyl formiate up to isoamyl vale-
rate, he finds M\=20"8Tjinthe mean, the greatest departures
being 20*4 for propyl isobutyrate, and 21*1 for propyl for-
miate ; for 8 hydrocarbons of the benzene series he finds a
mean coefficient 20 with 19*8 for cymene and 20*6 for benzene
as the greatest departures. To these 37 examples we will
add the following from Trouton^s paper, doubling his numbers,
as he used density instead of M.
1
'able
. XXIIL— Values of MX/1\.
0,H,01.
OHOI3.
CCI4.
ASOI3.
SnCl,.
SO,.
CS,
21
22 21
1
21
20
23
21
(0,H,),0.
(CaH,,),0.
(CH3),C0.
CioH,3. 1 (C,H3),0,0,.
22
24
23
22 23
The mean value of the coefficient is higher than that deduced
theoretically above (19*4), because in round fraction we wrote
Tj = 2T^/3, but the general truth of the relation is well enough
brought out.
Fourth method of finding the virial constant I : from the
critical temperature and pressure. Now we have (Section 4),
T^ = 120//409R^, p^=3Qll^09k%
... TJp^=10k/3R and Z=409R2T2/400p^,
this is for compounds; for elements / = 27R^T^/64;;^. Where
both the critical pressure and temperature are known, this
gives I theoretically with accuracy, but practically the diffi-
culties in measuring the critical pressure introduce inaccuracy.
In the relation T^//>^=10A:/3R, as R varies inversely as the
molecular weight, we see that the molecular domains (Mole-
cular volumes) of bodies at the critical temperature are
24^ Mr. William Sutherland on the
proportional to the quotient of critical temperature by critical
pressure, a relation which Dewar has proved experimentally
(Phil. Mag. xviii. 1884) for 21 volatile bodies, for which he
has determined and collected the data. These with other
data since published enable us to determine values of / for
certain bodies for which the other methods are not available.
As there are many more critical temperatures determined
up to the present than critical pressures, and as we have seen
that an error in the critical temperature is of less relative
importance than an error in the critical pressure, we can
make ourselves independent of critical pressures with ad-
vantage, by employing the approximation that has already
been useful to us, that k is proportional to the volume of the
liquid at the boiling-point or Ic = 2'S3vi. Then
Ml = 409MIIT//120 = 800T^vi
approximately, with the megadyne as unit of force. This is
a more accurate form of the relation M/ = 1190Tj^i given
above, in which we assumed the approximation Tj = 2Tp/3.
8. Fifth or Capilla7'y Method of finding the Internal Virial
Constant, with digressions on the Brownian movement in
liquids and on molecular distances. — So far we have been
proceeding on a purely inductive path, with two deductive
guides in Clausius's equation of the virial and in the law of
the inverse fourth power, which requires that the internal
virial should vary inversely as the volume. But now, in
passing on to our fifth and most useful method of finding I
from surface-tension, we must employ a deductive relation
between I and surface-tension, furnished by the law of the
inverse fourth power. In a previous paper (Phil. Mag. July
1887) it was shown that if the law of force between two
molecules of mass m, at distance r apart is 3A?n^/?'*, then the
internal virial for the molecules in unit mass is birAp log L/a,
p being density, and L a finite length of the order of magni-
tude of the hnear dimensions of the vessels used in physical
measurements, a being the mean distance apart of the mole-
cules. The ratio L/a remains the same for a given mass
whatever volume it occupies, but I also assumed that L/a is
so large a number that log L/a would hardly be affected by
such large variations as might occur in the value of L when
the behaviour of a kilogramme of a substance was compared
with that of a milligramme. To remove the haziness of this
assumption, I will now make a more accurate evaluation of
the internal virial.
By definition it is ^ . ^ . '^%3AnL^/r^j and we will evaluate it
Laws of Molecular Porce, 249
for a spherical mass. To cast it into the form of an integral,
take any molecule m amongst the number n in a spherical
vessel of radius R ; gather it to its centre as a true particle
and spread the remaining n — 1 in a uniform continuous mass
separated from m by a small spherical vacuum of radius a, so
chosen that the virial of m and the continuous mass is the
same as that of m and the n—1 molecules.
Suppose m at the point 0, and the centre of the vessel at
C, and let 00 = c. Take 00 as axis of x and any two rec-
tangular axes through 0 as axes of y and z. Let polar
coordinates r 0 cj) he related to these in the usual manner.
Then the equation to the surface of the sphere is
{x-c)^i-if + z^ = W or 7'2-2crsin(9cos^ + c2-R2=0.
Let Ti and r^ be the two roots of this equation in r so that
ri7^2 = ^^-~c^. Then nv^/r^ can be replaced by
mpr'^ sin 6 cW d(f) dr/r^^
and
%nv^lr^ by ftX^/o sin Q dd d(j) dr/r.
If we integrate with respect to r on one side of the plane i/2 from
a to 7\ and on the other from a to ^2 and add the two results,
then we have to take 6 and (/> each between 0 and tt, thus :
%77i^/r^= yCmp sin ddO # [T' dr/r+ \\ir/rl
The two integrals in brackets give
log r^Tz/a^ =log (R2-c2)/a2.
Hence
^m^/r^ = 27rmfj log (R^ + c^)/a\
To perform the second summation we can first add the values
of the last expression for all the molecules at distance c from
the centre of the vessel, and write the result in the form
27rp4:7rpc^dciog {R^-c')/a%
and we then have
i . ittSAmyr^ = GAttV C'"' c'dc log {W-d^)la^.
■n 1 • ^^
Evaluating the integral, this becomes
A7ry|g(R-a)^log— ^ y(R-a)3-gR2(R-a)+-g-log — ^
in which, neglecting unity in comparison with the large
number R/a, we get
A7ryR3{4 log 2R/a-16/3}.
Phil, Mag, S. 5. Vol. 35. No. 214. March 1893. !S
250 Mr. William Sutherland on the
But if W is the total mass in the vessel, then
and we get
Wa7rp{31og2R/a-4}.
^
When W = 1 the first term of this becomes identical with the
value of the internal virial previously given, with 211 written
instead of L. Eeplaciug E by its value in terms of W and p,
we get for the internal virial of mass W,
WA7rp(log QW/irpa' - 4) .
As pa^ is constant, we see that for a given mass the internal
virial for molecular force varying inversely as the fourth
power is rigorously proportional to the density, but it is not
purely proportional to the mass. Although the number
6W/7rpa^ is a large one, and has a logarithm varying slowly
with W, yet large enough variations in W can afPect it
appreciably, as we see if provisionally we accept Sir W.
Thomson's estimate of 2 x 10~^ centim. as the lowest possible
value for a. Suppose 7r/? = 3, then 6W/7rpa^ is lO^^W/4, and
if W is 4000, 4, or -004 grm., then the values of log 6W/7rpa'
are as 30, 27, and 24, and we have a larger mass variation of
the internal virial than is likely to have escaped detection in
its effects, such as a difference in the density, expansion, com-
pressibility, latent heat, and saturation-pressures of a liquid as
measured on a milligramme, from their values as measured on
a kilogramme. The raising of this difficulty suggests to us
in passing that there exists a department of microphysics in
which little has as yet been done by the experimenter, and
that great interest would attach to a research determining
when a mass variation of the properties usually spoken of as
physical constants actually sets in.
But meanwhile we must scrutinize more closely the meaning
of our last result. According to the views of Laplace (and
of the early elasticians), if a plane be drawn dividing a mass
of solid or liquid into two parts, then, in consequence of mole-
cular force, the one part exercises a resultant attraction on
the other, and this has to be statically equilibrated by a
pressure (called the molecular or internal pressure) acting
across the plane, a conception which is necessary in any purely
statical theory of elasticity. Adopting for the moment this
mode of viewing things, we see that our result amounts to
this, that the internal pressm-e is measurably greater at the
centre of a kilogramme than of a milligramme.
But if we try to carry out the kinetic theory in its integrity,
Laws of Molecular Force m 251
we must reject the idea of a statical pressure, and replace it
by its kinetic equivalent of a to and fro transfer of momentum ;
this may take place as a quite indiscriminate traffic of indi-
vidual molecules across the plane, or as such a traffic modified
by the existence of streams of molecules in opposite directions.
If streams, or motions of molecules in swarms, actually exist
in fluids, then our interpretation of the equation of the virial
would have to take account of their existence. The kinetic
energy of the motion of a swarm as a whole would not count
as heat but as mechanical energy, and for the amount of it
we should have approximately the kinetic energy of the swarm
motion equal to the virial of the forces between the swarms.
Therefore we require to divide the energy into two parts,
that of molecular motion inside the swarm constituting heat
and that of the swarm ; in the same way tlie internal virial is
divided into two parts, one within each swarm, the other be-
tween the swarms. But the swarms could on the average be
regarded as equivalent to spheres of radius L, where L must
be supposed nearly independent of mass and liable to the same
variations with temperature and pressure as the linear dimen-
sions of any quantity of the liquid, so that L is proportional
to a, and our expression (theoretical) WA7r/3(31og2L/a— 4)
for the internal virial becomes purely proportional to the
mass and purely proportional to the density, as 3WZ/4?; our
experimental internal virial for a mass W of a compound
liquid is.
This hypothesis would affect somewhat the rigorousness of
certain thermodynamical relations as usually interpreted, such
as JX=(v3— -Vi)Tc?p/(iT, since it provides a supply of internal
mechanical energy not taken account of ; but if this supply
is only slightly variable with pressure and temperature it
would make little difference in most parts of thermodynamics.
With the addition of this hypothesis of molecular swarms,
which will be used only in calculating molecular distances,
and will not affect at all the rest of our work, the law of the
inverse fourth power is brought into strict harmony with the
behaviour of compound liquids and of elements both as liquids
and gases. We must therefore inquire what experimental
evidence there is for the existence in liquids of a motion of
swarms of molecules, possessing the remarkable property of
not being degraded to heat as ordinary visible motions are.
In the motion long familiar to microscopists as the Brownian
movement we have such evidence. Gouy has recently
(Compt. Rend. cix. p. 103) recalled tlie attention of physicists
to this remarkable ceaseless motion of granules in liquids.
He states that it occurs with all sorts of granules, and with an
IS 2
252 Me. William Sutherland on the
intensity less as the liquid is more viscous and as the granules
are larger. It occurs when every precaution is taken to en-
sure constant temperature, and to ensure the absence of all
external causes of motion. Granules of the same size but as
different in character as solid granules, liquid globules, and
gaseous bubbles, show but little difference in their motions — a
fact which proves that the cause is to be looked for not in the
granules themselves but in the liquid, the granules being
merely an index of motions existing in the liquid. The most
pronounced character of the motion is its rapid increase with
diminishing size of granules, so that all that is seen under the
microscope is the limit of movements of unknown magnitude.
Gouy considers the Brownian movements to be a remote
result of the motion of the molecules themselves, but according
to what we know of molecular dimensions I fancy that the
Brownian movement must be considered rather as a sign of
the motion of swarms of molecules. If swarms of molecules
are weaving in and out amongst one another, so that the
average transfer of momentum at a point is the same in all
directions, then the vibratory agitation of granules amongst
the swarms is just what we should expect. The striking fact
about the Brownian movement is that it is ceaseless ; it is
never degraded into heat. This alone forces us to conceive a
form of motion existing in liquids on a larger scale than
molecular motion but possessing its character of permanence ;
in other words, the motion of swarms of molecules.
The existence of swarms would not affect our views of the
rise of liquids in capillary tubes as a purely statical question;
so that, for the connexion between molecular force and
surface-tension, we can use the calculation given in another
paper (Phil. Mag. April 1889) (rather badly affected with
misprints), where I have shown that the surface-tension of
liquids that wet glass, measured in tubes so narrow that the
meniscus-surface is a hemisphere, is given by the equation
where p is the average density of the capillary surface-film
(to be written also 1/v), and e is the distance which we must
suppose to be left between a continuous meniscus and the base
of a continuous column raised by its attraction, if the action
between the continuous distributions is to be the same as in
the natural case of discontinuous molecular constitution of
meniscus and column. The distance e is not identical with
the length a which occurs in our theoretical value of the
internal virial of unit mass,
A7rp(31og2L/a-4) = |i,
Laws of Molecular Force. 253
but it is closely proportional to it. If we can find the relation
between e and a^ then from capillary determinations we can
obtain relative values of the viriai constant I which, as we
haA^e already found some absolute values of I, can be converted
to absolute values ; at the same time, too, we shall be able to
find a value of a the mean distance apart of the molecules.
To find the relations between e and a we can proceed thus.
If we have a single infinite straio-ht row of molecules at a
distance a apart, the force exerted by one half of it on the
other is
ra=oo p=oo 3A??i^
which can easily be evaluated as approximately 3'6Am7<^*.
Two infinite continuous lines in the same line, of density ml a
with distance e between their contiguous ends, would exert a
force
3Am^r"r" dxdy
^' J Jo (^+3/)'
on one another ; this is equal to Kir^j^o^e^* If, then, the
continuous distribution is to be equal to the molecular, we
nave
e2=a77-2, e = al2'l.
Again, if along two infinite axes one at right angles to the
other and terminating in an origin 0 at its middle point
molecules are arranged along each at distance a apart starting
from 0, then the force exerted by the unlimited row on the
other is
which can be evaluated at about ^Kii^ja^,
Replace the rows by two continuous line-distributions of
density mja^ the one terminating at a distance e from 0 : it is
required to fmd e so that the force may be the same as this.
The force is
Hence in this case
e2 = a75, e=al'l'2.
From these two simple cases we get an nlea of the relation
between e and a. The case of a mcxiiscus attracting the
column which it raises in a capillary tube is more nnalogous
to the second than to the first, and it is easy to see that in the
254 Mr. William Sutherland on tJie
case of the meniscus we can say that e is not less than a/2'2.
It will suffice to write e = a/2'2.
Now according to the definition of a in our theoretical
expression for the internal virial, it is the radius of the sphe-
rical vacuum artificially used to represent the domain of a
molecule ; hut as it occurs in the expression log 2Jj/a, where
2L/a is a very large number and the value of L is indefinite,
we see that there is no inaccuracy in making it identical with
a the mean distance apart of the molecules. However, for
the sake of formal completeness, we can easily find the rela-
tion between the two quantities which we have denoted by
the one symbol a. Let us now denote by ,v the mean dis-
tance apart of the molecules, that is the edge of the cube in
a cubical distribution of the molecules ; then, from the defi-
nition of a, X and a are connected by the relation
SAmVr3= r ^tirkiMrlx^
J a
the summation being extended to all the molecules in a sphere
of radius R. By actual summation up to R = 5.2? we find
approximately a = '^x.
With our previous estimate of e as a/2*2, which we must
now write xl2'2 on account of our change of symbol for mean
distance apart, we have the two equations,
Z=A7r(4:log2L/-9.i^-16/3),
a = 7rp2A^/2-2(2+N/2).
We can replace p by p, the diff'erence between them being
necessarily very shght. Then for ethyl oxide we have the
following data : a at the boiling-point according to Schiff is
1-57 grammes weight per metre, or I'57/IO^ kilog. per cm. ;
I is 7500 kilog. cm.*, and Vj = 1*44 cm.^ Eliminating A from
the two equations, we have a relation between x and L,
namely
^=7-5?;i2|(9-2logio2L/-9^^-16/3).
L being hypothetical is not known to us, but we can give it a
series_ of possible values, and calculate by trial from the last
equation the corresponding series of values for x, mth the
following results : —
1/10^ cm. 4-6/10^ cm.
1/lOJ „ 7-7 „ „
1/10^ . 11-0 „ „
I
Laios of Molecular Force, 255
As it is subsequently to be shown that in liquid ethyl oxide
and in all regular compound liquids the molecules are paired,
and that each pair acts on the others as if it were a single
molecule, we may estimate it as likely that the mean distance
apart of the pairs in ethyl oxide is between 1 and 10 micro-
millimetres (1/10^ mm.). This result, though 100 times as
large as Sir Wilham Thomson^s limits for the distance apart
of molecules in liquids, namely 2 x 10"^ cm. and 7 x 10"^ cm.,
is yet in better agreement with the estimate of molecular dis-
tances arrived at by Riicker (Journ. Ghem. Soc. 1888) as the
most probable result obtainable from the most important
attempts yet made to measure the range of molecular forces.
