Van 't Hoff equation

Van 't Hoff plot

For a reversible reaction, the equilibrium constant can be measured at a variety of temperatures. This data can be plotted on a graph with ln K_eq on the y-axis and ⁠1/T⁠ on the x axis. The data should have a linear relationship, the equation for which can be found by fitting the data using the linear form of the Van 't Hoff equation

${\displaystyle \ln K_{\mathrm{e}\mathrm{q}}=-\frac{{\mathrm{\Delta }}_r{H}^{\ominus }}{RT}+\frac{{\mathrm{\Delta }}_r{S}^{\ominus }}{R}.}$

This graph is called the "Van 't Hoff plot" and is widely used to estimate the enthalpy and entropy of a chemical reaction. From this plot, −⁠Δ_rH/R⁠ is the slope, and ⁠Δ_rS/R⁠ is the intercept of the linear fit.

By measuring the equilibrium constant, K_eq, at different temperatures, the Van 't Hoff plot can be used to assess a reaction when temperature changes. Knowing the slope and intercept from the Van 't Hoff plot, the enthalpy and entropy of a reaction can be easily obtained using

${\displaystyle \begin{array}{cc}{\mathrm{\Delta }}_{r}H& =-R\times \text{slope},\\ {\mathrm{\Delta }}_{r}S& =R\times \text{intercept}.\end{array}}$

The Van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. This change in enthalpy can be positive or negative, leading to two major forms of the Van 't Hoff plot.

Endothermic reactions

For an endothermic reaction, heat is absorbed, making the net enthalpy change positive. Thus, according to the definition of the slope:

${\displaystyle \text{slope}=-\frac{{\mathrm{\Delta }}_{r}H}{R},}$

When the reaction is endothermic, Δ_rH > 0 (and the gas constant R > 0), so

${\displaystyle \text{slope}=-\frac{{\mathrm{\Delta }}_{r}H}{R}<0.}$

Thus, for an endothermic reaction, the Van 't Hoff plot should always have a negative slope.

Exothermic reactions

For an exothermic reaction, heat is released, making the net enthalpy change negative. Thus, according to the definition of the slope:

${\displaystyle \text{slope}=-\frac{{\mathrm{\Delta }}_{r}H}{R},}$

For an exothermic reaction Δ_rH < 0, so

${\displaystyle \text{slope}=-\frac{{\mathrm{\Delta }}_{r}H}{R}>0.}$

Thus, for an exothermic reaction, the Van 't Hoff plot should always have a positive slope.

Error propagation

At first glance, using the fact that Δ_rG^⊖ = −RT ln K = Δ_rH^⊖ − TΔ_rS^⊖ it would appear that two measurements of K would suffice to be able to obtain an accurate value of Δ_rH^⊖:

${\displaystyle {\mathrm{\Delta }}_r{H}^{\ominus }=R\frac{\ln K_1-\ln K_2}{\frac{1}{T_2}-\frac{1}{T_1}},}$

where K₁ and K₂ are the equilibrium constant values obtained at temperatures T₁ and T₂ respectively. However, the precision of Δ_rH^⊖ values obtained in this way is highly dependent on the precision of the measured equilibrium constant values.

The use of error propagation shows that the error in Δ_rH^⊖ will be about 76 kJ/mol times the experimental uncertainty in (ln K₁ − ln K₂), or about 110 kJ/mol times the uncertainty in the ln K values. Similar considerations apply to the entropy of reaction obtained from Δ_rS^⊖ = ⁠1/T⁠(ΔH^⊖ + RT ln K).

Notably, when equilibrium constants are measured at three or more temperatures, values of Δ_rH^⊖ and Δ_rS^⊖ are often obtained by straight-line fitting. The expectation is that the error will be reduced by this procedure, although the assumption that the enthalpy and entropy of reaction are constant may or may not prove to be correct. If there is significant temperature dependence in either or both quantities, it should manifest itself in nonlinear behavior in the Van 't Hoff plot; however, more than three data points would presumably be needed in order to observe this.

Applications of the Van 't Hoff plot

Van 't Hoff analysis

In biological research, the Van 't Hoff plot is also called Van 't Hoff analysis. It is most effective in determining the favored product in a reaction. It may obtain results different from direct calorimetry such as differential scanning calorimetry or isothermal titration calorimetry due to various effects other than experimental error.

Assume two products B and C form in a reaction:

a A + d D → b B,

a A + d D → c C.

In this case, K_eq can be defined as ratio of B to C rather than the equilibrium constant.

When ⁠B/C⁠ > 1, B is the favored product, and the data on the Van 't Hoff plot will be in the positive region.

