70 BELL SYSTEM TECHNICAL JOURNAL 



I can do for the present is to follow Tetrode by saying that he deemed (26) 

 as it stands to be an overstatement of the entropy, arising because in 

 advancing from the underlying theory to equation (18) one assumes the 

 atoms of the gas to be distinguishable, whereas actually for a substance of a 

 single kind they are indistinguishable. In somewhat the same way one 

 might overstate the entropy of a crystal by supposing it to be composed of 

 atoms no two of which were alike, while actually it was a substance of a 

 single kind. The subtraction of k In N\ was Tetrode's manner of correcting 

 the overstatement. He did better than perhaps he knew, for while his 

 reasons never commanded universal assent, his choice of a term to be sub- 

 tracted was ratified first by experiment and then by the "new statistics" 

 which made their appearance in physics some fifteen years ago. 



Returning to (26): writing the last term of the right-hand member as 

 nR In e^'~, and consolidating it with the third term; introducing Tetrode's 

 subtractive term; augmenting this last by a term —nR In k, and compensat- 

 ing by adding -\-nRln k to the third term — doing all this, one finds, 



/^ \3/2 7 5/2 5/2 



S = nRlnV + nC, In T - nR In (nR) + nR In ^ ""^^ ,, — (30) 



Now the additive constant is filled out completely, and ready for whatever 

 test experiment may impose. 



To prepare it for the test, we turn back first to equations (12) and (13), 

 and note that the constant there denoted by nl is none other than the fourth 

 term in the right-hand member of (30): 



/r, \3/2 ,5/2 5/2 



I = Rln^^ '"^J ' (31) 



Continuing onward to (15) we are reminded that no theoretical statement 

 about / is worth anything by itself, since all that data can supply is the 

 value of the combination (I- So). A hopeless situation, in appearance! 

 But now it is high time to hearken to what the data say. The data say, 

 to begin with: 



For many monatomic gases, the right-hand member of equation (15) is equal 

 to the ^^statistical'" value of I. 



This may be taken as meaning two things at once: that (a) the statistical 

 theory of the entropy of a gas is right, and (b) the entropy of a solid (crystal- 

 line and of a single kind, for such are these solidified gases) is zero at the 

 absolute zero. It is taken as meaning these things. It might of course 

 also be taken as meaning that both statements are wrong by about the same 

 amount, the errors compensating one another. But so unlikely does it seem 

 that two such different theories should both be wrong and yet by precisely 



