THE NEW STATISTICAL MECHANICS 377 



obvious from inspection of (44) when one remembers that L and H^ are 

 both proportional to N. I rewrite that equation accordingly, and put 

 No for the number of atoms in a gramme-molecule, R and Cv and RKo for 

 the values of L and Hj, and C appropriate to A^'o atoms: 



6* = (N/No)R In V + (N/No) C„ In T + (N/No)RKo (45) 



6* is doubled if N is doubled while V and T stay the same, which is what was 

 intended. 



This equation will not suit the second alternative; for if V is doubled 

 along with N, S will be more than doubled. Over and above the doubling, 

 .S" acquires an extra term 2(A^/iVo) R In 2. Now if the constant C just 

 happened to include a term —2(N/No) R In 2, the extra term would be 

 obliterated, and S would just be doubled if N and V were to be doubled 

 while P and T remained the same. Such a term is provided by replacing 

 (45) with the equation: 



S = {N/No)R In V + (N/No)Cv In T - iN/No)R In TV + (N/No)RKo (46) 



where the last two terms on the right are to be regarded as forming the con- 

 stant C. This then is the dependence of C on iV which is demanded by the 

 second alternative. 



To guide the choice between the two alternatives there is, so far as I 

 know, but the one argument; it is, however, a powerful one, and seems likely 

 to hold the field unchallenged. 



We have been thinking of two samples of identical gas at identical temper- 

 ature. Think of them now as divided by a removable partition. When 

 the partition is taken away, what happens? If the initial pressures are not 

 the same, there is a swirling and a surging, dying away in time into a state 

 in which the pressure is the same throughout the volume now common to the 

 samples, but is not the same as it was before in either separate gas. This is 

 just the sort of trend of events with which one likes to think that an entropy- 

 change, and indeed an entropy-gain, is linked. Notice also that if the 

 partition is replaced, the state of afifairs on either side does not become the 

 same as it was before ! But now suppose the initial pressures to be the same. 

 The partition can be removed and replaced without entailing any perceptible 

 change in the gas such as one likes to associate with a change in entropy. 



The second alternative is in harmony with these facts, the first is not. 

 So to the question "is the entropy of a gas of 2N atoms double the entropy 

 of a gas of N atoms?" the acceptable answer is: "yes, if the volume of the 

 double gas is twice that of the single gas, their temperatures being the same." 

 Now, this is also the answer given by the new statistics; for as we shall 

 presently see, it leads to a formula like (46). It is not the answer given 

 bv the old statistics, which (as I said in the earlier article) leads to a formula 



