May 22, 1902] 
i NA TORE 
89 
have in these experiments. All that is possible in the present 
instance is to adopt a linear formula. The usual formula is 
vr = Up (1 + aT), where the value v ato C. becomes vy at 
T° C. when a is the coefficient of expansion. If we use densities 
(d@) instead of volumes (z) this formula becomes 
ora = Gy = dr, 
a ——D 
Tay 
Another formula, when T and T’ are the temperatures dealt 
With, is 
d, =dz(1 + aT), a = 0°000538. 
ay = dy\t + a(T’ — T)}, ora = Tas a = 0'000595. 
Again 
, , ty — ay ; 
ay = dy {1 — «(T’ —- T)}, ora = a = Tie} a = 07000558. 
Also we may choose a mean formula 
a= CLES: ; a= 0000576. 
(a gz Gees + dy 
2 
The differences in the results of applying these formule are 
shown in the numerical values attached to each, which are 
calculated from the first experiment on solid carbonic acid, 
coupled with the specific gravity 1°53 of the solid at — 78° C. 
Perhaps as a matter of general convenience, the first of these 
formule is the best; however, the second was chosen to 
conform with the old work of Playfair and Joule, and it is the 
results of this formula which are mentioned below. 
The temperature range is taken from about — 186° C. to 
17° C., unless otherwise stated. 
Zce.—In determining the density at the temperature of liquid 
air of pieces of clear ice cut from large blocks, both the silver 
and copper balls already referred to were used as indicated. 
The true weight zz vacuo of the silver ball was 132°2855 
grammes, and that of the copper ball was 38-0802 grammes. 
The mean of the three densities obtained at — 188°°7 C. is 
0°92999. 
Recently Vincent (Roy. Soc. Proc., 1901) has redetermined 
the density of artificial ice at the freezing point, and also its co- 
efficient of expansion. He finds the density to be 0-916, or 
from histabulated results 0°91599. Playfair and Joule find the 
mean of the densities given by eight observers previous to them 
to be o'919, and they themselves get 09184; Bunsen found it 
to be 0'9167. If we take this most recent determination, 
namely, 0°91599 at 0, and 0°92999 at — 188°'7, and use the 
formula 
d)=ar(t+aT) 
we get a=o0'00008099. 
Vincent refers to ‘‘ only one” estimate for natural ice, namely, 
0'0001125, adding that ‘‘ the mean of three available results for 
artificial ice is 0°000160” ; finally, he gives the mean of four 
determinations of his own, namely, 0000152. Apparently, 
then, we may take 0020155! as the mean coefficient of expan- 
sion of ice between 0° and (say) —20° C. Thus the mean co- 
efficient of expansion between 0° and — 188°C. is about half of that 
between 0’ and —20°C. The mean coefficient of expansion of 
water in passing from 4° to — 10° is ~0’000362, and from 4” to 
40 C. it is 0’0002155. Hence the mean coefficient of expan- 
sion of ice between o and — 188° C. is about one-fourth of that 
of water between o° and —10° C., and half of that between 4° 
and 100° C. 
If we had the densities of ice at still lower temperatures, the 
values of the coefficient of expansion thence determined would, 
there is every reason to believe, be less than what we have found. 
We shall therefore not be overstraining the argument if we use 
the value just found to determine an upper limit to the density 
of ice at the absolute zero. The result is 0°9368, corresponding 
to a specific volume 1°0675. Now the lowest density of water, 
namely, at the boiling point, is 0°9586 (corresponding to specific 
volume 1°0432), so that ice can never be cooled low enough to 
reduce its volume to that of the liquid taken at any temperature 
under one atmosphere pressure. In other words, ice molecules 
can never be so closely packed by thermal contraction as the 
water molecules are in the liquid condition, or the volume of 
ice at the absolute zero is not the minimum volume of the water 
molecules. It has been observed by Prof. Poynting (‘‘ Change 
of State, Solid, Liquid,” PAz/. A7ag. 1881) that if we supposed 
water could be cooled without freezing, then taking Brunner’s 
NO. 1699. VOL. 66] 
coefficient for ice, and Hallstrom’s formula for the volume of 
water at temperatures below 4° C., it follows that ice and water 
would have the same specific volume at some temperature be- 
tween —120 and — 130°; applying the ordinary thermodynamic 
relation, then no change of state between ice and water could be 
brought about below this temperature. On the other hand, 
Clausius (‘‘ Mechanical Theory of Heat,” p. 172, 1879) has 
shown that the latent heat of fusion of ice must be lowered with 
the temperature of fusion some 0°603 of a unit per degree. If 
such a decrement is assumed to be constant, then about — 130° 
the latent heat of fluidity would vanish.' Baynes discusses the 
same subject (‘‘Lessons on Thermodynamics,” p. 169, 1878) 
and arrives at the conclusion that at a temperature of —122°°SC. 
