706 BELL SYSTEM TECHNICAL JOURNAL 



was coupled with the assumption that there are as many free electrons 

 as atoms, let us first of all find out whether this difficulty remains. 



Any distribution-law for an assemblage leads immediately to a 

 formula for the total energy E thereof as function of the temperature, 

 which is: 



E = eF{e)de. (68) 



Jo 



Putting the distribution-in-energy (27) derived from the classical 

 statistics, we find: 



E=^NkT (69) 



and putting the one just derived from the Fermi statistics with the 

 first-approximation value of ^, we find: 



E = Eo + hyVT\ 

 Eo = 



2t VG¥ / 3n y 



2m \4tGJ ' (70) 



two exceedingly different formulae. 



The derivatives dE/dT oi these expressions are the formulae supplied 

 by the two statistics for the specific heat of the electron-gas. The 

 classical theory predicts for the specific heat a constant value, while 

 the Fermi statistics makes it proportional to the temperature — being 

 thus in harmony with Nernst's heat theorem, while the other is not — ■ 

 and gives it even at room temperatures but a small fraction of the 

 classical value. 



Experimentally the specific heat of the electron-gas cannot be 

 measured separately from that of the lattice of atoms, which constitutes 

 the metal — an admission which seems to condemn as vain all hope 

 of testing these formulae. Nevertheless one can conclude with fair 

 certainty that the classical expression is inadmissible; for the specific 

 heat of an ordinary metal agrees so well with the value attributed 

 by statistical theories both old and new to the atoms alone, that there 

 is simply none left over for the electrons — no such great excess, that 

 is to say, as the amount 3nkJ2 which the classical theory requires. 

 The device of reducing n to so low a value that Snkjl would be in- 

 appreciable makes trouble in other directions, as I have intimated. 

 But with the new statistics the theoretical value yVT is inappreciable 

 even when n is made as great as the number of the atoms and T as 



