Finding relation between Gibb's free energy and equilibrium constant for a homogeneous gaseous phase reaction

Question

I was trying to find relation between Gibb's free energy change and equilibrium constant for the following homogeneous gaseous phase reaction at constant temperature $$\ce{aA +bB }\rightleftharpoons \ce{cC + dD}$$

Let the initial partial pressures are $P_{1,i}$ and final partial pressures are $P_{2,i}$ for $i=A,B,C,D$

Now Gibb's free energy change of reaction is given by

$$\Delta_rG=\Delta G_{reac} + \Delta G_{prod}$$

However when I tried to find Gibb's free energy change for $\ce{A}$, I got stuck at first step

Using the relation $dG=VdP-SdT$

$$\Delta G_{A}=\int_1^2VdP=\int_1^2\frac{n_ART}{P_A}dP$$

However I cannot perform this integration as $n_A$ is not constant.

I saw other answers relating to same question but cannot gather much from them

Is my method correct ? If yes, then how can I continue ?

Answer 1

It's sometimes not obvious because of the way the subject may be taught, but $\Delta_rG$ is what's called a differential form, so it actually describes how the free energy changes as the reaction progresses by an infinitely small change in the amount of reactants and products at a given composition, $T$ and $p$.

The equation presented for the chemical potential $\mu_i=\mu^o_i+RT\ln p_i$ is correct, but the partial pressures are constant when evaluating $\Delta_rG$. In fact, that equation can be used to derive the relation between the reaction quotient and the Gibbs free energy of reaction:

$$\Delta_rG = \sum_i c_i \mu_i= \sum_i c_i \left( \mu^o_i+RT\ln p_i \right)\\ = \sum_i c_i \mu^o_i + RT \sum_i c_i\ln p_i \\ = \Delta_rG ^o + RT \ln Q $$

where $Q = \sum_i c_i\ln p_i$ is the reaction quotient and the $c_i$ are the coefficients of the reaction (not to be confused with the amount $n_i$ of each reactant or product, which remains constant).

This idea can be conceptually difficult at first. Another way to think of it is to imagine a very (=infinitely) large reaction vessel for which you measure the change in free energy upon changing the amount of reactants and products by $c_i$ (see for instance the textbook by Klotz and Rosenberg). Since the $c_i$ are finite and the vessel infinite, the composition remains constant. It's a mathematical sleight of hand equivalent to adding an infinitely small amount of substance to a finite system.