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set of compressibility-curves for various mixtures at a given temperature 

 and ask the question : in what manner does the isothermal of the 

 associating substance run through this set of curves, i. e. how does 

 the degree of association vary with the volume at a given tempe- 

 rature? For the solution of this problem we shall follow the way 

 given by van der Waals (loc. cit., p. 121): we first establish the 

 expression for the free-energy of a given quantity (.¥grras) of a 

 mixture with constant a and then write down the conditions that 

 for every elementary change of condition (change of the molecular 

 constitution) at constant temperature and volume the free-energy 

 remains unchanged (the free-energy having a smallest value in the 

 condition of equilibrium). 



In our case the expression for the free-energy is ^) 



F=^- RTf{v)^''^-~{2Kx,y + RT2~logx,, + :EE,^x„-T2HnX,, , (8) 

 n V n 



where En is the internal energy of M grms ?z-fold molecules in the 

 ideal gas-state, Hn its entropy in volume 1'); f{v) stands for 



r dv 



I — ^, so that according to § 4: 



f{v) = log{v-bii,„) ^^-^ (8') 



The condition dF=0, connected with :i(ivn = 0, by the usual 

 method gives an infinite series of equations of the form 



RT .^ 2a RT RT 



;(") K>.Kx,,-V logx,,-] ^E,—TH,—ti=0 . (9) 



n V n n 



The constant n^) can be determined by the condition .2/(;„ = l. 

 In order to make this summation possible it is necessary to make 



^) It is only necessary to generalize the expression given by van der Waals 

 (loc. cit., p. 121). 



~) Eu and H,, are functions of the temperature, connected by the relation 

 dE„ _ dH„ 

 ^Tif — ~Fr~ ~ ^ "' ^^^^ specific heat of M gr. «-fold molecules in the ideal state 



at constant volume. 



3) Putting Z = F -\- pv (thermodynamic potential) the equation may also be put 



m the form n = ) , which shows the indeterminate constant n to re- 



present the so called molecular thermodynamic potential of the «-fold molecules. 

 Equations ( 9) thus express the law that in the condition of equilibrium the molar 

 thermodynamic potential is the same for all the different kinds of molecules ; this 

 is a well-known theorem, which we might have used straight away to establish the 

 equations (vid. for instance van Laar, loc. cit.). 