and under a pressure of 16,632 atmospheres there is no distinc- 
tion between the solid and liquid forms of water. At tempera- 
tures below this limit, no amount of pressure would transform ice 
into water. We are thus relieved from a difficulty that would 
follow but for this demonstration of Clausius, namely, that the 
application of enormous pressures to ice, even at temperatures 
below that of liquid hydrogen, might cause the transformation 
of ice into water. 
Carbonic Acid.—Two experiments were made with this sub- 
stance, the masses in each case being about 20 grammes. These 
were compressed cylinders; the former was compressed dry, 
while the latter was slightly moistened with ether. The density 
at —188°:8 C. was found to be 1°6308 and 1°6226. 
The density of solid carbonic acid at its boiling point was 
formerly given as 1°5 (see Proc. Roy. Inst., 1878, ‘‘The 
Liquefaction of Gases”), but the mean of my results at the time 
came to 1°53. Recently the same value has been found by 
Behn. Taking this value and 1°6267, the mean of the above 
results at — 188°°8 C,, and using the formula 
dy = ady\t + a(T’ —T)} 
we get a = 0°0005704. 
This is a very large coefficient of expansion, being greater 
than that of any substance recorded in the accompanying table, 
and comparable with that of sulphur between 80° and 100°, 
which, according to Kopp, is 0'00062. The coefficient of 
liquid carbonic acid at its melting point taken from the recent 
observations of Behn (Chen. Soc. Journ., 1901) is 0'002989, so 
that the rate of expansion of the liquid at its smallest value is very 
nearly five times that of the solid. 
Solid Mercury.—One experiment 
mercury, and the result is given below. 
Mallet determined with great accuracy the density of solid 
mercury at — 38°°85, his result being 14°193 ; coupling this with 
the density found for the liquid-air temperature, we find that the 
value of the coefficient of expansion between the melting point 
and — 189° C. is o'0000887. For fluid mercury above o° C. 
the mean value is about o*000182, so that in the solid state 
this coefficient is about half of that in the fluid state. 
The coefficients of expansion (a) obtained were as follows :— 
was made with solid 
a 
0°00 
Sulphate of aluminium (18) ............... o8Ir 
Biborate of soda (10). ........ 1000 
Chloride of calcium (6)..... Sen Lee IIQI 
Chloride of magnesium (6).. ............... 1072 
Potash alum (24) 0813 
Chrome alum (24), large crystal. 0365 
0478 
1563 
Phosphate of soda (12)...........-000:0se0 0 0787 
Hyposulphate of soda (5) Satkeese 0969 
Ferrocyanide of potassium ( ; 1195 
Ferricyanide of potassium....... bbandopaanh 2244 
Nitro-prusside of sodium (4) .... 1138 
Chloride of ammonium, sample i 1820 
45 7 sample ii 1893 : 
Oxalicracid(2)iasc:.<<eccessseeseseacse 2643 
Oxalate of methyl... seen 3482 
PATA Men ccs seers ccaelestis + tes ceeeins ainemceat 3567 
1 In my paper ‘‘On the Lowering of the Freezing-point of Water by 
Pressure” (oy. Soc. Proc., 1880), it was proved that up te 700 atmospheres 
the rate of fall was constant and equal to the theoretical value within the 
range of pressure if the difference between the specific volumes of ice and 
water remains constant ; thence the latent heat of fusion must diminish just 
as Clausius had predicted. 
2 The figures in brackets refer to the number of molecules of water of 
crystallisation. 
