400 Fundamental Properties of the Equation of State. 



But even then the expression will represent only approxi- 

 mately the facts. The value of b cannot be constant since 

 the chance of one molecule actually colliding with another 

 must be affected by their attraction upon one another. Thus 

 let ci-j denote the radius of the sphere of action of each 

 molecule which gives rise to actual collisions between them, 

 and <r 2 the actual radius of each molecule. The ratio of the 

 chance of an actual collision occurring when the molecular 

 attraction is acting to that when it is supposed not to be 



2 



actino- is —^. We must therefore write for b the ex- 



o"i 2 16 - 16 „ . . , c A 4 , 

 pression — 9 X -77 ir<T 2 = — 7royc- 3 instead ot -±x -770-9 . 

 <r 2 6 6 6 



The modified equation of state is then 



p + r n - y / W \-\3m) / 16 , \ 



.... (24) 



since "V = ( I according to the kinetic theory of gases. 



The value of a 1 must depend on the density of the substance 

 when the molecules are under each other's influence, since the 

 effect of the attraction of a molecule upon another is then 

 modified by the surrounding molecules. Thus, for example, 

 a molecule passing midway between two molecules at right 

 angles to the line joining their centres, is not affected by 

 either of them. It is evident from this example that o^ 

 decreases with increase of density of the substance. This 

 explains why the value of b in van der Waals' equation is 

 approximately constant, for as a x decreases V B increases, the 

 joint effect of which could conceivably be represented by 

 van der Waals' expression. It is obvious that b and 

 1 (\ 



-^-17(7^2 must be of the same order of magnitude. 

 o 



The fact that van der Waals* equation gives values of b 

 from which the radius of a molecule can be determined which 

 agrees well with that obtained by other methods, is not con- 

 clusive evidence of the correctness of the assumptions made. 

 One reason is stated above. Another is that when an equa- 

 tion contains more constants than one, their values can usually 

 be varied over a considerable range without seriously affect- 

 ing the agreement of the equation as a whole with the facts. 

 The values of constants determined by simultaneous equations 

 are therefore of less value than those determined directly. 