When ⁠B/C⁠ < 1, C is the favored product, and the data on the Van 't Hoff plot will be in the negative region.

Using this information, a Van 't Hoff analysis can help determine the most suitable temperature for a favored product.

In 2010, a Van 't Hoff analysis was used to determine whether water preferentially forms a hydrogen bond with the C-terminus or the N-terminus of the amino acid proline. The equilibrium constant for each reaction was found at a variety of temperatures, and a Van 't Hoff plot was created. This analysis showed that enthalpically, the water preferred to hydrogen bond to the C-terminus, but entropically it was more favorable to hydrogen bond with the N-terminus. Specifically, they found that C-terminus hydrogen bonding was favored by 4.2–6.4 kJ/mol. The N-terminus hydrogen bonding was favored by 31–43 J/(K mol).

This data alone could not conclude which site water will preferentially hydrogen-bond to, so additional experiments were used. It was determined that at lower temperatures, the enthalpically favored species, the water hydrogen-bonded to the C-terminus, was preferred. At higher temperatures, the entropically favored species, the water hydrogen-bonded to the N-terminus, was preferred.

Mechanistic studies

A chemical reaction may undergo different reaction mechanisms at different temperatures.

In this case, a Van 't Hoff plot with two or more linear fits may be exploited. Each linear fit has a different slope and intercept, which indicates different changes in enthalpy and entropy for each distinct mechanisms. The Van 't Hoff plot can be used to find the enthalpy and entropy change for each mechanism and the favored mechanism under different temperatures.

${\displaystyle \begin{array}{cccc}{\mathrm{\Delta }}_r{H}_1& =-R\times {\text{slope}}_1,& {\mathrm{\Delta }}_r{S}_1& =R\times {\text{intercept}}_1;\\ {\mathrm{\Delta }}_r{H}_2& =-R\times {\text{slope}}_2,& {\mathrm{\Delta }}_r{S}_2& =R\times {\text{intercept}}_2.\end{array}}$

In the example figure, the reaction undergoes mechanism 1 at high temperature and mechanism 2 at low temperature.

Temperature dependence

If the enthalpy and entropy are roughly constant as temperature varies over a certain range, then the Van 't Hoff plot is approximately linear when plotted over that range. However, in some cases the enthalpy and entropy do change dramatically with temperature. A first-order approximation is to assume that the two different reaction products have different heat capacities. Incorporating this assumption yields an additional term ⁠c/T²⁠ in the expression for the equilibrium constant as a function of temperature. A polynomial fit can then be used to analyze data that exhibits a non-constant standard enthalpy of reaction:

${\displaystyle \ln K_{\mathrm{e}\mathrm{q}}=a+\frac{b}{T}+\frac{c}{T^2},}$

where

${\displaystyle \begin{array}{cc}{\mathrm{\Delta }}_{r}H& =-R\left(b+\frac{2c}{T}\right),\\ {\mathrm{\Delta }}_{r}S& =R\left(a-\frac{c}{T^2}\right).\end{array}}$

Thus, the enthalpy and entropy of a reaction can still be determined at specific temperatures even when a temperature dependence exists.

Surfactant self-assembly

The Van 't Hoff relation is particularly useful for the determination of the micellization enthalpy ΔH^⊖
_m of surfactants from the temperature dependence of the critical micelle concentration (CMC):

${\displaystyle \frac{d}{dT}\ln \mathrm{C}\mathrm{M}\mathrm{C}=\frac{\mathrm{\Delta }{H}_{\mathrm{m}}^{\ominus }}{R{T}^2}.}$

However, the relation loses its validity when the aggregation number is also temperature-dependent, and the following relation should be used instead:

${\displaystyle R{T}^2{\left(\frac{\mathrm{\partial }}{\mathrm{\partial }T}\ln \mathrm{C}\mathrm{M}\mathrm{C}\right)}_P=-{\mathrm{\Delta }}_r{H}_{\mathrm{m}}^{\ominus }(N)+T{\left(\frac{\mathrm{\partial }}{\mathrm{\partial }N}\left(G_{N+1}-G_N\right)\right)}_{T,P}{\left(\frac{\mathrm{\partial }N}{\mathrm{\partial }T}\right)}_P,}$

with G_{N + 1} and G_N being the free energies of the surfactant in a micelle with aggregation number N + 1 and N respectively. This effect is particularly relevant for nonionic ethoxylated surfactants or polyoxypropylene–polyoxyethylene block copolymers (Poloxamers, Pluronics, Synperonics). The extended equation can be exploited for the extraction of aggregation numbers of self-assembled micelles from differential scanning calorimetric thermograms.