D11.2 Effect of Temperature on Equilibrium

Recall that:

ΔrG° = ΔrH° – TΔrS°     and     ΔrG° = −RT(lnK°)

Therefore:

RT(lnK°) = ΔrH° – TΔrS°

This equation can be used to calculate K° at different temperatures, if we assume that ΔrH° and ΔrS° for a reaction have the same values at all temperatures. This is a good, but not perfect, assumption. It is not a good assumption if there is a phase change for a reactant or a product within the temperature range of interest.

Dividing both sides of the equation by -RT gives:

lnK° =  -\dfrac{{\Delta}_rH^{\circ}}{R}\left(\dfrac{1}{T}\right) +  \dfrac{{\Delta}_rS^{\circ}}{R}
y = mx + b

A plot of lnK° vs.  \frac{1}{T} is called a van’t Hoff plot. The graph has a slope of  -\frac{{\Delta}_rH^{\circ}}{R} . Therefore, if the concentrations of reactants and products are measured at various temperatures so that K° can be calculated at each temperature, the reaction’s ΔrH° can be obtained from a van’t Hoff plot.

Figure: Van’t Hoff plots. The reaction N2 + O2 ⇄ 2 NO has ΔrH° = 180.5 kJ/mol. For this endothermic reaction, as T increases (1/T decreases) the equilibrium constant increases. The reaction N2 + 3 H2 ⇄ 2 NH3 has ΔrH° = −92.2 kJ/mol. For this exothermic reaction, as T increases (1/T decreases) the equilibrium constant decreases.

Based on the equation for the van’t Hoff plot, an exothermic reaction (ΔrH° < 0) has K° decreasing with increasing temperature, and an endothermic reaction (ΔrH° > 0) has K° increasing with increasing temperature. The magnitude of ΔrH° will dictate how rapidly changes as a function of temperature.

For example, suppose that K°1 and K°2 are the equilibrium constants for a reaction at temperatures T1 and T2, respectively:

 \text{ln}K_1^{\circ} = -\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T_1}\right)\;+\;\dfrac{{\Delta}_{\text{r}}S^{\circ}}{R}
 \text{ln}K_2^{\circ} = -\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T_2}\right)\;+\;\dfrac{{\Delta}_{\text{r}}S^{\circ}}{R}

Subtracting the two equations yields:

 \begin{array}{rcl}\text{ln}K_2^{\circ}\;-\;\text{ln}K_1^{\circ} &=& \left(-\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T_2}\right)\;+\;\dfrac{{\Delta}_{\text{r}}S^{\circ}}{R}\right) - \left(-\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T_1}\right)\;+\;\dfrac{{\Delta}_{\text{r}}S^{\circ}}{R}\right)\\[1em] \text{ln}\left(\dfrac{K_2^{\circ}}{K_1^{\circ}}\right) &=& -\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T_2}\;-\;\dfrac{1}{T_1}\right)\\[1em] \text{ln}\left(\dfrac{K_2^{\circ}}{K_1^{\circ}}\right) &=& \dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T_1}\;-\;\dfrac{1}{T_2}\right) \end{array}

Thus calculating ΔrH° from tabulated ΔfH° values and measuring the equilibrium constant at one temperature would allow us to calculate the equilibrium constant at any other temperature (assuming that ΔrH° and ΔrS° are independent of temperature).

Using this equation, you can also estimate ΔrH° without constructing a van’t Hoff plot if K° was determined at only two temperatures (but the result will not be as accurate as a van’t Hoff plot constructed from numerous data points).

Left-click here for different forms of the equation

You can multiply both side of the equation by -1 and obtain:

 -\text{ln}\left(\dfrac{K_2^{\circ}}{K_1^{\circ}}\right) = -\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R} \left(\dfrac{1}{T_1}\;-\;\dfrac{1}{T_2}\right)

 \text{ln}\left(\dfrac{K_1^{\circ}}{K_2^{\circ}}\right) = \dfrac{{\Delta}_{\text{r}}H^{\circ}}{R} \left(\dfrac{1}{T_2}\;-\;\dfrac{1}{T_1}\right)

Whichever form of the equation you use, keep careful track of the subscripts on the variables K1°, T1, K2°, and T2.

Exercise: Temperature Dependence of the Equilibrium Constant

Left-click here to read more about Van’t Hoff plots and ΔrG° vs. T relationships

You may have noticed that when a reaction is product-favored or reactant-favored at all temperatures, it appears that ΔrG° and K° have opposite trends with respect to changes in temperature. This is a rather subtle and complex issue, and one which we will not delve deeply into in this course. But we can unravel this a little bit by considering what we already know.

The relationship between ΔrG° and K° has a temperature dependence:

ΔrG° = −RT(lnK°)

While ΔrG° itself has a temperature dependence:

ΔrG° = ΔrH° – TΔrS°

and K° itself also has a temperature dependence:

 \text{ln}K^{\circ} = -\dfrac{{\Delta}_{\text{r}}H^{\circ}}{R}\left(\dfrac{1}{T}\right)\;+\;\dfrac{{\Delta}_{\text{r}}S^{\circ}}{R}

Therefore, temperature dependence of ΔrG° is informed by the rising impact of the entropy term (ΔrS°). In contrast, temperature dependence of K° is informed by the shrinking impact of the enthalpy term (ΔrH°). Usually, the two impacts point in the same direction, but this is not necessarily the case in the always product-favored or always reactant-favored situations.

Keep in mind that these temperature relationships we are discussing are simplified: we are not accounting for the possible temperature dependence of ΔrH° and ΔrS° terms.

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