.
A + B <--> C + D.
Assume that the standard Gibbs energy for this reaction at 25 C is 1 kJ/mol, also assume that the activity coefficients of all compounds are 1. Calculate a) the equilibrium constant for the reaction and b) the composition of the solution when the reaction reaches equilibrium.
a) The standard Gibbs energy for any reaction with the equilibrium constant K is expressed as
.
Rearranging this equation, one can easily get that the equilibrium constant is
.
Therefore, the equilibrium constant for the above reaction is:
.
b) The equilibrium constant on the other hand can be expressed in terms of activities of the compounds:
.
Taking into account that the activity coefficients of the compounds are all 1, the activities can be replaced by the numerical values of molalities or molar concentrations (at 25 C water density is about 1 g/ml, therefore they are the same). Thus, the equilibrium constant is
,
where [A], [B], [C] and [D] are the molar concentrations of the compounds at the equilibrium. Due to the reaction, the compounds A and B are converted into C and D (or vice versa), therefore, taking into account the stoichiometry of the reaction, one can express the equilibrium constant as
,
where A0, B0, C0 and D0 denote the initial concentrations of the compounds and x denotes the change in the concentrations. Rearranging this equation one get
.
Solving this equation for x gives two answers:
.
However, only one has physical meaning (we will see it in a moment). Substituting the numerical values into the equation for x, one gets the answers:
x1= -0.011 and x2= -1.0.
Obviously, the second solution does not have a physical meaning, since using it one get negative final concentrations. Therefore, the right solution for x is -0.011. Take into account that it is negative, it means that our initial assumption that A and B is transformed into C and D was wrong. Finally, the concentrations of the compounds at the equilibrium:
[A]= 0.111, [B]= 0.111, [C]= 0.089, [D]= 0.089.