AND OF THE SECOND LAW 63 



Step b 



Determination of a General Expression for the Entropy S of any 

 given Natural State 



This step is an easy one. We have in Eq. (10) the relation 

 expressing the universal dependence of entropy S on probability 

 IF. Substituting and writing out the logarithm of the quotient 

 given in ( 1 6), we have 



(17) 



The summation S must be extended over all the elementary 

 regions a. With the help of STIRLING'S formula, and remembering 

 that both a and JV=S/^ are constant for 'all changes of state, 

 the above expression (17) is reduced to the form 



Entropy 5 = constant k^f-logf-a. . . . (18) 



This magnitude S is numerically the same as H for which BOLTZ- 

 MANN proved that it changed in a one-sided way in all changes 

 of state. We must bear in mind too that function / represents, 

 for every state of the gas, the given space and velocity distribution 

 of the gas molecules. The permanent, stationary, state of the 

 gas known as thermal equilibrium is only a special case of the 

 general case (18), this special case being widely known as MAX- 

 WELL'S Law of Distribution of Velocities. 



Step c 



Special Case of (b), Namely, Determination of Entropy S for the 

 Thermal Equilibrium of a Monatomic Gas 



This case PLANCK derives very easily from the general case 

 represented by (18). As the desired result has already been 

 found in another way in pp. 48-53 when dealing with MAXWELL'S 



