40 Mr. S. A. Shorter : Application of the Theory 



If the solution is in equilibrium with the solid solvent at a 

 temperature T, we have 



/oOi,%, ...,s n ,p,T) = <l> (p, T). 



Hence we have 



A (*i,* 2 » ...,Sn,i>,T)= I T Q {p,d)d6. 



JT 



The value of the integral in terms of experimental data has 

 been found in Part II. Substituting- this value in the above 

 equation we obtain the result 



Aofo,*, ... ,sn,p,T) = ^f^ -f ° 7 ^,T)rfT + Tf °^L^dr. 



J-o Jt Jt t 



.... (20) 



Duhem's expression for the heat of dilution 

 A)( 5 i> s -i) • • • •> s >i, p, 0) may be written in the form * 



d f Agfa, $2, • ■ • , s n , p, 0) \ _ k(si,s 2 , ... s n ,p, 6) 



-del e j " e 2 ' ' [ ~ L) 



Hence if &i and 6 2 are any two temperatures, we have 

 A fa, s 2 , ... ,s n ,p,6 2 ) A (s u s 2 , ... , s n ,p,6i) _ C Ql lo{si,s 2 , ... ,s n ,p,d) 



e 2 d 1 ~~ Je & 



r 01 



• • • t 



We can obtain a formula involving the value of the heat of 

 dilution at one temperature only, by making use of the 

 relation f 



Z '(s l3 s 2 , . . . , s», /?, (9) =7o(p, 0) — y(s 1; 5 2 , . . . , s;,^, <9) 



+ ( 1 + S 1 + . . . + Sn) T S^ 7 ... (23) 

 1=1 O^ 



We will not write this formula out, as the necessary altera- 

 tions in equations (12) of Part II. are obvious, as are also 

 the alterations in equations (15), (16), and (17) of Part II. 

 necessary to make them applicable to a solution containing 

 any number of solutes. 



* La Mecanique Chimique, vol. iii. p. 49. Duhem establishes the formula 

 by the aid of a general theorem relating to the heat evolved in any 

 change of a system, but the formula is easily proved from first principles. 



t This relation may be proved by equating the total heat evolved 

 when an infinitesimal mass of the solvent is added to the solution, and 

 the solution then heated from 6 to 0+80, to the heat evolved when the 

 solution and solvent are heated separately from to O-\-$0, and then 

 mixed. 



(22) 