The most suo-o-estive of these is Reinold and Rilcker's dis-
covery, that the equilibrium of a soap-solution film becomes
unstable when its thickness is reduced to between ^% and 45
micromillimetres, but again becomes stable when the thickness
is still further reduced to 12 micromillimetres. At the latter
t^'ickness the film shows black in reflected light. If the
intermolecular distances are nearly the same in soap-solutions
as in liquid ethyl oxide, then the black film must be regarded
as consisting of a single layer of molecules or groups of mole-
cules (in the case of water the molecules will subsequently be
shown to go in double pairs). This is an intelligible result,
and gives the simplest explanation of Reinold and Riicker's
beautiful discovery of a stable thickness supervening on the
unstable, for we recognize a single layer of molecules as a
stable configuration. Of course it is to be understood that
what we mean by the thickness of a single layer of molecules
is the one nth part of the thickness of n layers ; and if the
black film is really only a single layer, it is in this sense that
Reinold and Riicker's estimate of 12 micromms. is to be taken,
for they did not measure an actual distance from the front to
the back of a black film, but only estimated from accurate and
accordant measurements, made in entirely difierent manners,
that the number of layers in the black film is to the number
in a thickness of 1 centim. as 12 micromms. is to 1 centim.
If the black film consists really of only a single layer of
molecules, it is surely a hopeful sign for molecular physics
that measurements should have been possible on it, though
only visible through its invisibility.
If, encouraged by this experimental support, we say that in
round numbers the mean distance between the pairs of mole
cules in liquid ethyl oxide is 10 micromms., then one gramme
contains 2^^ x lO-"^^ molecules, or the mass of a single moleculo
is
]/2-88xlO^«grm. =:3^5/10^^
256 Mr. William Sutherland on the
and so the mass of an atom of H is
3-5/74 X 10^^=5/102^ grm. nearly.
It would lead us too far from oiu' present purpose to discuss
other estimates of molecular distance, especially as Reinold
and Riicker^s measurement of the black film is the most defi-
nite and striking yet made of these minute distances ; hut the
question of the range of molecular force is of special import-
ance to us.
Quincke (Pogg. Ann. cxxxvii.) determined what thickness
of silver it is necessary to deposit on glass so that the capil-
lary effect on water may be the same as that of sohd silver;
that is, at what distance the difference between the molecular
attractions of glass and silver for water becomes too small to
be measured. He found the thickness to be about 50 micromms.
Now, according to the law of the inverse fourth power, the
attraction of a cylinder of radius c, length h, and density p on
a particle of mass m on the axis at a distance z from the
nearest end is easily calculated as
2Amp7r ( j = H =. )
^ \z z + h ^c^^z" Vc2 + (^ + A)V-
If the cylinder consists of a length li^ of silver with a length
7^2 of glass, the silver being near the particle, then, the suffixes
1 and 2 applying to silver and glass, the attraction of the
composite cyhnder is
Making the circumstances correspond to Quincke^s experi-
ment, we have z nearly equal to the mean molecular distance
in water, about 10 micromms. ; li^ is small compared to li^ and c,
and, according to Quincke, is 50 micromms. when the composite
cylinder exerts the same force on m as if it were all silver ;
accordingly the last expression reduces to the two terms
IK^mp^iTJz - %nir (Aj/?^ - K<^p^ /{z + Ai),
which permit us to compare the molecular force range hi with
the molecular distance z. That the second of these terms
should become negligible when hi is 50 micromms. is a result ,
quite in accordance with the value 10 micromms. for z. I
Let us briefly compare the magnitudes of molecular and I
gravitational force. The most convenient plan will be to I
compare the two forces in the case of two single ethyl-oxide
Laws of Molecular Force. 257
molecules at a distance of one centim. apart ; that is, to
calculate Am^ and Gw^, where m is the actual mass of the
molecule, and G the constant in the expression Gm^/r'^ for
gravitation.
In the expression
a=7rp2A^y2'2(2+x/2),
using the value 10 micromms. for a; and the values previously
given for the other quantities, we can find A, and then using
the value 3 '5/ 10^^ for the mass of a molecule of ethyl oxide we
find Am^=9/10^^ in terms of the dyne. To calculate G we
have 981 as the acceleration of gravitation ; the mass of the
earth is 6 x 10^ grm. and its radius is 6*37 x lO^cm., so that
G7?i^ = 2'l/10^^ in terms of the dyne. Hence at a distance of
Z' 1 centim. the gravitation of two ethyl-oxide molecules is
about double their molecular attraction, or, allowing for un-
I certainties in our calculation, we may say that at about
1 centim. apart two molecules exert the same gravitational as
I molecular force on one another.
We now return to the main business of this section, which
is the Fifth Method of finding the virial constant I. This
consists in using the equation already used for calculating
molecular distances in the form
/ = 7-5v2a(4 log 2L/-9,2?~16/3)/^.
Now a;, the molecular distance for different Hquids, varies as
mi v^, and the expression in brackets may be assumed to be the
same for all bodies ; hence lr=cuv^/m^, where c is a constant
whose value can be obtained on substituting in the case of
ethyl oxide the known . values of /, a, v, and 7?i, or, more
safely, by taking a mean value from several substances.
But we must remember that we are using v the volume in
the body of the liquid, instead of v that in the surface-film ; a
replacement which is not justified by experiment, seeing that
for a given liquid av^ measured at different temperatures is
not constant, the reason being that v varies much more rapidly
with temperature than v. But, in our ignorance of the rela-
tion between surface and body-density, all that we seek for
from the above equation for /, is true values for I from mea-
sured values for a. Accordingly the question arises. Can we
choose temperatures at which to measure u for different sub-
stances, so as to get true relative values of I irrespective of
our ignorance of v ?
As we have seen (Section 6) that at equal fractions of their
critical temperatures, and under equal fractions of their critical
pressures, one liquid is approximately a model of another on
258 Mr. William Sutherland on the
a different scale, we conclude that if we use the value of the
surface-tension measured at a constant fraction of the critical
temperature, and under a constant fraction of the critical
pressure, we ought to get correct relative values of Z ; as
surface-tension is not appreciably affected by pressure, we
can dispense with the condition as to pressure and use mea-
surements of a made under a pressure of one atmosphere at a
constant fraction of the critical temperature. I have chosen
the fraction as two-thirds, because it gives a temperature near
to the boihng-point of most liquids.
Schiff's abundant measurements (Ann. der Chem. ccxxiii.,
and, further, Wied. Beihl. ix.) include not only the height to
which different liquids rise in a capillary tube at their boiling-
points, but also its temperature-coefficient, which is such as
to show that the height in every case vanishes near the critical
point.
Let H be the height to which a liquid rises in a tube of
radius 1 millim. ; then if H really vanished at the critical
temperature and varied linearly with temperature, we should
at 2Tc/3 have H = Tt.&/3, where Z> is the temperature-coefficient.
But to use this would be to depend too much on the accuracy
of h.
If H^ is the value at T^ the boiling-point, then Tc = Ta + H6/Z>,
and
H = H, + (T, - 2T,/3) 5 = Hi/3 + %h/^,
which depends partly on Hj, measured by Schiff, and partly
on h. Now a = Hp/2 = R/2v ;
.'. /or cut4/ml = cH.v§/2mij
Z=c(H,/3 + Tbh/3)t4/2mh
If H is measured according to the usual practice as the
height in millimetres for a tube of radius 1 millim., that is, if
a is measured in grammes weight per metre^ then if / is
desired in terms of the megadyne, gramme, and centimetre as
units, c/2 = 5930, a mean value. Apart from all hypothesis
about molecular force, our last relation between the virial
constant and the constants of capillarity will be am.ply con-
firmed by the extensive comparisons soon to be presented in
Tables XXIY. and XXY. Meanwhile a few consequences of
the relation may be glanced at.
9. Establishment on Theoretical grounds of Eotvos^s relation
between surface-tension, volume, and temperature. — Accord-
ing to the modified equation of the fourth method of finding
Z, M/ = 800TcVi; and according to that of the fifth method,
l = cuvi/7n^. The first of these equations would be more
accurate if we replaced Vi by v, which in the second means
the volume at 2Tc/3 ; so MZ=800Te?^; and mthe actual mass
' Laws of Molecular Force. 259
of the molecule is proportional to M its molecular weight ;
so that from the second we have MZ proportional to uQKvY^v,
and hence a (Mr) I measured at 2T^./3 is for all bodies pro-
portional to Te- Now in our notation the relation discovered
by Eotvos (Wied. Ann. xxvii.) is
^{a(MiO*}/rfT='227,
or a(Mi;)^=-227(T-T/),
where Tc' is a temperature very close to the critical ; and this
is only a more general statement of the relation we have just
deduced.
As Eotvos has verified his relation experimentally for a
large number of bodies, his result is a verification also of our
general principles. The form of his relation also induces us
to examine a little more closely an important consequence of
the form of our infracritical equation, which, when multiplied
by M with the pv term removed at low pressures, becomes
2v \ 13 V— p/
Now E,'M is constant, and M-ljv is proportional to a(Mv)^, if
a and V are measured at 2X^/3, or any other constant fraction
of the critical temperature^ and imder these circumstances
a(Mv)l has been shown to be proportional to T^ ; hence if T
is aT^, a being a constant fraction, we get Tg proportional to
/. ^ ^T V-v\ „
so that 1+ -fs- pr, measured at a constant fraction of the
13 v—fi
critical temperature, is approximately the same for all bodies,
a result which our study of Van der Waals's generalization
showed us to be approximately true. This shows that Eotvos's
relation is rigorous only to the same extent as the constancy
of this last expression is rigorous ; as a matter of fact, exclu-
ding the alcohols and water, Eotvos finds the constant whose
mean value is taken as '227 to depart from this mean value
by not more than 5 per cent, in any individual case. This
brief discussion of Eotvos^s relation has therefore furnished
us with additional proof of the general accuracy of the ap-
proximations we have been forced to make in parts of our
work.
One of the main difficulties in the way of pushing on with
the many interesting inquiries opened up by these relations
lies in the fact that we do not know the relation between the
densities in the body and in the surface-layer of a liquid.
260 Mr, William Sutherland on the
We have to replace our relation
a varies as Ap^rn^
by the less accurate one,
6 1
u varies as Ap^m^,
Multiplying by (Mvf or (M/pf, we get the u(Kv)^ of
Eotvos proportional at all temperatures to AMp ; and as
d{a{'M.vY}dT is constant, and has been shown by Eotvos to be
constant almost right up to the critical temperature, and to be the
same for all bodies, we ought, if our assumptions were rigorous,
to have AMdp/dT constant almost up to the critical tempe-
rature and the same for all bodies (whence another approximate
method of finding A or I). Now dp/dT has been shown by
Mendeleeff to be constant for many substances within ordinary
temperature-ranges ; but the constancy does not hold up to
the critical temperature, and the ultimate meaning of the
apparent contradiction between Eotvos^s result and this is
that, while for most purposes we may safely enough assume
IP proportional to p, we cannot so accurately proceed to the
consequence dp^/dT proportional to dp^/dT ; in fact, a change
of temperature being accompanied by a change of stress in
the surface-layer, the change of p with temperature is more
complex than that of p. But within the range of temperature
for which dp/dT is approximately constant, we have the
important result that
is constant and. is approximately the same for all bodies.
As I now consider the term differentiated to be not two
thirds of the translatory kinetic energy of the gramme-
molecule, but two thirds of the sum of the total kinetic energy
and the chemic virial, I must replace the verbal statement of
the last result as given in my paper (Phil. Mag. April 1889,
p. 312) by the following : — The temperature-rate of variation
of the sum of the total kinetic energy and the chemic virial
of a molecule, measured at low constant pressure, is the same
for all bodies (approximately).
10. Tabulation of Values of the Virial Constant, determined
bv four of the five methods described, and multiplied by the
square of the molecular weight. — In the first place, I will
give a comparison of the values of MH for those bodies to
which existing data allow the application of three or four of
the previously described methods. The multiplication by M^
is for future convenience. The values obtained by the second
^
Laws of Molecular Force,
261
(liquid compression and expansion), third (latent heat) , fourth
(critical temperature), and fifth (capillarity) methods are
entered in the columns marked 2, 3, 4, and 5. (The modified
fourth method was used, namely, M/ — SOOT^^p)
The units are the megamegadyne (10^^ dynes), grni.,
and cm.
Table XXIY.
Substance.
CS,
POL
001,
OHOl,
O2H3CI
O2H.Br
CA^ ,.
OeHu
OeHe
OA
(OH3)2CO
Methyl butyrate
Ethyl butyrate .
0
3.
25-7
4.
5.
26-5
27-2
26-9
39-7
41-7
43-4 i
46-3
46-2
45-6 1
330
38-2
36-1
36-8 ;
22-3
28-3
26-5
1
32-1
31-5
29-0
48*6
45-3
47-1
57-0
58-5
59-3
400
431
42-7
43-8
59-0
55-8
56-2
56-4
36-0
31-3
311
65-0
60-4
55-6
56-1
84-0
74-6
69-5
71-3
The satisfactory agreement of these values, calculated for
such diverse bodies from such diverse data, must be taken as
the verification of the main principles so far unfolded — the
chief of which as regards molecular force is that for most
compound bodies the internal virial term of the characteristic
equation is Iftv below the volume ^, and l/{v-\-k) above that
volume.
We see, too, now how important for molecular dynamics is
the detailed study of each of the constants k, B, and /3 in the
characteristic equations ; but for the present we must refrain
from entering on such a study, and must consider the com-
parison in the last table as closing for the present the general
discussion of the characteristic equation.
Our immediate object is now to ascertain the law connecting
the value of M^Z for a body with its chemical composition.
On the hypothesis of the inverse fourth power (with the sub-
sidiary one of molecular swarms),
niH = 7rm2A(4 log 2L/a- 16/3) ;
so that, the bracketed expression being the same for all
bodies, 7nH is proportional to m^A, and the law of ?rrl or M-7
will be the law of m^A in the expression SAm^/r^ for the force
between two molecules. In the Phil. Mag. for April 1889 I
announced a law for the parameter A, calculated from Schifi"s
capillar}^ data, which applied fairly well to a large number of
organic compounds, but was affected with exceptions subversive
of its generality. Appl^dng now the more accurate method of
262
Mr. William Sutherland on the
calculation described as the fifth to all Schiff's data, we obtain
ample material for generalization, which can be supplemented
bj values calculated by the other methods.
In the following Table the units are again the megamega-
djne, grm.j and cm. For brevity the radicals methyl, ethyl,
and propyl, &c., will be denoted by the first two letters of
their names, while the acid radicals— -formic, propionic, &c. —
will be denoted by Fot, Prt, and so on, so that PrPrt stands
for propyl propionate. The numbers entered under the
heading S will be explained when we are discussing the law
of M.H,
Table XXV.— Values of MH.
First Method.
MH.
S.
1
M^;.
s.
ELO
40-2
7-1
15-0
8-5
8-8
4-5
1-07
2-05
1-25
1-3
H,
•22
1-23
1-16
2-2
6-5
•04
■205
•195
•35
1-0
CO,
^r:::::::::::::::
n!
0^
CH,
N.o
CA
Third ]
Method.
M^^.
S.
1 1 M^^.
1
S.
SnOl,
61-3
49-0
23-5
47-3
27-4
40-0
88-3
61-8
48-7
35-0
90-4
1090
49-4
109-0
44-1
35-6
36-0
47-6
47-5
50-2
GO -4
63-7
62-4
62-8
61-8
74-6
74-1
61
5-2
2-95
5-05
3-3
4-5
7-9
6-15
5-15
4-0
8-0
91
5-2
9-1
4-8
4-1
4-1
5-1
51
5-3
6-04
6-3
6-2
6-2
6-15
7-0
7-0
IsoBuAct
76-2
76-9
77-3
78-7
89-1
89-5
92-5
93-1
92-5
105
106
109
110
125
126
129
147
43-1
55-8
69-5
71-4
85-9
86-1
87-1
101
1
7-1
7^15
7-2
7-3
8-0
8-0
8-15
8-15
815
8-9
8-9
9-1
915
9-95
10
10-15
11
4-7
5-7
6-65
6-8
7-75
7-75
7-8
8-6
AsOlg
EtBut ...
jBClg
PrPrt
SiCI^
IsoAmFot
PrisoBut
OH3I
EtI
EtVat
IsoBuPrt
AmI(C,H,,I) ...
AmBr
AmCl
IsoAmAct
PrBut
C-H^o ....
IsoBuisoBut
PrVat
Ol o-Hr>/^
P^V
IsoBiiBut
C^H^Br,
EtoCoO...
IsoAmPrt
IsoAmisoBut
IsoPrVat
Et,0
EtFot
IsoAmBut
MeAct
IsoAmVat
EtAct
CM,
MePrt
O-H3
PrFot
CgH-Et
MeButiso
cXo(meta)
CgH-Pr
IsoBuFot
EtPrt
CgH^2(i^esitjlene)
C9H12 (pseudo- 1
cumene) j
CioHu (cymene) .
PrAct
MeBut
EtisoBut
MeVat
1
Laivs of Molecular Force.
263
Table XXV. (continued)
Fourth Method.
M.H.
S.
M.H.
S.
CL
5-8
J 0-5
9-8
14-7
441
36-9
27-0
17-7
7-9
15-6
26-6
•9
^•b
1-4
2-0
4-8
4-2
3-3
2-3
1-15
2-1
3-25
PrCl
34-6
100
17-6
23-5
31-6
21-2
41-4
65-5
33-4
67-6
4-0
1-4
2-3
30
3-7
2-7
4-55
6-4
3-9
6-5
HgS
NH3
OA
C H.
