ENTROPY 67 



which reduces to the statement, 



fi = Z-'exp{-Ei/kT) (22) 



Z being a symbol for the sum, 



Z = ^iexp{-Ei/kT) (23) 



which, incidentally, is known as the "partition-function" or "state-sum" 

 of the system — here, of the individual atom of the gas. 



In these latest equations, £,: stands for the energy-value corresponding 

 to some point in the /th cell. It is kinetic energy of translatory motion 

 with which we are concerned; therefore Ei is given by the equation 



Ei^ {\/2m)[{pl)i+ ip]),^ {plU (24) 



where now the components of momentum are to be evaluated at some 

 particular point in the fth cell. But at which particular point? And to go 

 further back, just how are the cells of volume ^o to be designed in the 

 momentum-space? As cubical blocks with their edges parallel to the 

 coordinate-axes, or how? There are problems in which definite answers 

 must be given to these questions, but we shall be able to avoid them. It 

 will be adequate here to conceive of the cells as cubical blocks and the 

 value of Ei as the average value of the right-hand member of (24) in the z'th 

 cell. 



Now we require from (24) the value of Z as defined in (23), to establish 

 the values of /( as given in (22), to yield finally the value of entropy as given 

 in (18). 



Let us form the integral 



/// ''■'■^ (-^/^'I^') dp- dpy dp,, £ = (l/2w)(/^' + pl + pi) (25) 



the range of integration extending over the whole of momentum-space. 

 This integral may be described as follows. Let the momentum-space be 

 divided into cells of unit volume. Each of these cells of unit volume makes 

 a contribution 



exp{-E/kT) 



to the integral, E standing now for the average value of E in the cell in 

 question. The integral is the sum of all of these contributions. Now let 

 us inquire how much of a contribution is made by this same cell of unit 

 volume to the partition-function. This second contribution is made up 

 of \/Hf) terms, one for each of the cells of volume Ha which occupy the cell 

 of unit volume. The values Ei corresponding to these cells will not be 

 exactly equal to the value E corresponding to the entire cell of unit volume; 



