and Electronic Potential Energy. 2G1 



energy of two molecules of 2 would not enter into the ex- 

 pression for the potential energy of any molecule or of an 

 average molecule. 



Then again the cubical arrangement seems an unsuitable 

 one to assume for a mixture of unlike molecules unless the 

 volume occupied by a molecule of each is the same. Never- 

 theless, by the application of kinetic principles we get over 

 these two difficulties in the following way. If p is the density 

 of a mixture containing n x molecules of 1 per unit mass, it 

 will contain n^p per unit volume. It contains n 2 of 2 per 

 unit mass and n 2 p per unit volume. Let n ol denote the 

 number of molecules of 1 per unit mass in the pure liquid, 

 ?i 02 being the number for 2. Then according to the statistical 

 principles used in the kinetic theory we state that the time 

 for which a molecule of 1 in the mixture is one of the imme- 

 diate neighbours of a molecule of 1 is the fraction u^u^pi of 

 the corresponding time for the pure liquid 1. Now from (3) 

 we know that the average potential energy of a molecule of 

 1 having molecules of 1 for its neighbours all the time is 

 ^e^s^Ri 3 . Hence the potential energy of a molecule of 1 

 and the other molecules of 1 in the mixture is ^n x peiS 2 \Kin^p: 

 so the mutual potential energy of the n ± molecules is 4m 1 2 ^ 1 2 5 1 2 ( u. 

 In this way by making our cubical arrangement the standard 

 of reference where it was geometrically possible we have 

 been able to pass to the case of mixtures where it is 

 impossible. 



As to the mutual potential energy of the n x and the n 2 

 molecules we can find it most simply by considerations of 

 symmetry from the result just obtained. When n t and n 2 

 are large, the number of pairs of a molecule of 1 with a 

 molecule of 1 is n 2 \2 nearly, while the number of possible 

 pairs of a molecule of 1 with a molecule of 2 is n^n^. Hence 

 for the desired result there needs only to replace in 

 ^n^eiSip the n 2 by 2/1^2 and e^Sj 2 by e^s^^, obtaining 

 Sn 1 n 2 e l s^e 2 s 2 p. If we desire to get this from first principles we 

 may return and analyse the product 4:ii l (e^s l 2 IRi 6 )(n 1 pjn Ql p 1 ) 

 in the following manner. As Ji i s n Q ^p l = 1, we have n 1 pjn ol p l 

 equal to the total volume of the molecules of liquid 1 in unit 

 volume of the mixture or to Wip/2 times the volume of a pair 

 of molecules of 1 when they are neighbours. Thus the 

 mutual potential energy of the n x molecules is equal to three 

 times the energy of a pair of them as neighbours 4^ 1 2 5 1 2 /B 1 3 

 multiplied by iiipj'2, times the volume of a pair as neighbours, 

 multiplied by n 1 . Let jRg be the distance between a mole- 

 cule of 1 and of 2 in their average positions as neighbours, 

 then the mutual potential energy of one molecule of 1 and 



