States, and the Thermodynamic State- Equation. 153 



E £=— \e, t =— -Jc. The corresponding " critical " reduced 

 equation is 



e' = 2d'-6' 2 . 



For another example : in all the systems defined as homo- 

 geneous solutions in which a chemical reaction of the first 

 order is taking place at constant temperature, the relation 

 between concentration x of reacting substance and time t is 

 expressed by the equation kt = \og (a/(a — #)). The reduced 

 equation, common to all these systems, is 



T = l0g(l/(l-£». 



For reactions of the second and third orders, the initial 

 concentrations of the reacting molecules being equal, the 

 reduced equations are, respectively, 



T=f/a-a 



t=?(2-£)/(W) 2 - 

 Corresponding times and concentrations are thus defined : 



for 1st order reactions by equal values of kt and a/a, 

 „ 2nd „ „ ,, akt and a/a, 



„ 3rd „ „ „ 2a 2 kt and a/a. 



When no non-specific reduced equation exists, i. e,, when 

 there are more specific constants than independent dimensions 

 in the original equation, it may yet be of value to form the 

 reduced equation, which is simpler in form and may shed 

 light on the inner nature of the relation which is expressed. 



Thus, the (m, 5, r) or Energy-Entropy- Volume equation of 

 ideal gases : — 



e s - ! 'z=(u—ay.r 1 " l f 



where c is the specific heat at constant volume and wi = M/R, 

 contains four specific constants. It is reduced by putting : 



a = u l , l\m = c/\ = s l , h=—.i 1 (\ocrv 1 -\-\.\ogUi). 



Then w 1} s l9 Vi have the dimensions of it, s, v, respectively, 

 and writing m/wj = u, s/si = <r, v/v l = <f>, the reduced equation is 



e°=(v-iy.(f>. 



Here \ = cm is the specific constant of zero dimension 

 remaining in the reduced equation. 



