64 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



Law (of distribution of molecular velocities), we will not repeat 

 PLANCK'S derivation of the law from (18). It will suffice here to 

 give the results: The law of distribution is given by function 



where a and /? are constants and U the total energy. As this expres- 

 sion for function/ is free from all location co-ordinates x, y, z, we see 

 that this state of thermal equilibrium is independent of these 

 space co-ordinates and conclude that in this state the molecules 

 are uniformly distributed in space; only the velocities are variously 

 distributed, all of which accords with the earlier presentation. 

 Substituting the results of (19) in the general equation (18) there 

 results for the entropy S of the state of equilibrium of a monatomic 



gas> S = constant + kN($ log U + log F) . . . . (20) 



To make Eq. (20) practically serviceable we need to know the 

 constants k and N and they will be found later on. 



Step d 



Confirmation, by Equating this Probability Value of S with that 

 found Thermodynamically and Securing well-known Results 



We know from Thermodynamics that the change of entropy 

 is defined in a perfectly general way (for physical changes) 1 by 



dU + pdV 

 dS= -- 7p - ....... (21) 



Deriving the partial differential coefficients and making use 

 of (20), there follows: kNT RnT 



Py- > ...... ( 22 ) 



1 This differential equation is valid only for changes of temperature and volume 

 of the body but not for its changes of mass and of chemical composition, for in 

 in defining entropy nothing was said of these latter changes. 