NH^Me
NHMeg
c^f;-::.:;:::::::
NMOg
Is"H,Et
cs,
NHEt,
NEt^
0,No
HCl
NH.Pr
OH3OI
EtOl
NHPr2
Fourth Method modified (not using critical pressures) .
M^^.
S.
■
S.
(CHoCl),
38-0
37-7
34-0
I 38-5
} 68-7
41-8
51-2
80-5
81-1
20-3
29-9
52-4
4-3
4-3
4-0
4-3
6-6
4-6
5-4
7.4
7-4
2-6
3-6
5-45
EtOaHgO
HCl
48-6
7-3
16-8
26-4
32-6
7-6
15-6
23-5
31-0
24-4
43-5
664
31-0
68-6
5-15
1-1
2-2
3-2
3-8
M
2-1
3-0
3-7
3-05
4-75
6-5
3-7
6-6
CH30Heio
Ogii CI ".
MeCi...
EtCl
CH^COCHg),
(methylal)
(acetal)
PrCl
NH3
NHaMe
NHMe.
NMeg
0«Hn,
NHoEt
OgH,6
NHEt,
Me 0
NEt3
MeEtO
NHaPr
EtPrO
NHPr2
Fifth Method.
M.H.
32-1
35-6
44-2
45-4
44-8
56-9
57-9
68-5
58-2
56-1
71-6
70-6
73-3
8.
i
MH.
S.
6-85
6-8
6-7
7-8
7-85
7-95
7-8o
7-8
8-85
8-9
8-75
8-75
1
i
EtFot
3-8
4-1
4-8
4-9
4-85
5-8
5-85
5-9
5-9
5-75
6-8
6-75
6-9
1
EtBut ,.
EtisoBut ..
72-3
71-3
69-7
87-3
MeAct
PrFot
; MeVat
EtAct
IsoAmAct
MePrt
IsoBuFot
IsoBaPrt .
87-7
1 80-3
! 87-7
! 86-5
j 105
PrBut
PrisoBut
PrAct
EtPrt
1 EtisoVat
IsoAaiPrt
MeBut
MeisoBut
IsoBuBut
l£;oAmFot
1 IsoBuisoBut 1 103
j PnsoVat 1 103
IsoBuAct
PrPrt
1:
264
Mr. William Sutherland on the
Table XXY. {continued) ^
CioHie
CgHg ..
0,H, ..
12
O9H:
CiqHu
CiiCi,
OCl, ...
PrOl
IsoBuCl.,
IsoAmOl
CgHjCl .,
CoOl.
03H5C10 ....
0013COH ....
CH,0100.,Et.
GClfiOM
C.HjOCl .
O-HgClo ...
EtBr ..'.....
PrBr
IsoPrBr....
OgH^Br ....
IsoBuBr .
IsoAmBr .
CeH.Br....
0-H,Br ....
O.H.Br^
OgHeBr.,
Mel ..:
EtI
PrI
IsoPrI
IsoBuI
IsoAmI
Normal hexane
Diisobutyl
Diisoamyl
Amylene
Caprylene ,
Terpens ,
Diallyl
Benzene
Toluene ,
Orthoxylene
Metaxylene
Paraxjlene
Ethylbenzene
Normal propylbenzene
Ethyltoluene
Mesitylene
Cymene
Chloroform
Carbon tetrachloride ...
Ethylene chloride
Ethidene chloride
Propyl chloride
Isobutyl chloride
Isoamyl chloride
Chlorobenzene
Chlorotoluene
Benzyl chloride
ProjDylene chloride
Trichlorethane
Epichlorhydrin
Chloral
Ethyl chloracetate ...
Ethyl dichlor acetate
Ethyl trichloracetate
Benzoic chloride
Benzilidene cliloride
Ethyl bromide
Propyl bromide
Isopropyl bromide . .
Allyl bromide
Isobutyl bromide ...,
Isoamyl bromide ...
Bromobenzene
Orthobromotoluene ,
Ethylene bromide . . . ,
Propylene bromide .
Methyl iodide
Ethyl iodide
Propyl iodide
Isopropyl iodide
Allyl iodide
Isobutyl iodide
Isoamyl iodide
M^^.
59-3
90-6
125-6
45-0
91-8
107-4
54-8
43-8
56-4
69-7
()9-7
70-3
70-6
86-9
86-1
85-3
103-0
36-9
45-6
44-4
38-1
37-9
520
66-4
57-7
76-0
82-2
56-8
65-6
58-2
49-8
50-9
63-2
79-6
93-5
79-6
89-4
-28-9
41-4
41-2
39-9
53-3
66-2
66-8
81-5
51-2
61-5
25-6
38-1
50-9
49-8
50-2
62-4
80-0
6-0
8-0
10-0
4-9
8-1
90
5-6
4-75
5-75
6-65
6-65
6-7
6-7
7-8
7-75
7-65
8-75
4-2
4-9
4-8
4-3
4-3
5-4
6-45
5-85
7 1
7-5
5-8
6-4
5-9
5-25
5-35
6-25
7-35
8-2
7-35
8-0
3-5
4-6
4-6
4-45
5-5
6-45
6-5
7-45
5-35
6-1
3-15
4-3
5-3
5-25
5-3
62
7-35
Laws of Molecular Force,
Table XXY. (continued).
265
OeH.T
NH^Pr
^H^CaHj
NHgisoBu
"NHgisoAm
NHEtg
NEtg
I^H,0,H,
C^H^N
OgH.N
CgH.N
MeNO^
CCI3NO2
EtNOg
IsoAmNOo
OgH.CN
C4H9CN
CgH.CN
CS2
CSNC3H5
CNSCH3
CNSC,H,
Et^S
PClg
POCl
POCoHgCl
PSCl
(CH3),C0
CeH.^Og
CIl3CH(OCH3), ..
Et^O
(CH3CO)oO
CHgCO^aHj
MeisoAmO
EtgCaO^
OeHgCO^CHg
0(5HgCO2C2Hg
CH3(CO)20H,OEt
C6H,OCH3
CeHiOC^Hg
C,H70CH3
OeH,(OCH3),
C4H2OHCOH
C4H9COH
CgH.iCOH
C.oHuO
(CH3)3CCOCH3 .,
lodobenzene
Propyl amine
Allyl amine
Isobutyl amine
Isoamyl amine
Diethyl amine
Triethyl amine
Aniline
Pyridine
Piperidine
Chinoline
Nitromethane
Chloropicrin
Ethyl nitrate
Isoamyl nitrate
Isobutylnitrile
Capronitrile
Benzonitrile
Carbon disulphide
Allyl sulphocarbimid ...
Methyl sulphoeyanate
Ethyl sulphoeyanate ...
Ethyl sulphide
Ethoxyphosph. chloride
Acetone
Paraldehyde
Dimethylacetal
Diethylacetal
Ethyl oxide
Acetic anhydride
Allyl acetate
Methylisoamyl oxide ...
Elhyi oxalate
Methyl benzoate
Ethyl benzoate
Acetacetic ether
Anisol
Phenethol
Methyl paracresolate . .
Dimethyl resorcin
Furfurol
Valeraldehyde
Cuminol
Carvol
Pinakoline
Wl.
37-8
36-6
50-6
64-4
60-1
75-2
65-0
59-7
100-3
108-6
22-6
57-6
37-2
76-9
39-2
50-5
65-8
26-9
54-0
33-7
44-1
550
43-4
250
62-8
35-5
31-1
85-7
53-3
87-7
43-8
51-9
541
66-8
91-0
91-4
109-9
81-5
70-0
80-7
841
96-2
51-2
48-0
1130
117-8
63-5
S.
7-35
4-25
4-2
5-3
6-3
5-5
7-05
6-B5
60
8-6
905
2-75
5-85
4-2
7-2
4-4
5-3
6-4
3-3
5-55
3-9
4-8
5-6
4-75
3-1
6-2
4-1
3-7
7-7
5-5
7-85
4-75
5-4
5-55
6-5
8-05
8-1
9-15
7-45
6-7
7-4
7-6
8-35
5-3
51
9-3
9-6
6-3
The following ai-e the chief sources of the data from which the above
table has been constructed: — Latent Pleats from Berthelot {Chimie
Mecanique) and R. Schiff {Ann. der Chem. ccxxxiv.). Critical Tem-
peratures from Sajontschewski (Wied. Beibl. iii.), Pawlewski {Ber.deut.
chem. Ges. xv., xvi.), Nadejdine (Wied. Beihl. vii.), and Dewar (Phil.
Mag-, xviii.). Surface-tensions from R. Schift' {Ann. der Chem. ccxxiii. ;
Wied. Beibl. ix.).
Phil. Mag. S. 5. Vol. 35. No. 214. March 1893. T
266 Mr. AVilliam Sutherland on the
To make clear the two simple laws that rule these tabulaied
valnes, it will be ad\dsable to confine our attention at first
only to those values obtained by the fifth method, using those
by other methods only to fill gaps. It will also be well to
consider first a single chemical familj^, such as the paraffins,
for which we have the following values : — ■
CH4. OsHy. CgH^^. CjjHjg. C^oHga-
2-2 14-7 59-3 90*6 125'6
These show that there is not a constant difi'erenoe in the value
of M?l corresponding to the diff^erence in the nuuiber of CHg
groups contained. We can amplify this list of paraffins by
using the material furnished by Bartoli and Stracciati (^Ann. de
Cliim. et de Phys. ser. 6, t. vii.), who have determined the more
important physical constants for all the paraffins from C5H12
up to C16H34. Their values of the capillary constants were
found at about 11 degrees in every case. To obtain those at
two thirds of the critical temperature, we have to use the
values of the critical temperatures calculated by them from
Thorpe and Riicker's convenient empirical relation (Journ.
Chem. Soc. xlv.),
/?/(2T -T) = constant,
p being density at T.
The following are the values of VI thus obtained : —
C3H,,
CeH,,.
C,H,e.
CtHi6.
C^H^y.
C9H20.
^10^22-
47-2
58 1
77-1
79-6
91-2
110
127
C11H24.
^\^2G
:. 0,
LgH^y. C
'lAo-
C.,H32.
CieHgj.
147
171
L92
215
230
256
Considering the assumptions involved in the. calculation of
these values and the difficulty of obtaining the paraffins pure,
the agreement for CgHi^, OgHis, and C10H22 with Schiff's
numbers is excellent ; but the higher we go in the series the
larger is the temperature-interval for which we have to
extrapolate, and the more uncertain do the values become.
However, they are useful as giving a general idea of the
course of M^Z in an extended series.
On plotting these numbers as ordinates with abscissse
representing the number of CHg groups in the molecule,
a curve was constructed which proved to be the parabola
M2Z = 6S + -66S^
where S is the number of CH2 groups. It is only necessary ,
then to determine on this curve the abscissa corresponding to |
Laws of Molecular Force. 267
the value of M.^1 for any substance to obtain the number of
CH2 groups to ^Yhich its molecule is equivalent as regards
molecular force.
11. Definition of the Dynic Equivalent of a substance, and
determination of its value for several Radicals. — In the manner
just described were obtained the numbers entered in Table
XXy. under the heading S, which I propose to call the
Djnic Equivalents of the substances. The djnic equivalent
of a molecule is the number of CH2 groups in the molecule of
the normal paraffin that exerts the same molecular force as it.
The table shows that the atom of an element contributes
the same amount towards the dynic equivalent of all molecules
in which it occurs (except in the case of the simpler typical
compounds) : thus, for example, consider the iodides from
methyl to amyl iodide, and notice that each CHg group has
the same value unity as in the paraffins, and that the iodine
atom is equivalent to about 2'3(JH2 in every case. The same
law holds throughout the table ; so that each elementary atom
or radical has its own dynic equivalent, which can be easily
determined.
In the first place, it appears that the extra H3 in the
paraffins C^H2?i+2 ^^^ ^^ neglected in a first approximation,
because C5H10 has practically the same dynic equivalent as
C5H12, and OgHie the same as CgHig. It may be that the
double binding in the unsaturated compounds compensates
for the H2 of the saturated, but I think that the simpler idea
for the present is that the two terminal H atoms in a paraffin
chain have a dynic equivalent so small that we may neglect
it, or more generally the middle H in a CH3 group is negligible
in a first approximation. If there is really such a ditierence
between the middle H and the two others in CH3, we ought
to find the dynic equivalents for the iso- compounds smaller
than for the normal : and the table shows that the isobutyrates
have equivalents smaller by '1 than the butyrates, while the iso-
butyl salts of the fatty acids have dynic equivalents nearly all
less than 1 greater than the propyl salts. But this is rather a
matter for the chemist to work out in detail ; it suffices to
indicate the idea here, and to point out that it is in harmony
with the lowering of boiling-points among isomers with in-
creasing number of CH3 groups. Accordingly, as a matter of
detail, in the estimation of the dynic equivalents of the elements,
it was assumed that the equivalent of C,^H.,^^_|^2, O,^!!^,,^.!, and
C^^H^,^ is in each case n when normal and n — 'ljJ when the
molecule departs from normality by p CH3 groups. (In
constructing my curve for dynic equivalents, for the sake of
T2
268 Mr. William Sutherland on the
simplicity at first, I ignored the fact that two of the paraffins in
SchifF's data are iso-compounds.)
From the tabulated values for the alkyl salts of the fatty
acids, we get ihe following mean values for the dynic equi-
valent of C0"0': — in the formiates 1'85, acetates 1*92, pro-
pionates 1*91, hutyrates 1'92, isobutyrates 1'91, and iso-
valerates 1'83, the mean for all being 1*9.
From the oxides (ethers) and other compounds containing
single-bound 0, we find for its mean value '6, although in the
ethers containing the benzene nucleus it comes out '8, while in
the benzoates CO^^O' is about 2*5 ; so that the junction of the
benzene nucleus with other groups seems to be accompanied
with an increase in its value : witness also the bromide, iodide,
and amine of CgHg, equal to the bromide, iodide, and amine of
C5H11, although CgHe is less than C5H12 by "25. This slight
variation of the value of CgHg is the only anomaly amongst the
numbers of Table XXV. excepting the simple typical com-
pounds. The values '6 for 0' and 1*9 for CO' are in harmony
with the results for all the other compounds containing oxygen.
From the values for the benzene series of hydrocarbons we
can get a value for H2 in CH2, and also some light on the
important question of the structure of the benzene nucleus.
Thomsen has been led by his thermochemical investigations to
the conclusion that in benzene each carbon atom is bound
to three other carbon atoms by a single bond to each (the
" bond " phraseology is used merely for bre^dty, and not as
expressing definite statical or dynamical facts). If we accept
this conclusion, then the dynic equivalent of Ce^isTninus that
for CfiHe is equal to the dynic equivalent of SHg ; similarly,
from the other members of the benzene series w^e get the values
for 3H2, the mean result being 1'29 or "43 for H2.
The accepted structure for CgHio diallyl is (OH2CHOH2)2,
or four whole CHg groups and two CH2 groups with H removed
from each ; and as the dynic equivalent of CeHjo is 5 6, we find
that of H2 to be '4. Again, recent investigations on the ter-
penes ( Wallach, Ann. der Chem. ccxxv., ccxxvii., ccxxx. ; Briihl,
ccxxxv.) show that the two ordinary forms can have their for-
mulae written CH3C3H7(CH2)2(CH)2C2 ; that is to say, they
have 6 hydrogen atoms cut out of CH2 groups, and as the dynic
equivalent of CioHig is 9 we have that of 3Ho as I'O or 'that
of H2 -33. Since the value '43 is derived from 10 accordant
members of the benzene series, we will take it as the value of
the dynic equivalent of H2 in CH2.
By similar but simpler reasoning the dynic equivalents of
CI, Br, I, and other radicals are easily found, and the
following is a list of mean values : —
Laws of Molecular Force. 269
Table XXYI.— Djnic Equivalents.
CH,
1-0
II.
0.
CO'.
0'.
NH2.
1-23
CN.
•215
•57
19
•6
1-35
NO3.
CNS.
S'.
CI.
Br.
I.
2-2
2-85
1-6
1-3
1-6
2-3
To illustrate the applicability of these values, I furnish a
comparison of the values calculated by means of them for
tvventv substances with the values tabulated in Table XXY.
Table XXYII.
Comparison of calculated and tabulated Dynic Equivalents.
CeHg
C^oHu ...
03H,C1 ..
O^HgBr ..
CgH^iBr..
O^HJ
C.HuI ..
ISH^CgH-
Calc.
Tab.
4-75
8-75
4-3
6-45
3-5
645
4-3
7-35
4-25
6-3
4-8
8-8
4-3
6-3
3-6
G-6
4-3
7-3
4-23
6-23
04H9CN
O^H.oS
O2H5CNS ..
