21:4/ Thermodynamics and Biology 



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(The more general case is somewhat more complex. The most 

 general reaction may be represented by 



*i 



v 1 B 1 + v 2 B 2 + v n B n ^± i x 1 C 1 + /x 2 C 2 + ix n C n 



(18') 



where the vs and yu.'s are positive integers and the B t 's are distinct molec- 

 ular species reacting reversibly to form the molecular species C t . In 

 this case, the equilibrium constant becomes 



tf, w<. 



K 



C?C 





B v iB v 2 



1 2 



■ B v « 



n 



Equation 20 is unaltered; that is 



K — — 

 The expression for the Gibbs' free energy may be written as 



(19') 



(20') 



G = V 



2 md + 2 Kto 



(21') 



As in the earlier case, one can show that, if 



dV = 



dG = 



2^-2^ 



dx + RT 



2 ".-2 



H 



dx 



(24') 



For equilibrium, one must demand that dG vanish, which leads to 



K = e% v i ~ 2»y*-AG°/*T (27') 



where 



AG = 2 hQj - 2 Vi G? (26') 



If, instead of constant volume, the reaction were restricted to constant 

 osmotic pressure, Equation 27' would be replaced by 



K = e-* G °' RT (27") 



Although the latter is simpler, Equation 27' appears to be a more 

 realistic one for biologically significant reactions.) 



Equations 27 and 27' are general relationships for reactions in a 

 liquid. Neither these equations nor the equilibrium constants indicate 

 how rapidly a solution of reactants will reach equilibrium. Nonethe- 

 less, Equation 27' is used in the discussion of absolute rate theory in the 

 next chapter. A simple example of the application of Equation 27' is 

 considered in the following section. 



