9.2: Chemical Potential

Equilibrium can be understood as accruing at the composition of a reaction mixture at which the aggregate chemical potential of the products is equal to that of the reactants. Consider the simple reaction

$$A(g) \rightleftharpoons B(g) \nonumber$$

The criterion for equilibrium will be

$$\mu_A=\mu_B \nonumber$$

If the gases behave ideally, the chemical potentials can be described in terms of the mole fractions of $A$ and $B$

$$\mu_A^o + RT \ln\left( \dfrac{p_A}{p_{tot}} \right) = \mu_B^o + RT \ln\left( \dfrac{p_B}{p_{tot}} \right) \label{eq2}$$

where Dalton’s Law has been used to express the mole fractions.

$$\chi_i = \dfrac{p_i}{p_{tot}} \nonumber$$

Equation $\ref{eq2}$ can be simplified by collecting all chemical potentials terms on the left

$$\mu_A^o - \mu_B^o = RT \ln \left( \dfrac{p_B}{p_{tot}} \right) - RT \ln\left( \dfrac{p_A}{p_{tot}} \right) \label{eq3}$$

Combining the logarithms terms and recognizing that

$$\mu_A^o - \mu_B^o –\Delta G^o \nonumber$$

for the reaction, one obtains

$$–\Delta G^o = RT \ln \left( \dfrac{p_B}{p_{A}} \right) \nonumber$$

And since $p_A/p_B = K_p$ for this reaction (assuming perfectly ideal behavior), one can write

$$\Delta G^o = RT \ln K_p \nonumber$$

Another way to achieve this result is to consider the Gibbs function change for a reaction mixture in terms of the reaction quotient. The reaction quotient can be expressed as

$$Q_p = \dfrac{\prod_i p_i^{\nu_i}}{\prod_j p_j^{\nu_j}} \nonumber$$

where $\nu_i$ are the stoichiometric coefficients for the products, and $\nu_j$ are those for the reactants. Or if the stoichiometric coefficients are defined by expressing the reaction as a sum

$$0 =\sum_i \nu_i X_i \nonumber$$

where $X_i$ refers to one of the species in the reaction, and $\nu_i$ is then the stoichiometric coefficient for that species, it is clear that $\nu_i$ will be negative for a reactant (since its concentration or partial pressure will reduce as the reaction moves forward) and positive for a product (since the concentration or partial pressure will be increasing.) If the stoichiometric coefficients are expressed in this way, the expression for the reaction quotient becomes

$$Q_p = \prod_i p_i^{\nu_i} \nonumber$$

Using this expression, the Gibbs function change for the system can be calculated from

$$\Delta G =\Delta G^o + RT \ln Q_p \nonumber$$

And since at equilibrium

$$\Delta G = 0 \nonumber$$

and

$$Q_p=K_p \nonumber$$

It is evident that

$$\Delta G_{rxn}^o = -RT \ln K_p \label{triangle}$$

It is in this simple way that $K_p$ and $\Delta G^o$ are related.

It is also of value to note that the criterion for a spontaneous chemical process is that $\Delta G_{rxn}\ < 0$, rather than $\Delta G_{rxn}^o$, as is stated in many texts! Recall that $\Delta G_{rxn}^o$ is a function of all of the reactants and products being in their standard states of unit fugacity or activity. However, the direction of spontaneous change for a chemical reaction is dependent on the composition of the reaction mixture. Similarly, the magnitude of the equilibrium constant is insufficient to determine whether a reaction will spontaneously form reactants or products, as the direction the reaction will shift is also a function of not just the equilibrium constant, but also the composition of the reaction mixture!

Example $\PageIndex{1}$:

Based on the data below at 298 K, calculate the value of the equilibrium constant ($K_p$) for the reaction

$$2 NO(g) + O_2(g) \rightleftharpoons 2 NO_2(g) \nonumber$$

	$NO(g)$	$NO_2(g)$
$G_f^o$ (kJ/mol)	86.55	51.53

Solution

First calculate the value of $\Delta G_{rxn}^o$ from the $\Delta G_{f}^o$ data.

$$\Delta G_{rxn}^o = 2 \times (51.53 \,kJ/mol) - 2 \times (86.55 \,kJ/mol) = -70.04 \,kJ/mol \nonumber$$

And now use the value to calculate $K_p$ using Equation $\ref{triangle}$.

$$-70040\, J/mol = -(8.314 J/(mol\, K) (298 \, K) \ln K_p \nonumber$$

$$K_p = 1.89 \times 10^{12} \nonumber$$

Note: as expected for a reaction with a very large negative $\Delta G_{rxn}^o$, the equilibrium constant is very large, favoring the formation of the products.