mean of the results 0 and 02, since it has been shown that 0(01-02). In fact, the mean of the two readings given-0.333 cm.-agrees with the mean value obtained by tapping as used by the writers named, viz. 0·335 cm.

manner.

Note on Wetting.

Notwithstanding that "wetting" has become such an important factor in connexion with the Flotation of Minerals and also in lubrication, the term is often used in a very loose The writer has not been able to obtain a definite statement as to the meaning of "wetting." Many writers tacitly assume that a liquid that "wets" a solid makes with it a zero contact angle. Cp. Edser, General Physics,' p. 312, and Richards & Carver (Journ. Amer. Chem. Soc. xliii. 1921, p. 827 et seq.). Others assume that a liquid "wets" a solid when <90°, and does not "wet" it when 6 >90°.

Clerk Maxwell, in the article on Capillarity in the Encycl. Brittanica, states the question thus:-"If a small quantity of a fluid stand in a drop on the surface of a solid without wetting it, the angle of contact is 180°; if it spreads over the surface and wet the solid, the angle of contact is zero."

With regard to this statement, there is no known combination of liquid and solid where 180° (unless the solid surface be dusty): thus, in Maxwell's sense, there is no case where the fluid rests on the solid without wetting it. On the other hand, most combinations of liquid and solid give a value of 0 intermediate between 0° and 180°. The cases might therefore be distinguished as "non-wetting " (0=180°), “partial wetting" (finite), and " complete wetting" (0=0). Still this does not give any real physical significance to the term "wetting."

From the fact that 0, is >02, it follows that the surface energy of the air/solid interface has been increased by having been in contact with the liquid. From this it is concluded that the solid has absorbed or imbibed some of the liquid. It then follows that if the solid be moved relative to the liquid in a direction tangential to the interface, for example by a clockwise rotation of the waxed cylinder in the experiments previously described, that portion leaving the liquid will tend to drag the liquid with it up to the limit of the force of cohesion of the liquid or of the adhesion between liquid and solid, whichever is the greater. Thus the contact angle on the leaving

side will be decreased towards 0, and that on the entering side increased towards 01. When separation does take place, the contact angle in the former case will be the minimum value 02, and in the latter case the maximum value 1, for that particular speed.

Thus the results obtained strongly support the view that this variation in the contact angle is due to absorption or imbibition of the liquid by the solid. Such a phenomenon is true "wetting," and, as is evident from these experiments, is possible for obtuse as well as acute angles of contact-degree of wetting depending on the range (02-01), that is to say on the degree to which the surface energy of the solid is affected on coming into contact with the liquid.

These experiments are to be repeated with other combinations of solids and liquids.

The author desires to thank Professor L. R. Wilberforce, M.A., for his kindly criticism and advice during this investigation.

George Holt Physics Laboratory,

University of Liverpool,

XXVI. The Second Virial Coefficient of Gases. By ANGUS F. CORE, Chemical Department, The University, Manchester*.

THE

HE equation of state of a perfect gas is pr=RT, where p is the pressure, the volume occupied by the unit. of quantity, T the absolute temperature, and R the gas constant for the unit of quantity. This is a limiting relation which we may suppose that the behaviour of all real nondissociating gases approaches as the volume becomes very great.

The equation of state of any gas may be written in the form

where A, B, C, etc. are called the first, second, third, etc. virial coefficients. These coefficients are functions of T but

not of v. The value of A for a non-dissociating gas, with which only are we concerned, is necessarily RT; the same for the gram molecule of all gases.

* Communicated by Prof. S. Chapman.

It is the object of the present discussion to deduce the value of the second virial coefficient, in the first place of a gas with molecules of a spherically symmetrical type, and then of a gas with molecules of a general type. In the very special case of a gas consisting of hard spherical molecules exerting no attraction on each other, the third coefficient, C, is readily found. The results agree with those obtained by Keesom and others, but the method is comparatively simple.

The kinetic significance of what is known in thermodynamics as "activity" is shown in the paper.

In the last section an expression has been obtained for the second virial coefficient of a binary mixture of gases.

Preliminary.

