TEE NEW STATISTICAL MECHANICS 381 



In the first place, (61) enables us to eject either B oxU from the expression 

 for Sm- It seems more sensible to do away with B, leaving Sm expressed as a 

 function of the energy of the gas; but I will let the reader do that for his own 

 instruction. In view of the peculiar significance of B this is for our purposes 

 the better one to keep. 



Now it is high time indeed to show what is that peculiar significance. 

 Differentiating S witli respect to U, we find kB for the derivative. Turning 

 back to (32) we are reminded that this derivative is l/T by the definition 

 of the absolute temperature T. Now the temperature has stepped across 

 the threshold, and Sm assumes the form: 



Sm = (3/2) NklnT - Nk In N + Nk In [{2irmkf''e''yqm] (62) 



Notice that the term + kN has been absorbed into the final term, so that 

 e^'' has been replaced by e'' in the argument of the logarithm: this is a usage 

 with which the student must become familiar. (Some writers also incorpo- 

 rate — Nk In N into this final term, which thereby acquires a factor N 

 in the denominator of the argument; the term then ceases to be a constant, 

 which is why I do not follow this policy.) 



Comparing (62) with (46) we see that Sm embodies correctly the 

 dependence of entropy on temperature, provided that C„ (the specific heat 

 per gramme-molecule) is equal to {3/2)kNo. Since a value for k — to wit, 

 R/N'o — has already been forced upon us as a necessary and a sufficient 

 condition for making 5 depend correctly on the volume, this new require- 

 ment is that Cy should be equal to (3/2)R. Now this is a fact of experience 

 for the gases called monatomic ! 



I said that the relation of U and B expressed in (61) is valuable in two 

 ways. The second is only the first seen from a different viewpoint, for 

 which I rewrite (61) in the form: 



U/N = U = (3/2)kT (63) 



For the flock of mass-points distributed in momentum-space in the manner 

 indicated as the most probable by the new statistics (as, for that matter, by 

 the old) the average energy is {3/2)k times the absolute temperature. This 

 is the very result obtained from simple kinetic theory for the ideal-gas scale 

 of temperature. The statistical theory therefore identifies the absolute 

 scale of temperature with the ideal-gas scale, which is as it should be. It is 

 therefore an adequate theory of temperature and (as we lately saw) of the 

 specific heat of monatomic gases. 



Now I have given an expression for Sc, the "contribution of volume to 

 entropy," which is (50); and an expression for Sm, the "contribution of 

 temperature to entropy," which is (62); and it seems natural to proceed by 



