134 BELL SYSTEM TECHNICAL JOURNAL 



magnitude of the achievement, I will rewrite equation (31) with two altera- 

 tions. The first consists in replacing R with Nk, so that the expression shall 

 refer not to a gramme-molecule of gas but to any number N of atoms. The 

 second consists in following Tetrode by affirming that the entropy is not 

 k In W , but k times the logarithm of W /N\ I follow him still further by 

 using, not the super-Stirling approximation in which iV In A" is written for 

 \nN\, but the better approximation in which {N iniV — N) or {N\nN — N]ne) 

 is written for In N\ The result is: 



S = {3/2)Nk \nT + NkXnV - Nk\nN 



+ Nk In [{2Trmk'"\"^/V^E,] (35) 



This quantity newly chosen as the picture of "entropy" depends on volume 

 and on temperature in the right way, as did the other. The dependence on 

 N the number of atoms is now correct, and no wonder, for the new quantity 

 was chosen with that purpose. There is a fourth term in the right-hand 

 member which is proportional to N, and its value is completely determined 

 if the value of VqHq is fixed. The value which it takes when N is made equal 

 to iVomay be called "the chemical constant"; but this name has been spoiled 

 through being used with several different meanings, and should probably be 

 abandoned. 



When to Fofl^o , the volume of the elementary cell in six-dimensional space, 

 there is given the value h^ — the cube of Planck's constant — the resulting 

 value of the fourth term is excellently confirmed by experiments on all of the 

 noble gases, and (with less precision) by experiments on many of the mona- 

 tomic vapors of metallic elements. This is the achievement known as "the 

 verification of the Sackur-Tetrode formula" and it is indeed a grand one. 



Anyone versed in thermodynamics will probably regard this not as a grand 

 result, but as an incomprehensible one! Are we not taught in thermody- 

 namics that nothing is ever measured about entropy except the differences 

 between its values under different conditions, so an additive constant like 

 the one in question must drop out of every verifiable equation, and its 

 value can never be found? How then can it make sense to speak of con- 

 firming the value of the fourth term on the right-hand side of (35)? 



Well, actually it is a difference which is measured: the difference between 

 the entropy of the gas at any convenient temperature and volume and the 

 entropy of its solidified crystalline form at the absolute zero. This dif- 

 ference is found to be such, that if for the entropy of the gas one puts the 

 value (35) with h^ substituted for VoHq , then for the entropy of the crystal- 

 line solid at the absolute zero one finds the value: zero. This result — this 

 conclusion that the entropy of a crystal is zero at the absolute zero — is in 

 itself so desirable and welcome that it is taken as the confirmation of the 