Suppose that in a vessel of unit volume, kept at a uniform temperature T, there are v molecules of a gas. Then if the molecules are non-attracting mass points, and if no external forces act on them, except those exerted by the walls of the containing vessel, the molecular concentration is everywhere v. By this statement it is not meant that in any volume element, de, there are v dv molecules, since, evidently, the chances are against there being any of the molecules in the particular element at a given instant. What is meant is that the probability of there being one molecule in the element de at any instant is v do. That is to say, if we could take a very large number, A, of instantaneous views of the element de we should observe a point molecule in it Av dv times; or if there were A exactly similar unit vessels each containing v molecules, then at any instant there would be Avde of them with a molecule in a particular element dv.

Suppose, now, that external forces act on the molecules, of a kind that can be derived from a permanent field of potential. Then the molecules will no longer be distributed uniformly over the unit volume. If the work required to bring a single molecule from a region taken to be of zero potential to a given point is x, then x is the potential at that point, and according to the classical theory of statistical mechanics the molecular concentration there is equal to

where T is the temperature and the gas constant per molecule. Putting x=0 it will be seen that vo is the molecular

* W. H. Keesom, Proc. Sect. of Sciences, Amsterdam, vol. xv. (1) pp. 240, 256.

Phil. Mag. S. 6. Vol. 46. No. 272. Aug. 1923.

S

Xx

concentration in regions of zero potential. In this expression x is the potential due to an external field which does not vary with the time. This result can be extended in the following way.

Let the molecules be such that they exert forces on each other depending only on the distances separating their centres. Then round each centre there is a field of potential with respect to another molecule The instantaneous potential, X, at any point, due to another molecule or other molecules, is defined to be the work required to bring the centre of a single molecule from a region of zero potential up to that point while the other molecule or molecules are kept fixed in position. Consider, then, the v molecules at any instant. They are in definite positions and consequently there is a definite instantaneous potential with respect to another single molecule. Let another molecule be added to the vessel, so that there are now v+1 molecules. Then if we regard only those moments in the history of the first v molecules in which they have these same positions, it is shown by statistical mechanics that the probability that the centre of the single molecule is in an element of volume de, where the instantaneous potential is x, is proportional to

e ̄x/kT de,

just as when the field is a permanent external one.

The theorem may be further extended so as to include molecules which exert forces on each other depending not only on the distance separating them but also on their orientations in any way. Just as before, we choose instants when the molecules are always in the same positions, not now as regards only the points taken as their centres, but also as regards their exact orientations. We cannot now speak of the potential of the remaining molecule at any point, for this does not depend only on the position of the molecule's centre, but also on its orientation. Nevertheless, it is shown that if we further restrict the instants of viewing the system to instants when the single molecule is orientated in a particular way, then the probability of its centre being in the element of volume de is proportional to

e-x/KT dv,

X being the now perfectly definite quantity of work required to bring the centre of the single molecule orientated in this definite way into the element of volume dr.

Spherically Symmetrical Molecules.

Imagine a single molecule in a vessel of unit volume, this volume being very large compared with the molecular field. Then if the potential throughout the vessel is uniform, the probability that at any instant the centre of the molecule. lies in an element of volume dv is equal to dv. The molecular concentration is everywhere equal to 1.

Now suppose that there are v other identical molecules in the vessel. Then if we regard the first molecule only at those instants when the other v molecules are always in the same positions, its concentration is no longer uniform but is equal to

Poe-x/T,

where χ is the instantaneous potential due to the other molecules, and Po is the concentration in regions of zero potential, which will be taken to be those regions outside the field of any molecule. It is evident that, since this concentration refers to one molecule in a vessel of unit volume, if it be integrated with respect to the volume over the whole unit volume the result must be 1. Therefore for finding Po we have the equation

Pose-x/KT dv = 1,

the integration being taken over the whole interior of the vessel. Now, outside the field of a molecule the potential x is zero, and therefore

since the integration is through a unit of volume. Thus

This, therefore, gives the concentration of the single molecule at all points outside the influence of any other molecule for a particular arrangement of these other molecules. Now in any real case v is very large and the statistical arrangement of the molecules practically never varies appreciably from