Table of Contents
Connection Between Thermodynamics and Equilibrium
Chemical equilibrium (introduced in the parent chapter) can be described both:
- by an equilibrium constant $K$ (from the law of mass action), and
- by the standard Gibbs free energy change $\Delta_\mathrm{r}G^\circ$ (from thermodynamics).
This chapter explains how these two descriptions are linked mathematically and conceptually.
We will use the symbol $\Delta_\mathrm{r}G^\circ$ for the standard Gibbs free energy change of reaction and $K$ for the equilibrium constant defined using activities (or, in simple cases, appropriately scaled concentrations/partial pressures).
The Fundamental Relationship
For a reaction written in its stoichiometric form, for example
$$
\alpha\,\mathrm{A} + \beta\,\mathrm{B} \rightleftharpoons \gamma\,\mathrm{C} + \delta\,\mathrm{D},
$$
the key equation linking thermodynamics and equilibrium is
$$
\Delta_\mathrm{r}G^\circ = -\,RT \ln K
$$
where:
- $R$ is the universal gas constant ($R \approx 8.314\ \mathrm{J\,mol^{-1}K^{-1}}$),
- $T$ is the absolute temperature in kelvin,
- $K$ is the equilibrium constant at that temperature,
- $\ln$ is the natural (base $e$) logarithm.
Equivalently,
$$
K = \exp\!\left( - \frac{\Delta_\mathrm{r}G^\circ}{RT} \right).
$$
These equations are only valid when:
- $\Delta_\mathrm{r}G^\circ$ refers to the same reaction (same stoichiometry, same standard states) as the $K$ you use,
- $K$ is expressed using activities and is dimensionless (concentrations and pressures are divided by their respective standard values).
Sign and Magnitude: What Do They Tell Us?
The sign and size of $\Delta_\mathrm{r}G^\circ$ give immediate qualitative information about the position of equilibrium, through the link to $K$.
Qualitative interpretation
Use the relation $\Delta_\mathrm{r}G^\circ = -RT\ln K$:
- If $\Delta_\mathrm{r}G^\circ < 0$
⇒ $-RT\ln K < 0$ ⇒ $\ln K > 0$ ⇒ $K > 1$
⇒ products are favored at equilibrium (under standard conditions). - If $\Delta_\mathrm{r}G^\circ > 0$
⇒ $\ln K < 0$ ⇒ $K < 1$
⇒ reactants are favored at equilibrium. - If $\Delta_\mathrm{r}G^\circ = 0$
⇒ $\ln K = 0$ ⇒ $K = 1$
⇒ neither side is strongly favored under standard conditions.
So:
- Negative $\Delta_\mathrm{r}G^\circ$ ⇔ $K > 1$ ⇔ “product‑favored” equilibrium
- Positive $\Delta_\mathrm{r}G^\circ$ ⇔ $K < 1$ ⇔ “reactant‑favored” equilibrium
Quantitative sense of size
Because the relation is exponential, small changes in $\Delta_\mathrm{r}G^\circ$ can lead to large changes in $K$.
At $T = 298\ \mathrm{K}$:
- $RT \approx 2.48\ \mathrm{kJ\,mol^{-1}}$
A useful approximation (at room temperature) is
$$
\Delta_\mathrm{r}G^\circ \mathrm{(kJ\,mol^{-1})} \approx -5.7\ \log_{10} K.
$$
So approximately:
- If $\Delta_\mathrm{r}G^\circ \approx -5.7\ \mathrm{kJ\,mol^{-1}}$
⇒ $\log_{10} K \approx 1$ ⇒ $K \approx 10$. - If $\Delta_\mathrm{r}G^\circ \approx -11.4\ \mathrm{kJ\,mol^{-1}}$
⇒ $\log_{10} K \approx 2$ ⇒ $K \approx 10^2$. - If $\Delta_\mathrm{r}G^\circ \approx -34.2\ \mathrm{kJ\,mol^{-1}}$
⇒ $\log_{10} K \approx 6$ ⇒ $K \approx 10^6$ (very product‑favored).
In the other direction:
- If $\Delta_\mathrm{r}G^\circ \approx +34.2\ \mathrm{kJ\,mol^{-1}}$
⇒ $K \approx 10^{-6}$ (very reactant‑favored).
From Gibbs Free Energy of Reaction to the Equilibrium Constant
Gibbs free energy as a function of composition: reaction quotient
For a given mixture composition (not necessarily at equilibrium) the Gibbs free energy change of reaction is
$$
\Delta_\mathrm{r}G = \Delta_\mathrm{r}G^\circ + RT\ln Q
$$
where $Q$ is the reaction quotient constructed in the same way as $K$, but using current activities instead of equilibrium ones.
