386 BELL SYSTEM TECHNICAL JOURNAL 



SO as to give entropy 6" as function of pressure and temperature: 



5 = -y^iVInP + {5/2)kN[nT -\- ^.V In R-^^^^^^-^^'l (68) 



Notice that here every term is strictly proportional to TV, in accordance 

 with the "second alternative" of page 377. 



Let P and T be so chosen that the gas is in equilibrium with its solid 

 crystalline phase. To keep this choice in mind, I will replace T by T^, 

 signifying "temperature of sublimation" at pressure P. Let the N atoms 

 of gas now be cooled to the absolute zero. First they will condense, still at 

 temperature T^, into the crystalline solid. In so doing they will disgorge 

 the "heat of sublimation," L per gramme-molecule, amounting to NL/Na; 

 and their entropy will decline by NL/NoTg, since the process is reversible. 

 Let the cooling continue. As the crystal declines in temperature from any 

 T down to {T — dT), it disgorges heat in the amount of (N/No)CpdT and 

 entropy in the amount of (N/No)(Cp/T)dT; here Cp stands for the specific 

 heat (per gramme-molecule) of the crystal. The pressure is supposed to 

 remain the same throughout the entire process. When the crystal arrives 

 at absolute zero, its entropy has the value: 



So= - kN In P + (5/2kN In T, + kN In Ri^':^^tMl"1 



- (N/No)(L/Ts) - (N/No) / (Cp/T) dT 



Jo 



The right-hand member of this equation embodies the new statistical 

 theory of entropy. If on the left I put the value zero for .^o, I express what 

 is known as "Nernst's Heat Theorem" or the "Third Law of Thermo- 

 ' dynamics." If experiments say that the right-hand member of (69) is 

 equal to zero, they ratify not indeed the statistical theory by itself or the 

 Third Law by itself, but the assumption that both are true. Now, this is 

 what the experiments do say. Better to describe the situation, they say 

 that the first three terms on the right of (69) are equal to the last two terms 

 with sign reversed. All of the noble gases have been tested with suitable 

 accuracy, and eight or nine of the metals with accuracy not so high, yet 

 better than "order-of-magnitude accuracy." For further details I must 

 refer to my article already cited, 



* I cannot refrain from mentioning a detail of the statistical theories, which is amusing 

 if one sees it at once and confusing if one sees it belatedly (mine was the latter experience). 

 It pertains to the power to which e is raised in the third term on the right in (69). If in 

 the new statistical theory we leave out the term ^V in (49), thus stopping with a first 

 approximation instead of going on to the second, we arrive ultimateh' at e''^ instead of 

 e''^. If in the old statistical theory as modified b>- Tetrode we use the first-order Stirling 

 approximation instead of the second-order one for N !, we arrive ultimately at e^'^ instead 



