tude inversely proportional to the square of the volume containing the gas.1

Hence van der Waals's corrected formula for Boyle's law becomes

nm G2 = (p + av 2) (v — b),

wherein a and b are constants independent of the pressure and volume; and if, as before, we express the molecular speed by the temperature,

(p + av−2) (v - b) = R(1 + ad),

where R is a constant, and a, 9 denote, as before, the coefficient of expansion of gases and the temperature. Only by comparison with experiment can it be determined whether a and b depend on the temperature; van der Waals finds that it is tolerably sufficient to assume them both to be independent of the temperature, as well as R and a.

45. Comparison with Regnault's Observations

The theoretical formula obtained agrees nearly exactly in form with the formula of interpolation by which Regnault represented the results of his observations; for this has the form 2

wherein m denotes the ratio of two values of the volume reduced to the same temperature, say 0° C., or

m = V│v,

r that of the corresponding values of the pressure,3 or

r = p/P,

while A and B are constant coefficients. If, as in Regnault's memoir, that volume V is taken as unity

Ritter has proceeded by this method in determining the cohesion of gases in a memoir (Mém. de la Soc. de Phys. de Genève, xi. 1846, p. 99), with which I have become acquainted only through the notice of it by von Morozowicz in the Fortschritte der Physik, ii. 1846, p. 89.

2 Mém. de l'Acad. de Paris, xxi. p. 421.

3 See Table of Errors at the end of vol. xxvi. of the Mémoires.

which, at the pressure P=1 metre of mercury, is filled by the mass of gas used in the experiment, Regnault's formula can be more simply written

In this form it agrees with the theoretically deduced formula of van der Waals for the temperature 0°, viz.

if it is allowable to replace the pressure in the correction term bp by the reciprocal of the volume in accordance with the approximately correct Boyle's law

for then van der Waals's formula runs

pv + (a b)v-1 — abv -2 =

R,

so that it is equivalent to Regnault's if the constants calculated for unit of mass are related to each other according to these equations:

Under these circumstances a detailed comparison of the formula with the numbers obtained directly is not necessary. But it should be mentioned that from the values of A and B found by Regnault we shall afterwards be able to calculate the numerical values of a and b, which represent the magnitude of the cohesion and the extension in space of the molecules.2

46. Pressure- and Volume-coefficients

Van der Waals's formula is also suitable for explaining the variation in the values of the expansion-coefficients, Van der Waals, Continuïteit; Roth, Wied. Ann. xi. 1880, p. 1. 2 See Chap. X.

and especially the circumstance that the coefficient which determines the increase of pressure with temperature is not identical with that on which the increment of volume depends.

The value of the former coefficient, which for distinction. from the other-the expansion-coefficient proper-may be termed the pressure-coefficient, is the more easily obtained. The increase which the pressure of a gas undergoes when the temperature is raised from 0° to 9 while the volume remains unchanged is found by comparison of the formula

(p + av ̄2) (v — b) = R(1 + ad)

for the latter temperature with that referring to 0°,

when we give to v the same value in both. By subtraction we get

or

(pp)(v - b) = Rad

P - Po= (Po + ar-2)ad,

whence we obtain for the pressure-coefficient the corrected formula

This teaches that gases in which cohesion really exists have a greater pressure-coefficient a, than the ideal gases for which its value is a. Since this behaviour agrees with experience, the formula can be used to deduce the value of the constant a, which measures the strength of the cohesion, from the observations.

The same agreement between theory and observation is also shown when we calculate from the theoretical formulæ the value of the expansion-coefficient proper, i.e. that coefficient which determines the increment of volume. Since p is now to be taken as constant and v as variable, the formulæ

give for the expansion-coefficient defined by the formula

In order to gain an insight into this complicated formula let us apply it to extreme cases, and first of all to that, approximately realised with hydrogen, in which a = 0 or the cohesion is vanishingly small. With this assumption we have

so that

a = (1-bv1)a,

a1<a = a.

Now, on the contrary, take the cohesion to be so great that in comparison with it we may neglect the correction that arises from the size of the molecules, or put b = 0; then

Both of these conclusions from theory are in consonance with experiment; for according to Regnault's observations already mentioned in § 40 we have for hydrogen

a = 0·003661, a, = 0.003667;

but for all other gases the relation between the values of the coefficients required for the second case is fulfilled.

Van der Waals's theory thus agrees in all points with experiment in so far as it rightly expresses the general laws. This agreement speaks for the fact that to a certain degree the assumptions on which the theory rests correspond to reality. If we should consider van der Waals's theory also as probably not yet perfect, we are yet justified in the view that in it the first step is taken along the path by which we shall arrive at a completely satisfactory kinetic theory of

real gases and not of ideal gases only. The point in which it most especially needs improvement is the manner in which the law of cohesion is introduced.

47. Completions of van der Waals's Formula

Properly recognising this imperfection, Clausius' has attempted to improve van der Waals's formula by substituting another expression for the value of the cohesion. His formula, which has the form

-2

{ p + a(v + B) −2} (v — b) = R(1 + ad),

differs from that of van der Waals essentially in this: that the cohesion-pressure is put inversely proportional, not simply to the square of the volume v, but to the square of the volume v increased by a constant B. By this means Clausius obtains a better agreement of the formula with observations that have been made under high pressures, and therefore with small volumes.

Ramsay and Young2 think it more correct to substitute the more general expression of an nth power instead of that of the square of the volume v.

A second difference between the formulæ of Clausius and van der Waals consists in this: that the magnitude a is with Clausius not a constant, but a function of the temperature. As he first employed his formula only in the case of carbonic acid, he could be content with the assumption of the simple formula

a =

с

in which c is a constant and denotes the absolute temperature

Θ • = I + a1.

Later on, when trying to apply the formula to other gases and vapours, he assumed a more general expression,

a = (40" B) R(1 + a9),

Wied. Ann. ix. 1880, p. 337; xiv. 1881, pp. 279, 692; Mechanische Wärmetheorie, 1889-91, iii. pp. 184, 215, 227.

2 Proc. Roy. Soc. xlii. 1887, p. 5.