(0H3)2O
CeHiA
(OH3CO),0
C.HgCOH..
OAO,
C.H.A
Calc.
Tab.
5-35
5-3
5-6
56
4-85
48
2-6
2-6
6-6
6-5
7-8
7-7
5-2
5^4
5-3
51
4-9
4-9
7-9
7-9
I
If we now look at the dynic equivalents of the uncombined
elements given in the first part of Table XXV., we may notice
that they are remarkably small compared to the values in the
combined state ; thus, that of H2 is '04, of Ng '205, of O2
•195, and that of CH4 is small too, '35, instead of 1 as it
should be, seeing that CH4 is the first of the paraffin series.
Other typical compounds have small values : CO2 has 1*05,
while C0"0' in more elaborate compounds has a value 1*9,
C2H4 has 1, while C2H6 has 2 ; and so on. The same fact has
been noticed in connexion with the molecular refraction of
some of the typical compounds and some of their immediate
derivatives, and it will yet prove a most important one in
chemical dynamics. But meanwhile it is of greater importance
270 Mr. William Sutherland on the
for present purposes to notice that the dynic equivalents given
for various radicals in Table XXVI. are closely proportional
to their molecular refractions.
12. Close parallelism hetiveen Dynic Equivalents and Mole-
cular Refractions. — As is well known, there are two methods
according to vvhich \hQ molecular refraction is estimated, the
first by means of Gladstone's expression, (72 — l)M//3, where n
is index of refraction ; the other by means of Lorenz's,
(n2-l)M/(n2 + 2)p.
In a brief paper (Phil. Mag. Feb. 1889) I showed that the
experimental evidence taken as a whole is in favour of the
Gladstone expression, for which also a very simple theoretical
proof can be given ; and, further, it was shown that it is best
to measure (?2 — l)M/p if possible in the gaseous state. But
as comparatively few measurements have been made on
bodies in the vapour state I suggested that, as the Lorenz
expression had been empirically proved to give more nearly
the same value in the liquid and vapour states of a body, its
value as determined in the liquid state and multiplied by 3/2
could be taken as giving the value of (n — l)M//3 in the vapour
state. The result of the theoretical argument was that, if
M//0 is taken to measure the molecular domain u, and if (J is
the volume occupied by the molecule in the same units, and N
is the index of refraction for the matter of the molecule, then
{ii-l)u = (X-l)U.
Landolt, Briihl, and others have determined the values of the
atomic refraction for several elements (Ann. der Chem. ccxiii.
p. 235), and by means of these and Masini's data for sulphur
(Wied. Beibl. vii.) and Gladstone's latest determinations
(Journ. Chem. Soc. 1884), I have obtained the values of the
refraction-equivalents of the preceding radicals in terms of
that for CH2 as unity. Mascart has given [Compt. Rend.
Ixxxvi.) values of the refraction of a number of substances in
the vapour state, from which, for the sake of comparison, I
have calculated the refraction-equivalents for as many radicals
as possible.
The following Table contains in the second column the
dynic equivalent, in the third the refraction-equivalent calcu-
lated according to the Lorenz expression, in the fourth the
refraction-equivalent calculated according to the Gladstone
expression from Mascart's data for vapours, and in the fifth
that calculated by Gladstone from liquid data. The value for
CH2 i^i every case is 1.
Laws of Molecular Force.
Table XXVIII.
Comparison of Djnic and Refraction Equivalents.
271
1.
2.
3.
4.
5.
CH2
10
•215
•57
19
•6
1-23
1-35
2-2
2-85
1-6
1-3
1-6
2-3
ro
•23
•54
1-4
•35
M2
1-18
2-2
Ti"
1-3
20
31
ro
•19
•62
1-5
•4
'i'3"
"i'-s"
1-7
2-7
10
•17
•66
1-5
•37
101
1-2
1-9
30
1-9
1-3
20
3-2
H
0
C0"0'
0'
NHo
CN
NO3
CNS
S'
CI
Br
I
This table brings out the remarkable fact that the paral-
lelism between the djnic and refraction-equivalents is so
close as almost to amount to proportionality. I shall not
discuss the meaning of this relation until I have shown how
to obtain the dynic equivalents for the elements usually
occurring in inorganic compounds, and established the same
relation for them also.
Meanwhile it will be useful to compare the dynic and
refraction equivalents of the uncombined elements and of
those simpler typical compounds to which we have said the
summative law does not apply as regards dynic equivalents and
does not accurately apply as regards refraction-equivalents.
Table XXIX.
Ratio of Dynic to Refraction Equivalents, each measured
in terms of that for CHg as unity.
H,.
0,.
N,.
CI,.
CH4.
CJI,.
H^S.
C,H,
CHCI3.
cs.
•09
•21
•22
•38
•32
•45
•79
•75
•94
•72
C,N,.
HCl.
CH3CI.
NH3.
CO,.
SO,.
N2O.
PCI3.
CCl,.
(CH3),0.
•90
•83
•79
1-06
•76
•96
•83
•88
•89
•m
I
272 Mr. William Sutherland on the
We see that the ratio is small for the uncomhined ele-
ments and CH^ and C2H4, a result of the important fact that
while the refraction-equivalent of a non-metallic element is
almost the same in the uncomhined state as in the combined,
the dynic equivalent is much smaller in the uncomhined state
of an element. The meaning of this fact will be discussed
later on ; connected with it is the fact that the ratio for most
of the typical compounds in the last table is notably less than
unity. A molecule has to reach a certain degree of com-
plexity before the summative law holds as regards its dynic
equivalent ; the same maybe said about its heat of formation.
Further comment on the connexion between dynic equivalent
and heat of formation must be deferred for a little.
We can now give a formal enunciation of the law of the
virial constant : — If the molecule of an organic compound is
of a degree of complexity higher than that of the ordinary
typical compounds, then the virial constant for one gramme
of the compound is given in terms of the megamegadyne,
gramme, and centimetre as units by the relation
MH = 6S + -66S2
where M is the usual molecular weight of the compound, and
S is the sum of the dynic equivalents of the atoms in its
molecule (measured in terms of that for CH2 as unity).
According to the law of force SAm^/r^, A??!^ is propor-
tional to M^/ and therefore follows the same law.
13. Return to the Discontinuity during Liquefaction of
Compounds and proof that it is due to the pairing of Molecules.
— The interpretation of the form of the internal virial expres-
sion above the volume k is of the highest importance in the
theory of molecular force, and can now be attempted in the
light of the law for M.H, If the molecules of a substance do
pair to produce an actual chemical polymer of it, then its
molecular mass changes from M to 2M^ while S the dynic
equivalent changes to 2S, and consequently
I or (6 S + -66 S2)/M2
changes to a value given by
V = (6 S/2-f -66 S2)/M^
When S is small we see that the pairing of molecules to pro-
duce a polymer causes the virial constant (for one gramme)
to diminish to nearly the half of its value for free molecules ;
when S is larger this statement becomes less exact. In the
I
Itaws of Molecular Force. 273
cflse of CO2, for example, if from Table XXY. we take S
as 1*05, then
/ = -00367 and Z' = '00206,
but in the case of ethyl oxide, with S = 4*5,
/= -00738 and /' = -00491.
In both cases we see that the pairing of the molecules to
form a new chemical compound or polymer is attended
with a reduction of the virial constant towards one half,
but not exactly to one half of the original value. Now the
data given in Tables I. and III. and the form of virial term
taken to represent them, ll{v + k), along with the fact that
below volume k the form is //2v, mean that in the limiting
gaseous state the virial term is practically //v, just as below k
it is Z/2?; ; hence we must regard the pairing of the molecules
to be such as to cause the virial constant to become one half
of its higher limit — in other words, the pairing must be
different from polymerization. We are therefore led to
differentiate the chemical and physical pairing of molecules
by the statement that while chemical pairing alters the virial
constant in the ratio
(6 S + -66 S2)/(12S + 2-64 S2) or (1 + -llS)/(2 + -44 Sj,
physical pairing alters it in the ratio
6S/12S or 1/2.
The term 'QQ S^ would thus appear to have a certain chemical
significance.
And now as to the form l/{v-\-k) connecting the two ex-
treme cases. We can explain it in the following manner : —
It will be shown when we come to treat of solutions that if a
salt having a parameter of molecular force 3 A (proportional
to its virial constant /) is dissolved in a solvent with parameter
3 W so that there are n molecules of salt to one of solvent,
then the solution behaves as if it consisted of molecules
having a parameter 3 X given by the singular relation
X-^ = (W-'+nA.-')/{l + n).
Now, in a gas being compressed towards the volume k, let us
assume that there are a number of pairs of molecules propor-
tional to k and a number of single molecules proportional to
v^k, and that the same relation apphes to the mixture of
paired and single molecules as to the solvent and salt in a
solution, then, replacing W by / and A by 1/2 and 71 by
274 Mr. William Sutherland on the
Jc/{v — Jc), we get for the reciprocal of the virial constant of
the mixture
^-={'i^4j.-f)/(.%4jWi
v + k
,'. X=lv/{v + k),
and therefore the virial term is l/(v-i-k).
This would be no demonstration of the pairing of molecules
in compounds, if we did not already have it proved in the
case of the elements that the virial term varies inversely as
the volume at all volumes. Remembering this we can accept
our form l/{v-{-k) as indicating the existence of a mixture of
paired and single molecules, the number of pairs at volume v
being to the number of single molecules as ^ to z^— ^.
The form of the energy term,
RT
('-^.)
must also be partly determined by this existence of pairs, but
it would be foreign to our immediate subject to attempt to
investigate it.
14. Brief Discussion of the Constitution of the Alcohols as
Liquids. — To learn a little more on the subject of pairing, it
will be of some profit to consider briefly here the alcohols and
water, which so far have been left aside, after having been
proved in Table XII. to follow in the supracritical region a
different law from the usual one. But also in the liquid state
the alcohols and water, while conforming to the general liquid
laws in many respects, are still exceptional in others. Thus
Eotvos (Wied. Ann, xxvii.) has shown that the alcohols will
conform to his generalization if at the lower range of tempe-
rature, from 20° 0. to 170° C, the molecules be considered
complex (double relatively to ordinary liquids) and water
also conforms if from 100° upwards its molecules be con-
sidered double ; the molecular lowering of the freezing-point
of water produced by the solution of bodies in it as measured
by Raoult and compared with the molecular lowering for
other liquids proves that relatively to these the molecule of
water is double. Taking all the facts into consideration, it
seems to me that the alcohols and water may be assumed, in
the liquid state, to have the pairs of molecules again paired,
the second pairing, however, not being of so intimate a nature
Laws of Molecular Force. 275
as the first, consisting of a mere approximation of the first
pairs without any change in the values of A or of /.
According to this assumption we should expect the beha-
viour of the liquid alcohols to be represented by a form similar
to our infracritical equation, and we will assume
pv
where W is about 2R.
It was on the infracritical equation that our second motho d
of finding M^Z was founded, giving the relation
, 4/ a 25 ^\ ^
which can be applied to Amagat's data {Ann. de Chim. et de
Phys. ser. 5, t. xi.), and Pagliani and Palazzo's (Wied. Beild.
ix.) for the alcohols.
Again, our assumption enables us to apply the fifth or
capillarity method of finding M?l to the alcohols, if only we
remember that the molecular domain becomes twice as large
and its radius 2^ as large as it would be with only one pair-
ing; hence the equation for the fifth method becomes for
the alcohols
/ = ca?;*/(2m)i
SchifF^s data are available.
As the latent heat and critical temperature are largely
dependent on the supracritical equation, we ought not to
expect the formulae for WH furnished by the third or latent-
heat method and by the fourth or critical-temperature method
to apply to the alcohols. But as the relation
or, empirically,
MA, = 19-4T^,
MX = 21 T„
which was deduced in the discussion of the third method,
does apply approximately to the alcohols, the constant being
26, we may as well, for purposes of comparison, see what the
formulae of the third and fourth methods give in the case of
the alcohols.
276
Mr. William Sutherland on the
Table XXX.— M^/ for the Alcohols.
1
Method
Second.
Third.
Fourth.
Fifth.
Methyl
8
14
24
24
34
47
22 (11)
38(19)
82(41)
326 (163)
11 (5-5)
17 (9)
25 (17)
34 (26)
33 (25)
46 (38)
57 (49)
9
9-4
17-6
27-8
27-6
37-8
48-1
6
Ethyl..
Propyl
IsoproiDvl
Isobutyl
Isoamyl
Cetyl
Water
The second and fifth methods give results in substantial
agreement with one another; the third gives numbers which,
when halved as in the brackets, come into agreement with
the others; while if 8 be subtracted from the numbers furnished
by the fourth method as in the brackets, the resulting num-
bers are in very close agreement with those given by the
fifth method. That the two discrepant columns should be
capable of harmonization with the two others by the simple
operations of halving and of subtracting a constant is signi-
ficant of some really simple principle on which the abnor-
mality of the alcohols in the supracritical region depends.
But too much importance must not be attached to the
general agreement among the numbers, as they are all founded
on tentative assumptions ; they simply prove that our idea of
a second loose pairing of molecular pairs in the liquid alcohols
is not discordant with facts.
The following are the dynic equivalents corresponding to
the values of M.H found by the fifth method compared with
those calculated from the dynic equivalents of the elements
and placed in the third row.
Alcohols Methyl.
Dynic equiv. found 1*35
„ „ calcul. 1*8
For water the dynic equivalent found is *91, while 1*03 is
the calculated value. The values found for ihe alcohols are
about '5 smaller than those calculated. If from the capilla-
rity data we had calculated the values of M^Z and the dynic
equivalent as if the alcohols were not exceptional in any way,
we should have got; —
Alcohols Methyl. Ethyl. Propyl. Butyl. Amyl.
Wl 11-8 22-2 35 47-6 60-6
Dynic equivalent 1-65 2'^ 3*9 5-1 6'1
Ethyl.
Propyl. Butyl(iso).
Amyl(i9o).
2-35
3-4 4-3
5-1
2-8
3-8 4-7
5-7
Laws of Molecular Force, 277
These values of the dynic equivalents agree well with the
calculated ones above, but then the values of M.H being all
increased in the ratio of 2^ to 1 are no longer in agreement
with the values got by the second method.
Accordingly we see that the alcohols will require an
exhaustive study for themselves, if the interesting features of
their molecular structure are to be thoroughly made out. I
have merely sketched lines on which their abnormality may
be hopefully investigated. Those bodies, such as nitric per-
oxide, studied by the brothers Natanson, and acetic acid,
studied by Ramsay and Young, which have been proved to
have double molecules split up both by the action of heat and
reduction of pressure have not been touched on in this paper;
they also would require a special investigation in which our
characteristic equations would lend an assistance much
required.
15. Methods of finding the Virial Constant for Inorganic
Compounds y including a tlieory of the Capillarity and Com-
pressihility of Solutions. — So far I have secured only two
methods of finding the virial constants and dynic equivalents
of inorganic compounds from existing data, and only one of
these is practically useful, namely the first, in which the
surface-tensions of solutions are the source of the values ;
the second is based on the compressibility of solutions. Using
our expression for surface-tension in terms of molecular force,
a = 7r/52A6/(2 + \/2),
in the form « = A^o^m^/c,
where c is a constant, I was led to imagine that it could be
adapted to the case of a solution by means of the following
suppositions : — First, that in a solution containing n mole-
cules of dissolved substance of molecular mass p to one of
solvent of molecular mass w, the solution may be assumed to
be a substance of molecular mass
m = {w + np)l{l-\-n) ;
second, that if 3W is the parameter of molecular force for the
solvent and 3A for the dissolved substance, then the parameter
3X for the solution is connected with W and A by the relation
This seems highly arbitrary, but will be completely verified
by the results to which it leads. I could make no progress
I
^
278 Mr. William Sutherland on the
in the handling of solutions until, in the course of some work
on the elasticity of alloys, I discovered a relation similar to
the above to hold, and this proved to be the immediate clue to
the treatment of solutions.
Let a^^ be the surface-tension of water, then we have the
following equations giving A : —
A-i = X-i+(X-i-W-i)/n.
These equations ought to give the same values of A~^ what-
ever the strength of the solution may be ; and herein lies a
first test of the truth of the principles involved.
The following values of cA~^ for NaCl are calculated from
Yolkmann^s data (Wied. Ann. xvii.) for its solution in water
at 20° ; IV is taken as 18, although we consider the water
molecule to be complex, but this does not affect the purely
relative comparison being made : —
71 . . . -105 -084 -052 -035 -017
cA-i . . 1-34 1-38 1-47 1*46 i-±2
Considering that the solutions range from saturation down
to considerable dilution, the approach to constancy is satis-
factory; but it will be noticed that, on the whole, there is a
tendency for the value of cA~"^ to increase with diminishing
concentration, and this same phenomenon is to be seen in
the case of almost all Yolkmann's solutions, most pronoun-
cedly in that of CaCU : —
n .
. -091
•068
•041
•021
•Oil
cA-i .
