Properties of the Equation of State. 397 



chosen. The fundamental form of the equation is therefore 

 that which is independent of the nature of the units. This 

 form is 



that is an equation which contains only the ratios 



X V 7) 



r jT , — , and — , where p c , v C} T c denote the critical values of 



p, v, and T. Now if we assume that the algebraical form 

 of this equation is the same for all pure substances, the law of 

 corresponding states at once follows. Thus in the ease of a 

 substance whose state is defined by two variables, it follows 

 directly from the above equation. When the slate of a sub- 

 stance is defined by one variable it follows from equations 

 (3), (4), and (5), each of which reduces to the same form as 

 the above equation. 



Or we may deduce the law thus. Let us assume, (1) that 

 the algebraical form of the equation for each pure substance 

 is the same, except that some of the constants it contains 

 depend on the nature of the substance, (2) that the con- 

 ditions it has to satisfy at the critical point and absolute zero 

 are the same for each substance, and are sufficient in number 

 to determine the nature of the foregoing constants. The 

 general equation of state is then of the form ^(T, v, p, 

 T c , v c , p c ) = 0. Now this equation must be independent of 

 the kind of units adopted, and therefore reduce to the same 

 form as equation (19). 



"We thus see that in obtaining the law of corresponding 

 states from the equation of state some very weighty assump- 

 tions are made which amount almost to assuming the law 

 without making any further deductions. There is nothing 

 from which we can argue that these assumptions must hold. 

 It appears that they do not hold for substances partly poly- 

 merized, since these do not obey the law of corresponding 

 states. But there is no obvious reason why this should be 

 so. At the end of the next section we will show that the 

 law can be made to depend on more fundamental assumptions 

 than those just considered. 



The General For?ns of U, u, and Z. 



The law of corresponding states enables us to obtain some 

 information about the general forms of U, u, and Z. Let us 

 assume that U = X + W, u = x-\-iv, and Z = Y + *, where W 

 and w may be functions of any form of T and v, z any 