At equilibrium, $\Delta_\mathrm{r}G = 0$ (no net driving force in either direction), and $Q = K$. Substituting:
$$
0 = \Delta_\mathrm{r}G^\circ + RT\ln K
$$
which rearranges to
$$
\Delta_\mathrm{r}G^\circ = -RT\ln K.
$$
Thus, the link between $\Delta_\mathrm{r}G^\circ$ and $K$ is a direct consequence of:
- The dependence of $\Delta_\mathrm{r}G$ on composition ($Q$), and
- The condition for equilibrium: $\Delta_\mathrm{r}G = 0$.
Temperature Dependence
Both $\Delta_\mathrm{r}G^\circ$ and $K$ are functions of temperature, and their temperature dependencies are linked.
Using the fundamental relation
$$
\Delta_\mathrm{r}G^\circ = \Delta_\mathrm{r}H^\circ - T\Delta_\mathrm{r}S^\circ
$$
and combining it with
$$
\Delta_\mathrm{r}G^\circ = -RT\ln K
$$
gives
$$
-RT\ln K = \Delta_\mathrm{r}H^\circ - T\Delta_\mathrm{r}S^\circ.
$$
Rearranging:
$$
\ln K = -\frac{\Delta_\mathrm{r}H^\circ}{R}\,\frac{1}{T} + \frac{\Delta_\mathrm{r}S^\circ}{R}.
$$
If $\Delta_\mathrm{r}H^\circ$ and $\Delta_\mathrm{r}S^\circ$ are (approximately) constant over a modest temperature range, this shows that:
- a plot of $\ln K$ vs. $1/T$ is approximately a straight line,
- slope $= -\Delta_\mathrm{r}H^\circ / R$,
- intercept $= \Delta_\mathrm{r}S^\circ / R$.
This linear relation (often called a van ’t Hoff plot) is one way to see how temperature changes the position of equilibrium, and it reflects the same underlying thermodynamic quantities that appear in $\Delta_\mathrm{r}G^\circ$.
Practical Use: Converting Between $\Delta_\mathrm{r}G^\circ$ and $K$
Because of the simple exponential relation, you can:
- Calculate $K$ from $\Delta_\mathrm{r}G^\circ$ (e.g., from tabulated standard Gibbs energies of formation)
- Calculate $\Delta_\mathrm{r}G^\circ$ from $K$ (e.g., from experimentally determined equilibrium compositions)
From tabulated data to $K$
- For the reaction of interest, compute
$$
\Delta_\mathrm{r}G^\circ = \sum_{\text{products}} \nu_i\,\Delta_\mathrm{f}G_i^\circ \;-\; \sum_{\text{reactants}} \nu_i\,\Delta_\mathrm{f}G_i^\circ,
$$
where $\nu_i$ are stoichiometric coefficients and $\Delta_\mathrm{f}G_i^\circ$ are standard Gibbs energies of formation.
- Insert this $\Delta_\mathrm{r}G^\circ$ into
$$
K = \exp\!\left( -\frac{\Delta_\mathrm{r}G^\circ}{RT} \right).
$$
From an experimental $K$ to $\Delta_\mathrm{r}G^\circ$
If $K$ is determined at temperature $T$:
$$
\Delta_\mathrm{r}G^\circ = -RT\ln K.
$$
This gives a standard thermodynamic description of the reaction based purely on equilibrium measurements.
Standard States and Dimensionless $K$
For the relationship $\Delta_\mathrm{r}G^\circ = -RT\ln K$ to be valid in its simple form, $K$ must be dimensionless and defined using activities $a_i$:
- For solutes: $a_i = c_i / c^\circ$ (concentration divided by a standard concentration, usually $c^\circ = 1\ \mathrm{mol\,L^{-1}}$),
- For gases: $a_i = p_i / p^\circ$ (partial pressure divided by a standard pressure, often $p^\circ = 1\ \mathrm{bar}$),
- For pure solids and pure liquids: $a_i \approx 1$.
For the general reaction
$$
\sum_i \nu_i\,\mathrm{A}_i = 0
$$
the thermodynamic equilibrium constant is
$$
K = \prod_i a_i^{\nu_i},
$$
which is dimensionless by construction. When approximate expressions using concentrations or partial pressures are written as $K_c$ or $K_p$, they should be understood as shorthand for these activity-based, dimensionless quantities.
The standard Gibbs free energy change $\Delta_\mathrm{r}G^\circ$ always refers to these same standard states; the connection to $K$ assumes consistent definitions of the standard state on both sides of the equation.