How Does Temperature Affect the Equilibrium Constant Through Entropy?

Temperature affects an equilibrium constant through the standard Gibbs energy change of the reaction. Writing that Gibbs energy in terms of reaction enthalpy and entropy gives a direct expression for $K$ and shows why entropy alone does not tell you whether heating increases or decreases the equilibrium constant.

Relating $K$ to entropy and enthalpy

For a reaction at a specified temperature and standard state, the dimensionless thermodynamic equilibrium constant satisfies

$$\Delta_r G^\circ = -RT\ln K.$$

The standard reaction Gibbs energy is also related to enthalpy and entropy by

$$\Delta_r G^\circ = \Delta_r H^\circ - T\Delta_r S^\circ.$$

Combining these expressions gives

$$\ln K = \frac{\Delta_r S^\circ}{R} - \frac{\Delta_r H^\circ}{RT}.$$

Here, $R$ is the gas constant, $T$ is absolute temperature in kelvins, and $\Delta_r H^\circ$ and $\Delta_r S^\circ$ are the standard reaction enthalpy and entropy at that temperature. The equation is exact when those quantities are evaluated at the temperature in question. Treating them as constant is an approximation that is often useful over a moderate temperature range.

At a fixed temperature, a more positive $\Delta_r S^\circ$ makes $\ln K$ larger, and therefore makes $K$ larger. But the direction in which $K$ changes as temperature rises depends on the reaction enthalpy. The van ’t Hoff equation gives

$$\frac{d\ln K}{dT} = \frac{\Delta_r H^\circ}{RT^2},$$

where $\Delta_r H^\circ$ is evaluated at the temperature in question. So $K$ generally increases with temperature for an endothermic reaction ($\Delta_r H^\circ>0$) and decreases for an exothermic reaction ($\Delta_r H^\circ<0$). Entropy contributes to the value of $K$; it does not, by itself, set the sign of this temperature trend.

Comparing two temperatures

If $\Delta_r H^\circ$ is approximately constant between temperatures $T_1$ and $T_2$, integrate the van ’t Hoff relation:

$$\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta_r H^\circ}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right).$$

This form is convenient when you know one equilibrium constant and the reaction enthalpy. Use kelvins for both temperatures and use consistent energy units for $\Delta_r H^\circ$ and $R$.

Worked example

Suppose a reaction has $\Delta_r H^\circ=-50.0\,\text{kJ mol}^{-1}$ and $\Delta_r S^\circ=-100\,\text{J mol}^{-1}\text{K}^{-1}$. Estimate its equilibrium constant at 298 K and 350 K, treating both reaction quantities as constant. Convert the enthalpy to joules: $-50{,}000\,\text{J mol}^{-1}$.

At 298 K,

$$\begin{aligned} \ln K &= \frac{-100}{8.314} - \frac{-50{,}000}{8.314(298)} \\ &\approx -12.03 + 20.18 = 8.15,\\ K &\approx e^{8.15} \approx 3.5\times 10^3. \end{aligned}$$

At 350 K,

$$\begin{aligned} \ln K &= \frac{-100}{8.314} - \frac{-50{,}000}{8.314(350)} \\ &\approx -12.03 + 17.18 = 5.15,\\ K &\approx e^{5.15} \approx 1.7\times 10^2. \end{aligned}$$

The entropy change is negative, so its term lowers $\ln K$ at both temperatures. The reaction is exothermic, however, so raising the temperature lowers $K$ further. The equilibrium constant remains greater than 1 in both estimates, but its decrease indicates that equilibrium is less product-favored at 350 K than at 298 K. The value of $K$ describes the equilibrium position; it does not by itself give the amounts of substances in a particular mixture.

When the approximation needs adjustment

Reaction enthalpy and entropy can vary with temperature, especially over a wide range or near a phase transition. If the reaction heat-capacity change, $\Delta_r C_p^\circ$, is significant, the enthalpy changes with temperature and the simple two-temperature formula with constant enthalpy may be inaccurate. The differential van ’t Hoff equation still applies using $\Delta_r H^\circ(T)$ at each temperature. In that case, temperature-dependent thermodynamic data or an integrated equation that includes heat capacity is needed.

The thermodynamic $K$ is dimensionless and is defined using activities relative to standard states. Introductory problems often use concentration-based or pressure-based equilibrium constants; use the convention specified in the problem, and do not mix constants with different standard-state definitions.

Common mistakes

• Using Celsius in the equations. Temperature must be in kelvins, especially in terms containing $1/T$.
• Assuming entropy determines the heating trend. The sign of $\Delta_r H^\circ$ determines the trend when enthalpy is roughly constant.
• Mixing joules and kilojoules. If $R=8.314\,\text{J mol}^{-1}\text{K}^{-1}$, use enthalpy in joules per mole.
• Confusing a larger $K$ with a faster reaction. The equilibrium constant describes equilibrium composition, not reaction rate.