. 2-41
2-53
2-77
2-95
3^12
This case shows us that there is a certain amount of incom-
pleteness in our theory of the capillarity of solutions, as indeed
we ought to be surprised if there were not^ when we try to
apply our arbitrary definition of l;he molecular mass of a solu>
tion to one which contains 5() parts by weight of CaCla to 100
of H2O as the solution for which n = '091 does, and also when
We assume that the concentration in the surface-layer is the same
as in the body-fluid at all strengths up to saturation. If our
object were an exhaustive representation of the connexion
between the surface-tension of a solution and its concentration,
it would be easy to introduce a slight empirical alteration into
the above equations to make them exhaustive. For instance.
Laws of Molecular Fi
orce.
279
we might imagine that the effective value of W in a solution
experiences a small change proportional to the concentration ;
but the equations as they stand will prove to be sufficient for
our purpose if, in comparing solutions of different substances,
we calculate cA~^ for the same value of n throughout.
In all subsequent calculations n= 18/1000. The experi-
mental data for surface-tensions of solutions are abundant, the
chief that I know of and have used being those of Yalson
{Comjjt. Rend. Ixx., Ixxiv.), Volkmann (Wied. Ann, xvii.),
Eontgen and Schneider (Wied. Ann. xxix.), and Traube
{J our n. fur Chem. cxxxix.).
The following Table contains the values of cA~' for a certain
namber of compounds, the surface-tension being measured in
grammes weight per linear metre and the half molecules of the
salts of the bibasic acids being regarded as molecules.
Table XXXI.— Values of cAr\
Li ..
Na ..
K ..
I.
Br.
CI.
NO3.
OH.
*so,.
iOOg.
4-15
2-46
•83
1-61
•83
1-61
3-99
2-21
•70
1-54
...
1-45
4-74
306
1-45
2-30
1-37
2-30
1^95
4-99
3-31
1-78
2-55
1-71
2-72
2-21
The study of this table brings out the fact that the differences
of the numbers in any two rows or in any two columns are
constant: thus the differences for the iodide and chloride of
the four bases are in order 3*32, 3*29, 3'29, and 3*21, while
the differences of cA~^ for the Na and Li salts of the mono-
basic acids are in order '59, '60, '62, '69, and •54. Accordingly
each atom contributes a certain definite part to the value of
cA~^ for the molecule in which it occurs, and that part is
independent of the other atoms in the molecule. I shall call
this part the parameter-reciprocal modulus of the atom ; we
have not at present sufficient data to get its absolute value in
any case, but if we make Li our standard positive radical, and
CI the standard negative, we can calculate the average values
of the difference between the parameter-reciprocal modulus of
a radical and its standard, — thus in the case of iodine this
difference is 3'28, and in the case of Na '61, and so on.
Before tabulating these mean values I will give the values
calculated for the salts of some other metals with the values
for the same salts of Li subtracted.
I
280
Mr. William Sutherland on the
Table XXXII.
I.
Br.
01.
NO3.
•53
^Mg-Li
•39
•35
^Ca -Li
•44
•73
•67
•55
iSr -Li...
1-67
1-71
iBa -Li
2-49
2-52
...
iZn -Li
...
1-35
133
iCd -Li
2-24
2-42
2-15
2^29
AMn-Li
...
117
•93
r22
!
To these we may add the following values, obtained from
the sulphates JFe-Li 1-27, ^M-Li 1-19. iCo-Li 1-15,
iCu-Li 1-49, Ml-Li -6, iFe. " "
Li '5, and ^Ci\
Li 1-0
and the two following from the nitrates Ag — Li 3'91, and
iPb-Li4-51.
The following Table contains the values of the parameter-
reciprocal moduluses of the different metals minus that for
Li and of the negative radicals minus that for CI.
Table XXXIIL
Mean Values of Parameter-reciprocal Modulus for the Metals
with that for Li subtracted.
Na.
K.
NH.
\Mg.
^Oa.
iSr.
iBa.
iZn.
iCd.
iMn.
•61
•90
-•15
•42
■65
1-7
2^5
1^33
2-3
1-2
ous
iM.
iCo.
iCu.
Ag.
lAl.
ic
*^^ie-
1-27
1-2
115
1-5
3^9
4-5
•6
•5
ro
The same for negative radicals with that for CI subtracted: —
I. Br. NO3. OH. iSO^. iCOg.
3-28 1-56 -81 -04 -86 -46
It may be worth mentioning that these diiierences show a
pretty close parallelism to the corresponding differences of
the atomic refraction given by Gladstone (Phil. Trans. 1870)
but not close enough to be worth dwelling on.
In the case of the organic bodies studied there was nothing-
analogous to this singular property possessed by these inorganic
compounds of having the reciprocal of the parameter A of
molecular force separable into a constant and definite parts
Laws of Molecular Force. 281
contributed by each constituent of the molecule. Thus we
have characteristic of inorganic bodies in solution another of
those properties called by Valson modular, who discovered
that the density, capillary elevation, and refraction of normal
solutions (gramme-equivalent dissolved in a litre of water)
could all be obtained from the values for a standard solution
such as that of Li 01 by the addition of certain numbers or
moduluses representing invariable differences for the metals
and Li and for the negative radicals and 01. Other properties
of solutions have since been proved to be modular, as for
instance their heats of formation from their elements and
their electric conductivities. I think the modular nature of
some of these properties of solutions is the outcome of this
modular property in the parameter reciprocal of molecular
force along with the additive property in mass. To prove
this in the case of density would require a special investiga-
tion, but if we assume the property for density we can easily
deduce Yalson^s result that the property applies to capillary
elevation. Let h be the height to which a normal solution of
any salt RQ rises in a tube of radius 1 millim., then
A=2./,=2Xp»(^)*/o.
Let r and q be the density moduluses of the radicals R and
Q, being small fractions, then p — d + r + q where *t/o^+/(A). Let suffixes
a and b attached to symbols refer them to two different bodies,
then
KX--KXr^=yc.„p|-/.,pf+/(AJ-/(A.),
but
.-. Kn(Aji- A,-i)/(l + n) =H-,pl-^,pl +/(AJ -A\) ■
Hence selecting pairs of bodies for which A^ = A^ approxi-
mately, we ought to get /i^p|— /i,jjo|proportional to cA~^ — cA"^^
the values of the last expression being obtainable from
Table XXXL
To facilitate the comparison I furnish the following broad
statements about A founded on the study of data as to the
molecular volumes of sabs, both solid and in solution, given
by Favre and Yalson [Comp. Rend. Ixxvii.) Long (Wied.
Ann. ix.), and Nicol (Phil. Mag. xvi. and xviii.). The modular
property applies approximately to shrinkage on solution ; the
shrinkage of a gramme molecule of LiCl is 2, and the shrinkage
for a gramme molecule is increased when for Li is substi-
tuted
K Na. NH^. AOa. ^Sr. ^Ba.
by 8 7 --5 10 11 12;
and v/hen for CI is substituted
Br. I. NO3. ASO,. iCOg.
byO 0 0 8 14.
Laws of Molecular Force,
^^^
After giving the values of ^p^ in the next table we can pro-
ceed with the comparison.
Table XXXIV.— Values of 10V/>^.
CI.
Br.
I.
NO3.
iso,.
^003.
Li
432
441
454
443
464
497
521
484
496
507
493
541
550
557
544
457
468
480
466
433
444
450
442
428
437
Na
K
NH^
iOa
iSr
^Ba
The experimental data used are those of Rontgen and
Schneider (Wied. Ann. xxix.), and those of M. Schumann
(Wied. Ann. xxxi.) for CaOlg, SrCl2, and BaCl2, in the case
of which I calculated the compressibility for the half-gramme
molecule according to his result that /jL—fji^=cp where pis
percentage of salt, using a value of c got from the more con-
centrated solutions.
An inspection of this table shows that W/ubp^ possesses the
modular property ; it gives for instance the following dif-
ferences for Na and Li, 9, 12, 9, 11, 11, with a mean value 10,
and so on for the other metals ; the mean values for the metals
minus that for Li are : —
Na.
10
K.
20
NH,
32
65
89;
and for the negative radicals minus that for CI :-
Br.
52
I.
105
NO3
25
0
ICO3.
-15.
We can now select pairs of bodies for which A^= A^ and test
if f^apl-f^bPl is proportional to cA-'^-cA-\
The following are pairs of elements characterized by nearly
equal shrinkage on solution with the values of the differences
of lO^/i/3^ and of cA~^ and also their ratio.
Table XXXV.
Equal shrinkage pair...
K,Na.
10
•29
34
^Sr, ^Ca.
iBa,^Ca.
Br. CI.
I, CI.
NO3, CI.
MeandifF. oflOVp-...
Mean diif. of cA-\..
Ratio of differences ...
33
105
31
57
1-85
31
52
1-56
33
105
3-28
32
25
•81
31
U2
284 Mr. William Sutherland on the
The agreement here is excellent, as will be seen more clearly
if we compare bodies not having equal shrinkage, as K and
Li, for which we get the Wfjup^ difference 20, the cA-^ diffe-
rence '90 with a ratio 22, or K and N'H4, for which the two
differences are 12 and I'Oo with a ratio 11.
The agreement above is a verification of the theory of the
compressibihty of solutions, here barely outlined, and the
equation
10'WI-/.,pD =32(«A-i-cA-i)
when Aa = Ab nearly constitutes a second method of getting
values of cA"^; but we will not use it, as it adds no bodies to
our list. It suffices to have partly verified the principles on
which the first method is founded by their appHcation to quite
another physical phenomenon, and especially the principle
involved in the remarkable equation
X-i = (W-i + nA-i)/(l + n) .
With the values in Table XXXIII. and that for LiCl, namely,
•83, we can obtain the value of cA~^ for any salt whose con-
stituents are to be found in the table^ or we can if we like use
the actual values in Tables XXX. and XXXI.; we can then cal-
culate MVcA-\ which is proportional to M% or Wl = CM^ cA-^,
where C is a constant. To connect the values of M.H thus
found with those previously given absolutely in Table XXY.,
we must find the value of C/ = ^(N-l)U,
where n is the index of refraction and u the molecular domain
of a substance, N the index of the matter of an atom, and U
its volume in the molecule. Hence the refraction-equivalent
of an element is the product of the refracti^ity (Sir W. Thom-
son's name for index minus unity) of the substance of its atom
and the volume of the atom (the volume of the atom being
measured in terms of the unit in which the molecular domain,
usually called molecular volume, is measured) ; so that the re-
LaiGS of Molecular Force. 289
fraction-equivalent is a function of the two variables only,
namely^ the volume of the atom and the velocity of light
through it. Now we have seen that the expression WH, as a
whole, in one aspect appears to be not dependent directly on
the molecular mass M, seeing that M.H can be represented in
terms of certain quantities which we have called dynic equi-
valents. Hence, as I is proportional to A in our expression
BAm^/r^ for molecular force, we see that in one aspect mole-
cular force seems to be not directly dependent on the mass of
the attracting molecules ; and yet, on the other hand, in con-
sidering solutions we found that the quantity A asserted its
individuality as separate from the whole expression Am^, so
that in another aspect there does appear to be a mass action
in the attraction of two molecules. However, regarding
M-^l or Am? as a whole, the simplest hypothesis we can make
about the mutual action of molecules is that it depends most
on the size of the molecules. This would make Am^ a simple
function of U ; so that the dynic and refraction equivalents
would have this in common, that they are both simple functions
of U. Suppose, now, that the velocity of light through all
matter in the chemically combined state is approximately the
same, or that N is approximately the same for all atoms as
constituents of compound molecules, then the refraction-equi-
valents given by Gladstone are directly proportional to the
volumes of the atoms in the combined state, and then the
parallelism between dynic and refraction equivalents would
mean that S is nearly proportional to the volume. It is very
interesting, therefore, to inquire briefly v/hether there is any
evidence to prove that Gladstone's refraction-equivalents are
proportional to the volumes of the atoms ; and I think that
in Kohlrausch^s velocities of the ions in electrolysis we have
such evidence. If different solutions, such as those of KCl,
NaClj or ■|-BaCl2 sre electrolysed under identical circum-
stances, then we knosv, according to Faraday's law, that
each atom of K and of Na, and each half-atom of Ba, may be
considered to receive the same charge, so that they acquire
their ionic speeds under the action of the same accelerating
force. Accordingly, the ionic speed characteristic of an atom is
reached when the " frictional " resistance to its motion is equal
to this accelerating force ; hence the " frictional '^ resistance is
the same for all atoms, or rather for all electrochemical equi-
valents. Now the '' frictional '^ resistance will be a function
of the velocity of the atom and of its domain (atomic volume)
and of its actual volume as well as of the domain and actual
volume of the molecule of the solvent ; but if water is the
solvent in all cases, the only quantities which vary from body
290 Mr. William Sutherland on the
to body are the velocity and the domain and volume of the
ions, so that we can say
'' frictional " resistance = <^(V; w, U),
"""^ Y = F(u, U).
Now the simplest connexion that one can imagine between
the velocity and the domain and volume of the ion is that
the velocity wdll be greatest when the free domain or the
difference betw^een the domain and the volume is greatest,
or, to be more general, when the difference between the
domain and some multiple of the volume is greatest ; but
if N is the sanie for all combined atoms, then U is proportional
to the refraction-equivalent q. Hence the form of F is such
that it contains u—aq, w^here a is a constant. On studying
the experimental data I found that a might be considered
unity, and that Y is a linear function of u — q. There is a
little difficulty in determining with accuracy the domain of an
ionic atom in a solution. Nicol, in his work (Phil. Mag. xvi.,
xviii.) on the molecular domains of inorganic compounds in
solution, has confined his attention almost entirely to differ-
ences of domains, making the assumption suitable to his purpose
that the molecular domain of water is unaltered in solutions,
wdiereas w^e should expect that the greater part of the shrinkage
occurring on solution of an inorganic crystal happens in the
water, which is far more compressible than the crystal.
Accordingly I take the molecular domains of salts in the
solid state, as given by Long in his paper on diffusion of
solutions (Wied. Ann.iiL.), as nearer to the true domain when
they are in solution than Nicol's values ; but to allow to a
certain extent for the change of state on solution, I have assumed
that in each case the water experiences four fifths of the total
shrinkage and the dissolved salt one fifth. This is an arbitrary
adjustment, and is of no material importance to the comparison
to be made except as showing that the point has not been over-
looked. In the following Table are given under u the mole-
cular domains, under q the molecular refractions (Gladstone's),
in the next column their differences, under Jc the specific mole-
cular conductivities determined in highly dilute solution by
Kohlrausch and shown by him to be equal to the sum of the
velocities of the ions in each case. These are taken from his
paper (Wied. Ann. xxvi.), with a few additions from an
earlier one (Wied. A7in.\i.). Under k (calculated) are given
values of the conductivity calculated from the equation
A; = 68 4- 2-2 (w-^),
4
Laws of Molecular Force.
291
expressing the linear relation between conductivity or sum of
ionic velocities and u — q.
Table XXXIX.
u.
q-
u-^.
Jc.
Jc (calc).
KI
53-5
44
36
41
32
24
20-5
20
22
24
24
23
35-3
25
18-8
32
21-7
15-5
14-5
14
16
17-5
18-6
15-8
18
19
17
9
10
8-5
6
6
6
6-6
5-4
7
107
107
105
87
87
87
78
80
81
83
86
77
108
110
105
88
90
87
81
81
81
82
80
83
KBr
KCl
Nal
NaBr
NaCl
LiCl
AMeCl
lOaOL
|SrCl ....
^BaClo
AZnCls
The agreement is here such as to prove a true connexion
between conductivity and u — q, the more striking as no relation
can be seen between conductivity and u or q taken separately.
The only bodies I have omitted from Kohlrausch's latter list
are the nitrates of some of the above metals and of silver, the
hydrogen compounds of the halogens, and the ammonium
compounds. These do not give results in harmony with
those in the last table, and, indeed, we should hardly expect a
compound radical like NO3 to experience frictional resistance
in the same manner as a single atom like CI, and as to the
hydrogen compounds they form a class by themselves with
respect to many physical properties. It will be as well to
show the amount of departure in these cases in the following-
Table :—
u.
?.
u-q.
k.
Jc (calc).
HI
56
50
42
47
36
47-5
35-5
26
16
11
22
19
25-5
22-2
30
34
31
25
17
22
13
327
327
324
98
82
98
104
134
142
136
123
105
11(>
97
HBr
HCI
KNO3
NaNOg
NH.NOg
NH^Cl
Kohlrausch has pointed out that there is some difheulty in
determining the true connexion between ionic velocities and
conductivities in the case of the bibasic acids 80^ and CO3, so
that we must leave them out of the count for the present.
