128 BELL SYSTEM TECHNICAL JOURNAL 



Now we have attended to every term in (26) except the term JJ/T. Nearly 

 every reader will remember that the average kinetic energy of an atom of a 

 monatomic gas at temperature T is | kT. I therefore leave out the deriva- 

 tion of this result, except for showing the student how to begin on it: the 

 first step is to go back to equation (11) where an expression was given for 17, 

 and in that expression to replace the summation S Ei exp { — BEi) by 



(I/Hq) times the integral I I E exp {—BE)dpx dpy dp^. It follows that 



U/T is (3/2) iV^, which for one gramme-molecule of gas is (3/2)R, which I 

 write as R In e^''. 



The picture of entropy for a monatomic gas limned by the Boltzmann sta- 

 tistics, is now completed. Entropy is the function which follows: 



S = lR\nT-{-R\nV -\- R\n ^^""'"'^^ (31) 



2 K Ho 



The dependence on volume is correct, i.e., just the same as in the thermody- 

 namic formula. The dependence on temperature is correct, for {3/2)R is 

 the value of the specific heat at constant volume per gramme-molecule of a 

 gas, the quantity C„ of equation (18). The additive constant, as to the 

 value of which thermodynamics says nothing, is fij^ed when the volumes 

 Fo and Hq of the elementary cells in the ordinary space and the momentum- 

 space are fixed. 



Mixtures of Gases 



Now we will go through the mental operation which is called "considering 

 a mixture" of two different monatomic gases, N' atoms of the one and N" 

 atoms of the other, in the same box and (necessarily) in the same momentum- 

 space. Let me denote by U' and U" , respectively, the energies of these 

 two gases; and by Ni — N'wi and Ni — Nwi , respectively, the numbers of 

 atoms of the two kinds in the ith. cell of momentum-space. 



If we seek the most probable distribution of the first gas in the momentum- 

 space, making the stipulation that we will admit only such variations of the 

 quantities Wi as leave N' and U' unchanged — well, of course, we get the same 

 result as before, the distribution (8), with N' in place of N and (let me say) 

 A' in place of A and B' in place of ^. A' will depend upon B' and B' will 

 depend upon U'/N'. If we do the like with the second gas, we get anew to 

 the distribution (8) with N", A " and B" in place of N', ^ ' and 5'. A" will 

 not be the same as A' nor will B" be the same as B', unless it happens that 

 U" /N" is equal to U'/N' . There is no cause for surprise in this. In acting 

 this way we are only treating each gas by itself, and have as yet done nothing 

 which can be regarded as "considering a mixture". 



Let us however seek the most probable distribution of the two gases, mak- 



