MEMORIAL TO CLASSICAL STATISTICS 125 



tribiUion, multiplied by a constant (always denoted by k) wliich is adjusted to 

 bring about agreement with experiment: 



entropy ^ = ^ In IT. (15) 



To illustrate this doctrine and to evaluate k, I now take the student back 

 to the coordinate-space, where a box of volume V populated with N mole- 

 cules is divided mentally into M equal cells of volume Vq , and the most 

 probable distribution is characterized by the value N In M for the logarithm 

 of the probability. The entropy^ — or no, not the entire entropy of the gas, 

 but merely what I will call "the contribution of the volume of the entropy" 

 and denote by Sc — is then supposed to be kN In M, or: 



Sc = kN \nV - kN In Vo (16) 



Reverting to the equation (14) in which the definitions of entropy and 

 temperature were tangled up together, and rearranging it, we get: 



TdS = dU + PdV (17) 



Now, an "ideal gas" is defined by two attributes. First, there exists between 

 its pressure and its volume and its temperature the relation P = aT/V, 

 wherein a stands for a constant. Second, its energy U depends upon the 

 temperature only, and not upon any other variable, in particular not upon 

 the volume. Therefore we may write: 



TdS = CJT -f (aT/V)dV (18) 



Cv here standing for something of which we need only know that it is a 

 function of T alone. Integrating, we find: 



^ = i? In F + (function of temperature) + constant (19) 



and lo! it is seen that the dependence of entropy on volume is precisely of 

 the sort which the theory is fitted to explain. 



The next step is to adjust the value of the constant k. The constant a 

 aforesaid is proportional to the amount of gas in the box, proportional there- 

 fore to N : it is the constant ratio of a to TV to which k must be equated. For 

 the amount of gas let us choose one gramme-molecule. Then a assumes the 

 value always symbolized by R and called the "gas-constant", and N assumes 

 the value usually symbolized by No and called the "Avogadro number". 

 Both of these are known from experiment, and k is fixed by the equation 



k - R/No (20) 



The constant k is named in Boltzmann's honor, though in his time its value 

 was not known because the value of N'o was only vaguely apprehended. 

 Now we have settled what I called "the contribution of volume to en- 