I
292 Mr. William Sutherland on the
18. An Attempt to Determine the Velocity of Light through
the substance of the water-molecule. — In spite of the excep-
tions, the relation demonstrated in Table XXXIX. is suffi-
ciently striking. To explain it, let us replace q by its value
(N — 1)U ; then, in interpreting the expression z* — (N — 1)U
as occurring in our expression for the conductivity of a solu-
tion, there are two methods of procedure ; first, we can assume
that z^ — U, the free or unoccupied part of the domain, is the
most likely to occur, in which case iN'= 2 ; second, that u — cTJ
occurs in the expression for conductivit}^, and that c happens
to have the same value as i^ — 1, on which supposition it would
be desirable to determine N. At present I know of only one
way of attempting to find N or v/V, the ratio of the velocity
of light through free aether to its velocity through the matter
of an atom, namely by means of Fizeau's experiment, repeated
by Michel son and Morley, on the fraction of its motion com-
municated by flowing water to light passing through it.
Exactly in the manner of my paper (Phil. Mag. Feb. 1889,
p. 148) we can estimate the effect of the motion of matter on
the light passing through it. Let s be the distance travelled
by light in water flowing through the aether at rest with
a velocity h in the same direction as the light, v'^ the mean
velocity of light through the flowing water, v' the mean
velocity through still water, v its velocity through free aether,
y through a molecule of water, I the mean distance through
a molecule, and a its mean sectional area ; then the total loss
of time experienced by a wave of unit area of front or a tube
of parallel rays, or, briefly, a ray of unit section in passing
through the matter-strewn path s instead of a clear path in free
aether, will be equal to its loss in a molecule multiplied by the
number of molecules passed through in the path. This
number, when the matter is at rest, is proportional to s, to a,
and to p/M, or it varies as sap/M. ; but when the matter is in
motion it is reduced in the ratio 1 — S/v": 1. The loss of time
in each molecule is found thus : l/Y is the time taken to pass
through a molecule, and in this time the molecule moves
a distance IB/Y and the unit wave-front moves a distance
/(I 4- S/Y) , which in free aether would take a time /(I + 8/Y)/v ;
so that the loss of time in a molecule is l/Y — l{l + B/Y)/v,
Hence, the total loss of time in the path s may be written
1+^
M \Y ~r')\}^^)'
But the loss of time is also s/v"^s/v ; equating the two ex-
Laws of Molecular Force, 29S
pressions and putting M/p^u and aZ=U, v/v"=n", v/Y=l^
we get ,
as k is equal to 1, seeing that if 8 = 0 andU = ?i, then N must
be equal to n. This equation is the companion to that for
still matter, namely,
w(^-l) = U(N-l).
But to allow for deformation of the wave-front in passing-
through molecules it was shown (Phil. Mag. Feb. 1889,
p. 150) that this first approximation might be altered to the
form
u(n-l) = U{'N-l)+cp,
where c is a constant, and this form was verified, so that we
may write
^,(n"-l)=u(N-l-^N)^l-^7^"j+c/)
neglecting the term in 8^ ;
/. ^,(,^'/_,,) = ^|u(N-l)('J^+n'^)
n"-n_ 3U(N-1) n-lf N „\
^v8 xh
n V u[n — 1)
Now
where x is the fraction of the water's velocity imparted to the
velocity v' to change it to v" . Fizeau [Ann, de Ch. et de Pliys.
ser. 3, t. Ivii. ) found a value '5 for .^', while Michelson and Morley
(Amer. Journ. Sc. ser. 3, vol. cxxxi.), inamore extended and
accurate series of experiments, found a value .2;= '43 ±'02,
which we will adopt. U(N — 1) is equal to xi{n — \) measured
in the vapour of water, for which Lorenz (Wied. Ann. xi.)
gives the value 5*6 ; the vahie for water at 20° C, according
to his data is 6, and n is 1*333, which may also be taken as the
value for n" where it occurs ; all these values being substituted
in the equation
N n^ w(>i-l) ,,
^^X-'^n-V [J(N-l) '
294 Mr. William Sutherland on the
give the value N = 9. Hence the velocity of light through
the water molecule appears to be one ninth of that through
free aether. But before we could ascribe any degree of
accuracy to this estimate we should need to be surer of the
value of £c, whose measurement is attended with great experi-
mental difficulties. It is much to be desired that we had
similar measurements for other bodies than water, both liquid
and solid, to permit of other estimates of N, so as to see if it
is the same for all compound bodies, and also to decide be-
tween the theory here sketched and Fresnel's hypothesis,
that matter carries its own excess of gether with it, so that
,x= [)i^ — l)/n^j which in the case of water is '437, in excellent
agreement with Michelson and Morley's experimental number ;
but one such agreement is not sufficient to establish an
hypothesis founded on such artificial grounds. However, if
N = 9 then N — 1 = 8, and we have the electrical specific mole-
cular conductivity ^■ = 684-2'2 (i^ — 8U). It is only a coinci-
dence that this agrees so exactly in form with Clausius^s
calculation that the number of encounters experienced per
second by a molecule of volume U moving amongst a number
of others of volume U is greater than that experienced by an
ideal particle moving under the same circumstances in the
ratio u : u — S\J. Further experiment must elucidate the
subject-matter of these speculations.
19. Suggested relation between the cliange in the volume of an
atom on combination and the change in its chemical energy.
— Returning to the idea that the dynic equivalent furnishes
a measure of the volume of its atom, we can get a suggestive
ghmpse into the relation between the volume of an atom and
its chemical energy. Kundt has recently (Phil. Mag. July
1888) shown that the velocities of light through the metals
(uncombined) are as their electrical conductivities, being in
the case of silver, gold, and copper greater than through free
aether, and as in this case both n—1 and N — 1 are negative,
we see that {n — l)u or (N — 1)U for the metals changes
greatly when the metals pass from the combined to the free
state. Now this is in strong contrast to the behaviour of the
non-metallic elements, which have been shown in the case of
0, N, C, S, P, CI, Br, and I to possess nearly the same values
of (n — l)u in the combined and free states, and the same may
perhaps be said of H. Again, in contrast to this approximate
inalterability of (n— l)w for these non-metals we have the
fact, already pointed out, that the dynic equivalents of H, 0,
and N are much smaller in the free than the combined state.
If, then, the dynic equivalents give a measure of the volumes
I
Laws of Molecular Force. 295
of the atoms in both states, we must consider the volumes of
free H, ,0, and N to be smaller than when they are combined,
the change of volume corresponding to the change of energy
on combination. If this is true, then the elasticity and density
of the non-metallic atoms (or the equivalents of these proper-
ties in the electromagnetic or any other theory of light) are
so related that although the density changes (N — 1)U re-
mains constant, whereas in the metallic atoms the relation
between density and elasticity must be quite different, because,
as we have seen, (N — 1)U actually changes sign in some cases.
It would be possible to determine approximate values for
the dynic equivalents of the uncombined metals from Quincke^s
data for the surface-tension of melted metals, and also to get
some light on the constitution of salts from his measurements
of the surface-tension of melted salts, but these would be
most appropriately discussed in connexion with a general
study of the elastic properties of solids. I have, however,
satisfied myself that the dynic equivalents of the uncombined
metals are different from their values in the combined state.
To show the existence of an intimate relation between dynic
equivalents and chemical energy we can enumerate the follow-
ing propositions : — That in the great majority of inorganic
compounds the evolution of heat accompanying the passage
of an atom from the ui-hcombined to the combined state is
almost independent of the nature of the atoms it combines
with, similarly the change of dynic equivalent of an atom on
combination is almost independent of the nature of the atom
it combines with ; that in organic compounds with ihQ excep-
tion of the simpler typical forms the same proposition as this
applies both as regards heat and dynic equivalent.
These general remarks are intended to indicate the most
hopeful direction for the continuation of these researches to
open up new fields ; and yet in old fields there is abundance
of scope for the application of the law of molecular force
towards the acquisition of a knowledge of the structure of
molecules, in the elasticity of solids, in the viscosity of gases
and of liquids, in the kinetics of solutions, and many kindred
subjects.
Melboui-ne, February 1890.
[ 296 ]
XXVIII. The Fusion- Constants of Igneous Rock. -^Tart III.
The Thermal Capacity of Igneous Rock, considered in its
Bearing on the Relation of Melting-point to Pressure. By
Carl JBaeus*.
[Plate VI.]
1. JNTRODUCTORY.—ThQ present experiments are in
series with the volume-measurements of my last paper,
and the same typical diabase was operated upon. Since it is
my chief purpose to study the fusion behaviour of silicates,
more particularly the relation of melting-point to pressure,
the observations are restricted to a temperature-interval
(700° to 1400°) of a few hundred degrees on both sides of
the region of fusion f (§ H).
2. Literature. — Experiments similar to the present, but
made with basalt, were published quite recently J by Profs.
Koberts- Austen and Riicker§. The irregularities obtained
by these gentlemen with different methods of treatment
(heating in an oxidizing or a reducing atmosphere, repeated
heating, sudden cooling), the anomalously large specific heat
between 750° and 880°, where basalt is certainly solid, and the
absence of true evidences of latent heat||, contrast strangely
with the uniformly normal behaviour occurring throughout
my own results. Basalt is chemically and lithologically so
near akin to diabase (particularly after melting) that I anti-
cipated a close physical similarity in the two cases. Unfor-
tunately the account given of the basalt work is meagre.
Detailed comparisons are therefore impossible.
The elaborate measurements of Ehrhardt (1885) and of
Pionchon (1886-7) are less closely related to the present work.
Apparatus.
3. The Rock to he tested, — About 30 grammes of diabase
were fused in the small platinum crucible together with which
they were to be dropped into the calorimeter. Two such
charged crucibles were in hand, to be used alternately. The
molten magma, after sudden cooling, shows a smooth, appa-
rently unfissured surface, glossy and greenish black. After
* Commimicated by the Author.
t The geological account of the present work is begun by Mr. Clarence
King, in the January number of the American Journal.
X This was written some time ago. See American Journal, December
1891 and January 1892. A forthcoming Bulletin, No. 96, U.S. Geological
Survey, contains the work in full.
§ Roberts-Austen and Riicker : this Magazine, xxxii. p. 355(1891).
'I Supposing basalt to solidify (§ 13) below 1200°.
The Fusion Constants of Igneous Mock. 297
drying and weighing, the mass is often found to have gained
5 per cent, in weight. I was at first inclined to believe that
this was attributable to water chemically absorbed by the
viscous magma; but the water is only mechanically retained,
for it passes off after 24 hours of exposure to the atmosphere,
or by drying at 200° C. for, say, 30 minutes. Hence I
weighed my crucibles at the beginning of each measurement,
having previousl}' dried them at 200°. The solid glass, sud-
denly cooled from red heat, soon shows a rough and tissured
surface, and its colour changes from black to brown, possibly
from the oxidation of proto-to sesquisalt of iron, possibly from
mere changes in the optical character of the surface (§ 2).
Throughout the course of the work the charge of the
crucibles was neither changed nor replenished.
4. Thermal Capacity/ of Platinum. — Data sufficient for the
computation of the heat given out by the crucibles were
published in 1877 by Violle*, whose datum for the high
temperature {t) specific heat of platinum is '0317 + •000012^.
Hence the increase of thermal capacity from zero Centigrade
to the same temperature is ^(•0317 + '00000()^), which is the
allowance to be made per gramme of platinum crucible.
5. Furnace. — Inasmuch as heat is rapidly radiated from the
white-hot slag, it is necessary to transfer the crucible from
the furnace into the calorimeter swiftly. I discarded trap-
door, false bottom, and other arrangements for this purpose,
because the mechanism clogs the furnace, interferes with con-
stant temperature, and is too liable to get out of order. The
plan adopted is shown in figs. 1 and 2 (Plate VI.), in sectional
elevation and plan. The body of the furnace consists of two
similar but independent bottomed half-cylinders, A A and
B B, of fire-clay properly jacketed, which come apart along
the vertical plane c c c c. The lid, L L, however, is a
single piece, and fixed in position by an adjustable arm (not
shown). Each of the halves of the furnace is protected by a
thick coating of asbestos, C C, D D, and by a rigid case of
iron, E E, F F. Set screws, gggg, pass through the edges
of this in such a way as to hold the fire-clay and asbestos in
place. The horizontal base or plate of the casing E F is bent
partially around the two iron slides, G Gr, along which the
two halves of the furnace may therefore be moved at pleasure
while the lid is stationary ; as is also the blast-burner, K,
clamped on the outside (not shown), and entering the furnace
by a hole left for that purpose.
* Violle's calorimetiic work will be found in C. R. Ixxxv. p. 643 (1877),
Ixxxvii. p. 981 (1878), Ixxsix. p. 702 (1879) ; Phil. Mag. [4] p. 318 (1877).
Phil. Mag. S. 5. Vol. 35. No. 214. March 1893. X
I
298 ' Mr. Carl Barus o/i the Fusion
The charged crucible is shown at K (figs. 1, 2, and 3), and
is held in position by two crutch-shaped radial arms, ^, N, of
fire-clay, the cylindrical shafts of which fit the iron tubes
P, P, snugly, and are actuated by two screws, R, R. More-
over P, P are covered with asbestos (not shown), and thus
subserve the purpose of handles, by grasping which the two
halves of the furnace may be rapidly jerked apart. It is by
this means that the crucible is suddenly dropped out of the
furnace into the calorimeter immediately below (not shown).
Care must be taken to have the arms N, N free from slag.
6. Temperature. — As in the former work, the temperature
of the furnace is regulated by forcing the same quantity of
air swiftly through it at all times, but lading this air with
more or less illuminating-gas, supplied by a graduated stop-
cock. The amount of gas necessary in any case is determined
by trial, and observations are never to be taken except after 1 5
or 20 minutes^ waiting, when the distribution of temperature
is found to be nearly stationary. Nevertheless the tempera-
ture of the crucible is never quite constant from point to
point. I therefore measured this datum at three levels — near
the bottom, the middle, and the top of the charge, after the
stationary thermal distribution had set in (see Tables, § 10).
For this purpose the fire-clay insulator*, tt^ of the thermo-
couple, a 6, passing through a hole in the lid, is adjustable
along the vertical. Before dropping the crucible the thermo-
couple is withdrawn from the charge and suspended above it.
The cold junction is submerged in petroleum and measure-
ments made by the zero method.
When the charge is solid, a small platinum tube ])reviouslv
sunk axially into the mass (see fig. 3) enables the observer to
make the three measurements for temperature as before. In
my later work I also encased the insulator of the thermocou])le
in a platinum tube closed below (see fig. 1) when making
these measurements for the molten charge. Slag being a
good conductor at high temperatures^ hydroelectric distor-
tions of the thermoelectric data may not otherwise be absent.
I state, in conclusion, that when constancy of temperature
is approached the hole in the fid is closed with asbestos, and
the products of combustion escape by the narrow seam in the
side of the furnace, through which, moreover, crucible and
appurtenances are partially visible.
* These are cylindrical stems, 0 5 centim. thick, 25 centim. long, with
two parallel canals running from end to end, through which the plalinnm
wires are threaded. Cf. Bulletin U.S. Geolog. 'Sun-ev, No. 54, p. 95
(1889). ^ : ' ' ' > .
Constants of Igneous Rock. ^99
7. Calorimc4er. — A hollow cylindrical box, provided with a
hollow hinged lid, through both of which a current of cold
water at constant temperature continually circulated, sur-
rounded the calorimeter on all sides. Thus the temperature
of the environment was sharply given, and the correction for
cooling could be found and apphed with accuracy.
The calorimeter was a vessel of thin tinned sheet iron,
28 centim. long, 8 centim. in diameter, having a water-value
of 19 gramme-calories, and holding a charge of about 1200
grammes of water. The inside of the vessel was provided with
a fixed helical strip running nearly from top to bottom, and
was supported on a bard rubber stem. This could be actuated
On the outside of the outer case from below, and served as
a vertical axle around which the calorimeter could be rotated.
In this way the water within the vessel was churned, and
three small hard rubber rowels near the top gave steadiness
to the rotation. I pass the description of this apparatus
rapidly here, but shall recur to it in connexion with other
calorimetric work.
The box or outer vessel of the calorimeter, with its pro-
jecting stem, was movable on a small tramway, the tracks of
which lay at right angles to the slides G, G (figs. 1 and 2) .
Thus at the proper time the lid of the box was opened and the
calorimeter rolled directly under the furnace. After receiving
the crucible the calorimeter was again rolled away and the box
closed, whereupon the temperature-measurements were made
by a sensitive thermometer inserted through a hole in the lid.
Were I to continue work like the present I should make the
crucible bullet-shaped, and provided with a permanent central
tube much like fig. 3. The splashing of water by the drop-
ping crucible (an annoyance which is sometimes serious)
would then be to a great extent obviated.
Results.
8. Method of Work. — While waiting for stationary furnace
temperature I made the initial measurements for the cooling
of the calorimeter in time series. Knowing, therefore, the
time at which the body was dropped I also knew the tempo*
rature of the water into which it was dropped, accurately.
