394 Thermodynamics and Biology /2I : 4 



where V is the volume and where the partial molal free energy has been 

 defined by Equation 17' as 



G Bl = G° Bi + RT\n[B t ] 



Differentiating Equation 21, using the definition of G in Equation 17', 

 leads to the relationship . 



dG = {^jdV + V{G Bi d[B,] + G B2 d[B 2 ] + G C A C \] + GcAC*]) 



+ RT{d[B^\ + d[B 2 ] + dlC,] + d[C 2 ]) V (22) 



The various differentials are restricted by Equation 18 so that, if the 

 volume remains constant 



d[B{\ = d[B 2 ] = -d[C{\ = -d[C 2 ] = dx (23) 



in which x is an arbitrary parameter expressing the amount the reaction 

 in Equation 18 has progressed to the right. Substituting Equations 23 

 into 22 and setting dV = 0, one arrives finally at the desired equation 



dG = V[G Bi + G B2 - G Cl - G C2 ]dx (24) 



for the change of Gibbs' free energy for a small displacement of Equation 

 18 from equilibrium. 



For equilibrium, this last expression must vanish. Replacing the 

 G's by their expressions in Equation 17' and rearranging terms, leads to 



G° Bl + G% 2 - G° Cl - Gg 2 = 



RT (In [C\] + In [C 2 ] - In [B x ] - In [B 2 ]) (25) 



The left-hand side of this equation is the difference in the partial molal 

 free energies of Equation 18 when all substances are in their standard 

 state. This is usually denoted by — AG , defined by 



AG = G° Cl + G° C2 - G° Bl - G° 2 (26) 



The right-hand side of Equation 25 may be recognized as RT In K, 

 thereby allowing one to write 



AG = - RT In K 

 or 



A' = e -* G ° IRT (27) 



The preceding was carried through for two reactants on each side of 

 the equation. If one assumes the osmotic pressure is constant, then 

 the volume will be constant also. 



