Van 't Hoff equation

The standard pressure, ${\displaystyle P^{0}}$, is used to define the reference state for the Van 't Hoff equation, which is [2][3]

where ln denotes the natural logarithm, ${\displaystyle K_{eq}}$ is the thermodynamic equilibrium constant, and R is the ideal gas constant. This equation is exact at any one temperature and all pressures, derived from the requirement that the Gibbs free energy of reaction be stationary in a state of chemical equilibrium.

In practice, the equation is often integrated between two temperatures under the assumption that the standard reaction enthalpy ${\displaystyle \Delta _{r}H^{\ominus }}$ is constant (and furthermore, this is also often assumed to be equal to its value at standard temperature). Since in reality ${\displaystyle \Delta _{r}H^{\ominus }}$ and the standard reaction entropy ${\displaystyle \Delta _{r}S^{\ominus }}$ do vary with temperature for most processes,[4] the integrated equation is only approximate. Approximations are also made in practice to the activity coefficients within the equilibrium constant.

A major use of the integrated equation is to estimate a new equilibrium constant at a new absolute temperature assuming a constant standard enthalpy change over the temperature range. To obtain the integrated equation, it is convenient to first rewrite the Van 't Hoff equation as[2]

${\displaystyle {\frac {d\ln K_{\mathrm {eq} }}{d{\frac {1}{T}}}}=-{\frac {\Delta _{r}H^{\ominus }}{R}}.}$

The definite integral between temperatures T₁ and T₂ is then

${\displaystyle \ln {\frac {K_{2}}{K_{1}}}={\frac {\Delta _{r}H^{\ominus }}{R}}\left({\frac {1}{T_{1}}}-{\frac {1}{T_{2}}}\right).}$

In this equation K₁ is the equilibrium constant at absolute temperature T₁, and K₂ is the equilibrium constant at absolute temperature T₂.

Combining the well-known formula for the Gibbs free energy of reaction

${\displaystyle \Delta _{r}G^{\ominus }=\Delta _{r}H^{\ominus }-T\Delta _{r}S^{\ominus },}$

where S is the entropy of the system, with the Gibbs free energy isotherm equation:[5]

${\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K_{\mathrm {eq} },}$

we obtain

${\displaystyle \ln K_{\mathrm {eq} }=-{\frac {\Delta _{r}H^{\ominus }}{RT}}+{\frac {\Delta _{r}S^{\ominus }}{R}}.}$

Differentiation of this expression with respect to the variable T while assuming that both ${\displaystyle \Delta _{r}H^{\ominus }}$and ${\displaystyle \Delta _{r}S^{\ominus }}$ are independent of T yields the Van 't Hoff equation. These assumptions are expected to break down somewhat for large temperature variations.

Provided that ${\displaystyle \Delta _{r}H^{\ominus }}$ and ${\displaystyle \Delta _{r}S^{\ominus }}$ are constant, the preceding equation gives ln K as a linear function of 1/T and hence is known as the linear form of the Van 't Hoff equation. Therefore, when the range in temperature is small enough that the standard reaction enthalpy and reaction entropy are essentially constant, a plot of the natural logarithm of the equilibrium constant versus the reciprocal temperature gives a straight line. The slope of the line may be multiplied by the gas constant R to obtain the standard enthalpy change of the reaction, and the intercept may be multiplied by R to obtain the standard entropy change.

The Van 't Hoff isotherm can be used to determine the temperature dependence of the Gibbs free energy of reaction for non-standard state reactions at a constant temperature:[6]

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} },}$

where ${\displaystyle \Delta _{\mathrm {r} }G}$ is the Gibbs free energy of reaction under non-standard states at temperature ${\displaystyle T}$, ${\displaystyle \Delta _{r}G^{\ominus }}$ is the Gibbs free energy for the reaction at ${\displaystyle (T,P^{0})}$, ${\displaystyle \xi }$ is the extent of reaction, and Q_r is the thermodynamic reaction quotient. Since ${\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K_{eq}}$, the temperature dependence of both terms can be described by Van t'Hoff equations as a function of T. This finds applications in the field of electrochemistry. particularly in the study of the temperature dependence of voltaic cells.

The isotherm can also be used at fixed temperature to describe the Law of Mass Action. When a reaction is at equilibrium, Q_r = K_eq and ${\displaystyle \Delta _{\mathrm {r} }G=0}$. Otherwise, the Van 't Hoff isotherm predicts the direction that the system must shift in order to achieve equilibrium; when Δ_rG < 0, the reaction moves in the forward direction, whereas when Δ_rG > 0, the reaction moves in the backwards direction. See Chemical equilibrium.

Van 't Hoff plot for an endothermic reaction

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

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

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

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

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

Van 't Hoff plot for an exothermic reaction

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

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

For an exothermic reaction Δ_rH < 0, so

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

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

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^⊖:

${\displaystyle \Delta _{r}H^{\ominus }=R{\frac {\ln K_{1}-\ln K_{2}}{{\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}}},}$

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.[9] 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.

Van 't Hoff analysis

In biological research, the Van 't Hoff plot is also called Van 't Hoff analysis.[10] 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.[11]

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.[12] 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.

Van 't Hoff plot in mechanism study

A chemical reaction may undergo different reaction mechanisms at different temperatures.[13]

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.

${\displaystyle {\begin{aligned}\Delta _{r}H_{1}&=-R\times {\text{slope}}_{1},&\Delta _{r}S_{1}&=R\times {\text{intercept}}_{1};\\[5pt]\Delta _{r}H_{2}&=-R\times {\text{slope}}_{2},&\Delta _{r}S_{2}&=R\times {\text{intercept}}_{2}.\end{aligned}}}$

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

Temperature-dependent Van 't Hoff plot

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:[14]

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

where

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

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

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):

${\displaystyle {\frac {d}{dT}}\ln \mathrm {CMC} ={\frac {\Delta H_{\mathrm {m} }^{\ominus }}{RT^{2}}}.}$

However, the relation loses its validity when the aggregation number is also temperature-dependent, and the following relation should be used instead:[15]

${\displaystyle RT^{2}\left({\frac {\partial }{\partial T}}\ln \mathrm {CMC} \right)_{P}=-\Delta _{r}H_{\mathrm {m} }^{\ominus }(N)+T\left({\frac {\partial }{\partial N}}\left(G_{N+1}-G_{N}\right)\right)_{T,P}\left({\frac {\partial N}{\partial T}}\right)_{P},}$

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[16] or polyoxypropylene–polyoxyethylene block copolymers (Poloxamers, Pluronics, Synperonics).[17] The extended equation can be exploited for the extraction of aggregation numbers of self-assembled micelles from differential scanning calorimetric thermograms.[18]