Similarly the three measurements for the temperature of the
charge had just before this been made in time seiies.
The experiments showed that ten minutes after submer-
gence the crucible and charge might safely be considered
cold, for the maximum temperature of the calorimeter was
X 2
300 Mr. Carl Bams on the Fusion
reached after 5 minutes. Hence the time from 10 to 15
minutes was available for making the final measurements for
cooling ; knowing the extremes, I found the intermediate
rates in accordance with the law of cooling. Thus, while the
calorimeter was being constantly stirred, its temperature was
measured at the end of each minute. Hence I knew the mean
excess of its temperature above its environment during the
course of every minute, and was able to add the corresponding
allowance for radiation and evaporation at once. How im-
portant this correction is the Tables (§ 10) fully show. The
only drawback against sharp values is the lag error of the
thermometer ; but this is eliminated in a long series.
I have stated that both the calorimeter and the crucible
were weighed before and after each measurement. The latter
data were taken.
9. Arrangement of the Tables, — The two crucibles (§3)
and tubes (fig. 3) are designated I. and II. In all cases ni
is the mass of the charge, M the calorimetric value of the
calorimeter (corrected for temperature), r the temperature of
the environment. 0 is the temperature at the top, the
middle, and the bottom of the charge at the time of submer-
gence. The mean value is also given. The temperature of
the calorimeter at the time specified is given under 6, and a
parallel column shows the correction of 6 for radiation.
Finally, the computed thermal capacity of the platinum cru-
cible and appurtenances (correction Ji), and the thermal
capacity* h of the charge computed up to each of the con-
secutive times, are found in the last columns. A few obvious
remarks follow, x^ote that h reaches its true (constant) value
in proportion as the body is cold.
To avoid prolixity I have only given full examples of the
data here defined at the head of each table. The remainder
is abbreviated.
10. Tables. — In the data of the first series (^Table I.) only
one value of © is in hand for the liquid state. Moreover the
construction of the furnace was somewhat faulty, not being
flat-bottomed. Hence these results are of inferior accuracy
as compared with Series II. (Table II.), which are the best
obtained.
* The constant h is really the increase of thermal capacity above zero
degrees Centigrade.
I
Constants of Igneous Rock.
Table I. — Thermal Capacity of Diabase. First
Platinum crucible, I., 11' 169 g. ; Platinum tube, I.,
Platinum crucible, II., 11 '271 g. ; Platinum tube, II.,
301
Series.
•985 g.
•654 g.
' 1
Cruc.
Time. 6.
j Mean 6.
M. 9.
m.
Correc-
tion
Correc-
tion
h.
h.
i
I.
' 12
Minutes. 1 °C.
0 :
1 i
I 2
3
5
8
11
14
1367°
1202 g.
33-36 g.
°a
14-92
22-80
25-20
25-50
25-58
25-40
25-25
25-12
°C.
•02
•06
•11
-20
•33
-46
•60
g.-cal.
17-9
; g.-cal.
t
267
' 355
i 367
372
370
370
370
i
Immersion
Liquid.
I 370
I.
16
0
11
1306°
1145 g.
33-75 g.
25-90
36-04
i-io
i'6-6
"364
Immersion
Liquid.
II.
I.
12
0
14
1378°
1202 g.
29-32 g.
22-16
30-97
i-ii'
20-7
"385
Immersion
Liquid.
12
0
14
1337°
1196 g.
32-22 g.
14-55
24-40
"-69
18-0
"373
Immersion
Liquid.
II.
12
0
14
1274°
1196 g.
29-16 g.
21-92
29-98
i-'io'
iV-'i
"358
Immersion
Liquid.
' I.
12
0
14
1199
1163
1138
1166°
1 195 g.
32-22 g.
14-87
22-98
"•si'
16-7
"311
Immersion
Solid.
Immersion
Solid.
I II.
I.
12
11
0
14
1100
1074
1060
1078°
1196 g.
29-16 g.
1001°
1196g.
32-23 g.
1025°
1195 g.
2916 g.
21-16
27-25
"•74
i6-4
"263
0
11
1021
998
983
1489
21-31
"47
13-9
"242
Immersion
Solid.
II.
11
0
14
1035
102.5
1015
19-76
25-55
"•73
15-5
"253
Immersion
Solid.
I.
II.
11
0
11
889
880
872
880°
1198 g.
32-24 g.
16-19
21-59
"•4i"
12-0
"264
i
Immersion
Solid.
11
0
14
827
827
833
829° 20-07
1192 g. 24-39
2916 gJ
"•65
1
12-i' "l91
1 i
Immersion
Solid.
k
302
Table II.—
Mr. Carl Barus on the Fusion
Thermal Capacity of Diabase. Second Series.
i
!
Cruc!
No. i
II.
j
T. 1
i
e. 1
Mean 9.
M.
m.
\
[
Time.
e.
1
Correc-i
tion
6>.
1
Correc-'
tioD \ h. \
1 i
1
°c.
12
°c.
1265
1246
1241
1
i
1
1251°
1189 g.
26-39 g.
1
Minutes.
0
1
2
3
5
8
11
14
°c.
18-94
24-60
26-05
26-52
26-61
26-45
26-25
26-08
°G. '
"-04 i
-10 1
•16
•30
•50
-69
-88
g.-cal. 1
20-6
g.-cal.
j Immersion
236 Liquid.
305 !
329 j
339 ;|
341 IJ
I.
12
997
995
: 987
993°
1192 g.
32-22 g.
0
11
1407
20-61
1
, -22
13-8 1 238-3 ;
1
Immersion
Solid.
II.
II.
L
12
1260
1251
1243
1251°
1190 g.
26-07 g.
0
14
19-34!
26-49 1 -80
1
26-8
342-5 1
Immersion
L^"quid.
10
1354
1333
1319
1334° ■ 0
1190 g. 1 14
26-27 g.
13-78 i
21-79 ! -84
i
22-4 376-6
Immersion
Liquid.
10
954
948
942
948°
1186 g.
32-22 g.
0
14
20-24
25-81 i -94
I
i'3-0 j 226"6
!
Immersion
Solid.
n.
10 * 1364
i 1354
1 1339
1352°
1194 g.
26-05 g.
1 0
1 14
1718
24-82 -87 ! 23-1 367-0
i 1
Immersion
Liquid.
L
II.
II.
1
I.
10
1 877
i 873
1 870
873° j 0
! 1191 g. j 14
32-20 g.|
14-831 ; 1
j 2013 1 -46 1 11-9 202-1
Immersion
SoUd.
10
1176
1164
1158
1166°
1187 ff.
25-97 g.
0 ' 17-40 i !
14 1 24-21 *77 ! 19-2 ! 309-5
Immersion
Liquid.
10
1215
1191
1186
1197°
1192 g.
25-95 g.
0 14-38
14 21-13 -62
19-9
318-5
i
Immersion
Liquid.
10
1 782
780
780
1
781°
1189 g.
32-19 g.
0 19-36 '
14 1 23-611 -94
t j
i
16*4
179-7
j
Immersion
Solid.
IL
1
'lO
1204
1 1195
1183
1
1194°
1195 g.
25-90 g.
0 ; 14-54
14-5 ; 21-22 -66
19-9
i
317-9
Immersion
Liquid.
Constants of Igneous Rock,
Table II. (^continiied).
3oa
Cruc.
^0.
r.
e.
Mean 9.
M.
I'll.
1171°
1192 g.
32-20 g.
Time.
e.
Correc-
tion
0.
Correc-
tion
h.
h.
I.
1
10
°c.
1177
1170
1166
Miuiites.
0
14
°0.
19-88
27-37
°C.
i'-io
g.-cal.
16-7
g.-cal.
301-6
Immersion
Solid.
1— 1
11
11
1106
1094
1088
1096°
1195 g.
32-21 g.
0
14
16-28
23-24
■-68
15-5
268-"2
Immersion
Solid.
1262
1244
1238
1248°
1191 g.
25-49 g.
0
14
19-72
26-55
■-89
21-1"
338-8
Immersion
Liquid.
11
1237
1216
1202
1218°
1188g.
29-43 g.
0
14
13-67
21-60
"•69
17-7
330-3
Immersion
Liquid.
III.
1.
1
11
1224
1216
1205
1215°
1185 g.
25-57 g.
0
14
19-73
26-27
'"•95
264
326-"6
Immersion
Liquid.
For brevity the later observations were averaged per
0 minutes, and under li the mean value for the last 11 minutes
is usually given.
In Series I. the increase of temperature from top to bottom of
the crucible is as large as 60° at 1200°, usually much smaller,
however, and falling off pretty regularly to 6° at 829°. In
Series II. the corresponding mean difference is about 25° at
1300°, 14° at 1000°, 10° at 800°. The errors thus involved
cannot be greater than 2 per cent, in the extreme case ; but
since the distribution of temperature is measured^ it is probably
negligible except at very high temperatures. I am inclined
to infer that the greater constancy of the solid distribution
as compared with the liquid is due to greater thermal con-
ductivity in the former case (solid), convection being neces-
sarily absent in both.
Considering the observational w^ork as a whole, the data
are satisfactory, seeing that an error of 0*1° C. in the calori-
metric temperatures, initi^d or final, must distort the results
at least 1 per cent. But the real source of error is probably
accidental, and is encountered when thehotbodv falls through
the surface of the cold water.
Inferences.
11. Digest and Charts. — In Tables III. and lY. I have
summarized the chief results on a scale of temperature. The
304
Mr. Carl Barus on the Fusion
results of the latter (Series II.) are graphically shown in the
chart (fig. 4), in which thermal capacity in gramme-calories
is constructed as a function of temperature *. Straight lines
are drawn through the points, shov^ing the mean specific
heats for the intervals of observations, solid and Uquid. The
letter a marks the region of fusion.
Table III. — Thermal Capacity of Diabase. Series I.
Digest, cf. § 15.
Mean specific heat, sohd, 800° to 1100° . -304.
„ liquid, 1200° to 1400° . -350.
Latent heat of fusion, at 1200°, 24 g.-cal. ; at 1100°, 16 g.-cal.
Solid.
1
Liquid.
i —
Temp.
Thermal
capacity.
Temp.
Thermal
capacity.
Temp. Tl^erm^l
1^ 1 capacity.
Temp.
Thermal
capacity.
829
880
1001
191
204
242
1
1025
1078
tll66
1
253
263
311
0
1274
1306
i
1 1337
358
364
373
1367
1378
370
385
Table IV. — Thermal Capacity of Diabase. Series II.
Digest, cf. § 15.
Mean specific heat, solid, 800° to 1100° . -290.
„ liquid, 1100° to 1400° . -360.
Latent heat of fusion, at 1200°, 24 g.-cal. ; at 1100°, 16 g.-cal.
Solid.
Liquid.
Temp.
Thermal
capacity.
Temp.
Thermal
capacity.
Temp.
Thermal
capacity.
Temp.
Thermal
capacity.
o
781
873
948
993
180
202
227
238
1096
tll71
268
302
1166
1194
1197
1215
1218
310
' 318
319
327
330
1248
1251
1251
1334
1352
339
340
342
377
367
* The corresponding chart for Table III. is almost identical with this,
t Incipient fusion (?) at the base of the crucible.
Constants of Igneous Rock. 305
In both the tables, III. and IV., the soKd pomts lie on lines
which, if reasonably curved, would be nicely tangent to an
initial specific heat of aboufc 0*2 at °C. The grouping, in
other words, is so regular as to exclude the probability of
anomalous features, either in the observed or the unobserved
parts of the loci. The solid point near a (fig. 4, a similar point
occurs in Table III.) alone lies markedly above the curve ; but
inasmuch as in my volume work I found solidification to set in
at 1100°, it is altogether probable that the occurrence at 1170^
is incipient fusion (§ 13).
The regularity of the liquid loci (Tables III. and lY.) is
slightly less favourable ; bat the discrepancies which occur
are above 1300°, and obviously accidental (§ 10, end).
12. Specific Heat, — As regards the mean specific heats be-
tween 800° and 1100° in Tables III. and IV., it will be seen
that the intermediate datum would satisfy both groups of
points about as well as the individual data given. A tracing-
made of the first group practically covers the other. The
same remarks may be made for the liquid state. I have not
attempted any elaborate redactions, since the equations of
the necessarily curved loci would have to be arbitrarily
chosen, and since values for specific heat are of no immediate
bearing on the present inquiry.
13. Hysteresis. — Recurring to the suggestion of the pre-
ceding paragraph, it appears that the fusion behaviour of rocks
must be accompanied by hysteresis* of the same nature as
that which I observed with naphthalene and other substances :
for, whereas in my volume work with diabase I was able to
cool the rock down to 1095° without solidifying it, evidences
of fusion (at a, figs. 4 and 5) do not occur in the present
work until 1170° is reached. The magnitude of the lag is
thus of the order of (say) 50°, and its pressure-equivalent
may be estimated as 500 atmospheres.
14. Latent Heat. — In virtue of the fact that the (upper) end
of the solid locus (Tables III. and IV.) may be carried so
near the beginning of the liquid locus, the datum for latent
heat is determinable with some accuracy, in spite of its sur-
prisingly small (relative) value. Difficulties, however, present
themselves in the determination of the true melting-point, a
datum which can only be sharply defined when the tempera-
ture of the crucible is quite constant throughout. I have,
therefore, considered it preferable to state the conditions at
1200° and at 1100°, the former being nearer fusion and the
latter very near solidification. The latent heats for those
* Am. Journal, xlii. p. 140 (1891) ; cf. ibid., xxxviii. p. 408 (1889).
306 The Fusion Constants of Igneous Rock.
temperatures are 24 and 16 respectively. The coincidence of
results in both of the independent constructions (Tables III.
lY.) is in a measure accidental.
15. Tlie Relation of Melting-point to Pres.^ure. — The first and
second laws of thermodynamics leid to the equivalent of
James Thomson's fusion equation, which in the notation of
Clausius* is ^nd y, together with explicit functions of such
values, while the space- and time-variations of all these quan-
tities are absent from the equations, it is evident that the
conditions to be satisfied at the surface S are the same as if
Wi, pi5 U2^ p2y V were absolute constants. We conclude then,
that, with our assumptions^ a surface of discontinuity cannot
be propagated through a fluid with any velocity, uniform of
variable, except under that special law of pressure for which
progressive waves are of accurately permanent type.
3. What, then, becomes of waves of finite amplitude after
discontinuity has set in ? We may emphasize this difiiculty,
and at the same time obtain a cine to its solution, by con-
sidering the following case (fig. 2): — A is a piston fitting a
Z2
320
Dr. G. Burton on Plane and Spherical
Fio:. 2.
t
cvlindrical tube (or, if we
please, is a portion of an un-
limited rigid plane). All the
air to the right of A is initially
at rest and of uniform density,
and then A is impulsively set in
motion, and kept moving to the right with uniform velocity v.
Consider the speed with which the disturbance generated by
A advances into the still air to the right ; it is evident that
in all cases the front of the disturbance must advance faster
than A. Take, then, the case in which
V > a,
where a is the propagation-velocity of infinitesimal disturb-
ances. Two alternatives present themselves : —
(i.) If velocity and density are always either constant or
continuously/ variable in the direction of propagation, the rate
of propagation at any point will, in accordance with known
principles, be
^/
dp
and therefore at the front of the disturbance, where u = 0 and
p = the " undisturbed " density, the velocity of propagation
will be simply =a ; that is, less than the velocity with which
A is advancing. Ob^dously this will not do.
(ii.) If velocity and density are not always either constant
or continuously variable, that is, if one or more surfaces of
discontinuity are being propagated through the air, we are
met by the difficulty explained in the last section.
4. A simple mechanical analogy ^ill help to indicate the
actual motion. A number of equal spheres, of the same
material throughout, are capable of sliding without friction
Fig. 3.
along a straight bar (fig. 3) , and are connected together by a
number of very weak and exactly similar springs (not shown),
so that when there is equihbrium they are equally spaced
I
Sound- Waves of Finite Amplitude. 321
along the bar. If one of the spheres were moved backwards
and forwards through a small range, a disturbance would
travel through the whole system, but owing to the weakness
of the connecting springs it would travel very slowly. Sup-
pose, now, that the last sphere on the left hand is connected to
a movable piston by a spring half the length of the others,
but otherwise similar to them ; and let this piston be suddenly
moved to the right with a considerable velocity which is kept
constant, and Avhich we may call unity. The weak connecting
spring between the piston and the first sphere produces no
sensible effect until the two are almost in contact, when the
sphere rebounds with velocity 2. This first sphere then
strikes the second, imparting to it the velocity 2, and at the
same time coming to rest. The positions of the spheres after
successive equal intervals of time are represented in fig. 3,
where the number written on any sphere represents its velo-
city just after the impact which it is sufiering. No number
is written on those spheres which have not so far been affected
by the motion. From this it will be evident that when the
piston moves to the right with a constant velocity which is
very great compared with the propagation -velocity of infini-
tesimal vibrations of the system, the disturbance advances to
the right with twice the velocity of the piston, provided that
the diameters of the spheres are excluded from the reckoning.
