66 BELL SYSTEM TECHNICAL JOURNAL 



S = -kNZfilnfi (18) 



a definition which in the case of the even distribution of the gas throughout 

 the container reduces to this, 



S = kN InV - kN InVo (19) 



This is of course a tentative definition, to be eschewed if ever it should lead 

 us into contradiction with the known properties of entropy. As yet it leads 

 us into none, for the term in In V corresponds to a similar term in the 

 description of entropy which equation (13) has already supplied. To make 

 the two agree exactly, we have simply to assign a special value to the factor 

 k; and as will be seen at once, this value is 



k = R/No (20) 



No standing for the number of atoms per mole, the Avogadro number- 

 Though k is known as Boltzmann's constant, this evaluation was beyond 

 Boltzmann's powers, for in his time the value of No was not known. 



The expression (19) contains no allusion to temperature. It is in fact 

 not the entropy in full which has so far been defined, but only what I may 

 call the "contribution of volume to entropy." We have now to account 

 for the contribution of the kinetic energy of the molecules to the entropy of 

 the gas. Thus far I have been able to come by adducing the deeply- 

 inbred conviction that a gas in equilibrium in a container is evenly spread 

 throughout the container. There is no such widely-held conviction about 

 the distribution-in-energy of the molecules of the gas; but to everyone who 

 has studied physics for more than a year or two there will be nothing 

 surprising in the formula which follows. It must be introduced by asking 

 the reader to imagine a three-dimensional space, in which the variables 

 along the three axes are identified not as coordinates in ordinary space, 

 but as components of momentum p^,, py, p^. The momentum which is 

 meant is the momentum of the individual atom, and the axes x, y, z along 

 which its components are taken are axes of a coordinate frame in ordinary 

 space — they might be along three edges of the container, for instance. A 

 point in the "momentum-space" represents an individual atom in respect 

 of its momentum and therefore in respect of its energy, but not in respect 

 of its position. 



The momentum-space is now to be divided into compartments of equal 

 volume Ho; but we are not to besprinkle its compartments uniformly with 

 the dots representing the atoms! Instead, when comparing any two of 

 the cells, say the ith and the^'th, we are to write 



fi/fj^exp[-(Ei-E,)/kT] (21) 



