RECENT STATISTICAL THEORIES 705 



many free electrons as there are atoms, and test the combination of 

 these two assumptions. 



On putting for m the mass of the electron, for NjV the number of 

 atoms per unit volume of any metal, for T any temperature from 

 zero absolute up to several thousand degrees, and for G any small 

 integer, one finds that the quantity equated to A in (64) is very large. 

 Thus with NJV = 5.9 -lO^^ (the number of atoms in" a cc. of silver), 

 T = 300° K., G = 2, Sommerfeld computed: 



(nhyG){27rmkT)-"^ = about 2400 (65) 



a result which invalidates equation (64). 



We turn then to the other series-expansion of the integral J"F(e)de, 

 the one which Sommerfeld proved applicable for large values of A. 

 The first two terms of this expansion are as follows: 



r 



F{e)de = N= ^y (2mkT log A)''' (l + y O^g ^)"'+ • • •) ' (66) 



Taking the first term only of this expansion and putting the aforesaid 

 values of N/V and T and solving for log A, one finds a very large 

 value indeed (loge A = 325). Assuredly then we may use the first 

 two terms of this expansion by themselves when we are dealing with 

 the electron-gas in a metal, and indeed the first term will for some 

 purposes be amply sufficient. 



We have thus the following first — and second — approximation 

 formulae for A in terms of n or Nj V: 



2mkT log A = h^iSnl^irGy^ first approx 

 2mkT log A = h^(3nl4TrGy'^ 



_ {2TrmkTY ( 3n \-"' 



second approx. 



(67) 



(the second approximation being computed by putting the first- 

 approximation value of log A into the second term of the series 

 expansion). 



On substituting one or the other of these into the distribution- 

 functions (61) and (62), we have the postulated distribution of the 

 free electrons expressed to as high a degree of approximation as we 

 require, with no disposable constant except n; and we are ready for 

 the applications. 



The Specific Heat 



As it was the notorious difficulty with the specific heat which 

 spoiled the old electron-gas theory in which the classical statistics 