Now suppose that the spheres are too small and too close
together to be individually distinguished; then, at any instant,
the system will appear to be divisible into two parts, in one
of which the velocity is unity, while in the other it is zero ;
and in the moving part the spheres will appear to be twice as
thickly condensed as in the still part. That the constant
velocity of the piston is very great compared with the propa-
gation-velocity of small vibrations is of course only a sup-
position introduced for the sake of simplicity. If, on the
other hand, these two velocities are comparable, two adjacent
spheres will always remain finitely separated from one another,
and the velocity of any individual sphere within the disturbed
stretch will never be as small as zero, or as great as twice the
velocity of the piston ; the mean velocity within the disturbed
stretch being equal to that of the piston. When the spheres
are very small and very close together, we shall still have
apparently an abrupt transition from finite velocity and greater
density to zero velocity and smaller density; and the energy,
which is apparently lost as the spheres pass from the latter
condition to the former, exists as energy of relative motion
and unequal relative displacement amongst the spheres in the
disturbed stretch.
5. Let us now compare the case just considered with the
322 Dr. C. Burton on Plane and Spherical
case of § 3 (fig. 2) : and first, concerning the nature of the
analogy, it should be noticed that the individual spheres are
not the analogues of the separate gaseous molecules, but that
when both spheres and molecules are very small and very
numerous, the apparently continuous properties of the system
of spheres correspond to similar properties of the gas. The
connecting springs represent the elasticity of the gas, iso-
thermal or adiabatic as the case may be, and the energy of
relative motion and unequal relative displacement amongst
the distm'bed spheres suggests that there is a production of
heat over and above that which would be due to the (iso-
thermal or adiabatic) change of density : that is, a dissipative
production of heat. The motion considered in the last section
properly corresponds to the case where there is no conduction
of heat, so that the connecting springs are the representatives
of adiabatic elasticity, and the additional heat generated
remains wholly within the more condensed part of the air.
If we make the somewhat ^dolent assumption that the tempe-
rature of the air remains constant throughout, the additional
heat generated will be conducted away isothermally, and the
equivalent energy will be, for our purposes, entirely lost.
To represent this case by means of our spheres we should
have to regard the connecting springs as representing iso-
thermal elasticity, while the energy of relative motion and
unequal relative displacement among the disturbed spheres,
as fast as it is produced, is to be consumed in doing work
against suitable internal forces.
6. The mechanical system of spheres and springs, having
suggested a solution, has served its purpose, and it now
remains for us more closely to consider the aerial problem in
the light of this suggestion. We may take, first, the case
where the temperature is supposed to be invariable ; for
although such a supposition is necessarily far removed from
the truth, it leads to very simple results,"^ which indicate well
enough the general character of the motion. Let the piston
A (fig. 4) be mo^-ing to the
right with constant velocity jrio-. 4.
V (which may be either less
or greater than a, the velo-
city of feeble sounds in air).
Assume all the air between
A and a parallel plane sur-
face B to have the velocity v
and density p^^ while all the air to the right of B is at rest
and has the density /Oq. Let the plane B move to the right
with velocity V. Then the invariability of mass between A
Sound- Waves of Finite Amplitude, 323
and a plane C fixed in the still air gives
p^{Y-v)-p,v=0', (4)
while from the principle of momentum,
Piv{Y-v)=Pi-Pq; (5)
the pressure p being a function of p only, since the tempera-
ture is supposed to be constant throughout. If we assume
for this case the truth of Boyle's law, so that p = a^p always,
(5) becomes
p^(^a?^Yv + v^)=p,a\ .... (6)
which together with (4) is sufficient to determine V and pi
when V and /Oq ^i'^ given. Taking all these quantities to
remain constant throughout the motion, we see that at each
instant the following conditions are satisfied : —
(i) Every necessary condition between A and B, since
density and velocity are there constant with respect
to space and time ;
(ii) Every necessary condition to the right of B, since the
air there is at rest and in a constant uniform state ;
(iii) Equality between the velocity of A and that of the
air in contact with it ;
(iv) At B, the conservation of mass and momentum, which
are necessary conditions, and which, together with
our supposition that the temperature is somehow
maintained uniform, are sufficient to determine what
takes place at B ^.
Moreover, if at a time t (reckoned from the instant when
A was impulsively started into motion) we take the distance
of B from A to be (V — v)t^ so that initially B coincides with
A, the initial conditions are satisfied.
Thus the assumed motion satisfies all the necessary con-
ditions ; it is therefore the actual motion.
7. Let us now examine what occurs when no heat is
allowed to pass by conduction or radiation ; a state of things
much more nearly realized in practice. Suppose the motion
of A and the condition of the undisturbed air to be the same
as in the last section, while the (constant) velocity of B is
now called Y^, and the density and pressure of the air between
A and B (called p^, p' respectively) are also taken to be uni-
form and constant. At each instant, in place of (4) and (5),
we shall now have
p'{^T'-v)-p,v=0, (7)
p'v{y'-v)=p'-p, (8)
* Energy appears to be lost, because dissipatively produced beat is
conducted away isothermally.
I
324 Dr. C. Burton on Plane and Spherical
Since we assume that there is no transference of heat by
conduction or radiation, the raie at which the total energy of
the system increases must be equal to the rate at which work
is being done upon it by the piston A. Let Oq be the abso-
lute temperature to the right of B, that between A and B
being 6', and let us further assume for simplicity that
^ = a const. ;
while 7, the ratio of the two specific heats, is also supposed
constant. It can then be shown without difficulty that the
total energy per unit mass between A and B exceeds that to
the right of B by
{ry-l)po ^0 2 '
and multiplying this by pqV^, the mass of air which crosses
one unit of the surface B in each unit of time, we obtain the
rate (referred to unit area) at which the system is gaining
energy. Again, the rate at which unit area of the piston does
work on the system
p'0'
and equating this to the rate of gain of energy, we obtain
We may also write equation (8) in the form
p<^(Y<-v)=^{p<0-pA); ■ ■ ■ (10)
and (7), (9), and (10) will then serve to determine V'^ /o', 0'
when V, po, 6q are given. Since we have taken all these
quantities to remain constant throughout the motion, we see,
as before, that at each instant all the necessary conditions are
satisfied ; the principles of mass and momentum, together
with our supposition that there is no exchange of heat, being
sufficient to determine what takes place at B. Again, if at a
time t from the commencement of the motion we take the
distance of B from A to be (V' — v)ty so that initially B coin-
cides with A, the initial conditions are satisfied. The assumed
motion thus satisfies all the necessary conditions, and is there-
fore the actual motion.
8. If we compare the results of the last two sections with
Sound- Waves of Finite Amplitude. 325
those given by Eiemann"^, we shall find complete accordance
so far as § 6 is concerned, though with § 7 the case is dif-
ferent ; and this may he easily explained. We cannot in
general investigate the motion of a (frictionless) compres-
sible fluid by means of the equations of continuity and
momentum, without further making some supposition as to
the exchange or non-exchange of heat^ and so we usually
assume either that the temperature remains constant, or that
there is no exchange of heat : in either case (provided the
motion is continuous), the pressure is a function of the
density only. At a surface of discontinuity there is not only
the ordinary heating effect due to compression, but also, as
we have seen, a dissipative generation of heat, and so, when
applying the equations of continuity and momentum at such
a surface, we must know what becomes of this additional
heat. Now in all cases Kiemann makes the assumption that
the pressure is a function of the density only, and this is
necessarily equivalent to an assumption concerning the trans-
ference of heat. Throughout most of his treatment of waves
of discontinuity Eiemann assumes that temperature is
constant and that Boyle's law holds good j accordingly our
§ 6 is entirely in harmony with his conclusions, in fact (4)
and (5) are only particular forms of equations given by
Riemann. Of course the hypothesis that a portion of gas
can be instantaneously compressed to a finite extent without
any appreciable change of temperature, is not in accord-
ance with experience, but provided we accept the assump-
tion that the temperature rem^ains constant throughout, all that
Riemann says concerning the propagation of leaves of discon-
tinuity under BoyWs law will hold good.
The assumption made in § 7, that there is no appreciable
transference of heat, is probably much nearer the truth; but
this is not in accordance with any assumption made by
Eiemann. When pressure is assumed to be a function of
density only, and to vary with it according to the adiabatic
law, it is virtually assumed that at the discontinuity just so
much heat remains in the gas as would be due to sloio adiabatic
compression, while the further amount of heat ivhich is dissipa-
tively produced is completely and instantaneously removed by
conduction. But though Eiemann's results may thus be
justified by impossible assumptions concerning the diffusion
of heat, we may more reasonably, following Lord Eayleighj
regard them as involving a destruction of energy. The real
source of error lies in Eiemanu^s fundamental hypothesis.
At the outset he supposes the expansion and contraction of
* Loc. eit.
326 Dr. C. Burton on Plane and Spherical
the air to be either purely isothermal or purely adiabatic, and
thenceforward he treats the air as a frictionless and mathe-
matically continuous fluid, in which pressure and density are
connected by an invariable law. But in general the existence
of such a fluid is contrary to the conservation of energy ;
for as soon as discontinuity arises, energy will be destroyed.
9. It may not be out of place to conclude this portion of
the subject by a short reference to a paper by Dr. 0. Tum-
lirz ^ . This author starts, as Eiemann did, with the assump-
tion that the pressure is a function of the density only, the
law of pressure being further assumed to be the adiabatic
law ; and in order to avoid Eiemann's error, he explicitly
uses the principle of energy applicable to continuous motion,
in place of the principle of momentum. But the foregoing
discussion will have made it clear, I think, that the solution
of the difficulty is not to be sought for in this direction. In
addition to the assumptions common to his own work and to
that of Tumlirz, Hiemann uses only the principle of mass
and the principle of momentum ; and since by their aid alone
he arrives at a completely determinate motion, it follows
that any other motion consistent with the same arbitrary
assumptions, and with the condition of mass, must violate
the condition of momentum. We have seen, in fact, that
there is dissipation of energy at a surface of discontinuity, so
that the condition of energy applicable to continuous motion
ceases to hold good. We are acquainted, too, mth other
instances where loss of continuity involves dissipation of
energy ; for example, there is the case of one hard body
rolling over another.
As the result of his investigation. Dr. Tumlirz concludes
that as soon as a discontinuity is formed it immediately dis-
appears again, this eff'ect being accompanied by a lengthening
of the wave and a more rapid advance of the disturbance.
In this way, therefore, he seeks to explain the increased
velocity of very intense sounds, such as the sounds of
electric sparks investigated by Mach f . But it has already been
pointed out [§ 3 (i.)],that lulien density and velocity are every-
where co7itinuoiis functions of the coordinates, the frojit of a dis-
turbance advancing into still air must travel forward with the
velocity of infinitely feeble soimds. A greater velocity can
only ensue when the motion has become discontinuous.
* " Ueber die Fortpflanzimg ebenej' Luftwellen encUiclier Sc]i\nng-
ungsweite/' Sitzungsh. der Wien. Akad. xcv. pp. 367-387 (1887).
t Sitzungsh. der Wien. Akad. Ixxv., Ixxvii., Ixxyiii. Cf. also W.
W. Jacques I On Soimds of Cannoni, Amer. Joiirn. Sci. 3rd ser. xyii. f
p. 116 (1879). !
Sound-Waves of Finite Amplitude. 327
Paet II. — Spherical Waves.
10. When plane waves of finite amplitude are propagated
through a Motionless compressible fluid, discontinuity must
always occur sooner or later^ and a moment's consideration
will show that there are at least some cases when the motion
in spherical waves becomes discontinuous ; the question arises
whether in any case it is possible (in the absence of viscosity)
for divergent spherical waves to travel outward indefinitely
without arriving at a discontinuous state. This question was
suggested to me by Mr. Bryan, who at the same time kindly
handed me notes of his manner of attacking the problem.
His method was to write down the exact kinematical equation
for spherical sound-waves, and then to obtain successive
approximations to the integral of this equation. If it appears
that after any number of approximations the integral would
remain convergent for large values of the radius, we may con-
clude that our equation holds good throughout, and hence that
no discontinuity arises. If, on the other hand^ the second or
any higher approximation becomes divergent for large values
of the radius, it is probable that the motion becomes some-
where discontinuous. This method I have not followed out ;
but by another method which is, I hope, sufiiciently con-
clusive, I shall now endeavour to show that discontinuity
must always arise.
The case in which the motion loses its continuity compara-
tively early requires no further consideration here ; we have
only to concern ourselves with the case in which the initial
disturbance has spread out into a spherical shell of very small
disturbance whose mean radius is very great compared with
the difierence between its extreme radii. The equations
applicable to the disturbance are then, very approximately,
u^J-^, (11)
r
Cv -I
u or a-^cc - for a given part of the wave, . (12)
where p is the mean density, p + 8p the actual density at a
point where the velocity is w, and a is the velocity of infinitely
feeble sounds in air of density p ; r is as usual the distance of
a point from the centre of symmetry. Let us consider two
neighbouring points M and K, on the same radius, each heing
fixed in a definite part of the ivave, the point M being behind
N (z. e. nearer to the origin), and the air-velocity at M ex-
ceeding that at N by Au. Then, as the wave advances, each
^/\
328 Dr. C. Burton on Plane and Spherical
part of it will be instantaneously moving forward with (very
approximately) the velocity
dp
determined by the corresponding values of p and u ; so that
M will be gaining on N at the rate
approximately. We may admit then that the rate at which
M gains on N is
never < BAu,
where B is a constant suitably chosen.
Again, if AqU is the difference between the air- velocities at
M and N at the time ^ = 0, and Tq is the corresponding co-
ordinate of M, we may admit that
A • A?^A .
All IS never < ^ AqU,
Tq + at
where A is a constant not very different from unity. Thus
M gains on N at a rate which is
never < AB ^^ AaU :
rQ4-at
and between the times t = 0 and t=ti the distance gained by
M relatively to N will be
at least ABAowf'^^,
Jo '^'o + at
i.e. atleast ABAoW^log^^^i^. . . . (U)
If B is finite and positive this expression increases indefinitely
with the time, so long as the laws of continuous motion hold
good. If AqT was the distance between M and N at time
^ = 0, the time required for M to overtake^ N will be 7iot
greater than the value of t^ given by
-Ao7^=ABAoi.^log^^^±^;
or, when M and N are taken indefinitely close together at
starting, by
i.e., we have t^ 'Z-ie'^o^ V^(-^')o} __i\ ^ ^ ^^^
* Cf. Lord Rayleigli, ^Theory of Sound/ vol. ii, p. 36.
Sound-Waves of Finite Amplitude, 329
which gives us a finite upper limit to the time required for
discontinuity to set in, provided B is finite. As our assump-
tions only remain approximately true so long as the motion is
continuous, (15) will only give an approximation to the time
when discontinuity first commences, and accordingly the
relation must be taken to refer to that part of the wave for
which its right-hand side is a minimum. If B is negative
(which is not the case for any known substance), the appro-
priate part of the disturbance will be such that 'du/^r is
positive.
To determine approximately the value of B, we may refer
to (13) and the inequality immediately following. If we
assume Boyle's law of pressure, so that V {dp/dp) = cojist.j we
have evidently
B = 1 very nearly.
If we assume that the changes of density take place adia-
batically, so thatp oc/)^ and y is nearly constant, the approxi-
mate value of B becomes
'-Wi-^W'
by means of (11) ;
dp V dp V dp
___7 + l
If, then, viscosity be neglected, we must conclude that under
any practically possible law of pressure the motion in spherical
sound-waves always becomes discontinuous, and a fortiori the
same will be true of cylindrical waves. But inasmuch as
our result for spherical weaves depends on the existence of an
infinite logarithm in (14) when ^i is increased without limit,
we may conclude that for waves diverging in tour dimensions
(or, more generally, in any number of dimensions finitely
greater than three) there would be some cases where the
motion remained always continuous.
11. The general question of spherical sound-waves of finite
amplitude is by no means an easy one. In the case of plane
waves we can write down at once from Riemann's equations
the condition that the disturbance may be propagated wholly
in the positive or wholly in the negative dii.ection. The
respective conditions are * : —
where po is the density of that part of the fluid whose velocity
* Cf. also Lord Rayleigh, ^Theory of Sound/ vol. ii. p. 35 (3).
330 Dr. 0. Burton on Plane and Spherical
is reckoned as zero. No such simple criterion can be given
for the existence of a purely convergent or purely divergent
spherical disturbance : a fact which may be readily seen from
the equations for waves of infinitesimal amphtude. If (/> is
the potential of a purely divergent system of waves, we have
r^=f[at-T\ (16)
where / is a function whose form is unrestricted. Let p be
the ordinary density of the air, and p-^hp the actual density
at a point where the coordinate is r and velocity u.^ We
have, then, on differentiating (16) the well-known relations
__'b*