# van 't Hoff equation

The **van 't Hoff equation** relates the change in the equilibrium constant, *K*_{eq}, of a chemical reaction to the change in temperature, *T*, given the standard enthalpy change, Δ*H*^{⊖}, for the process. It was proposed by Dutch chemist Jacobus Henricus van 't Hoff in 1884 in his book *Études de dynamique chimique* (*Studies in Dynamic Chemistry*).[1] This equation is sometimes also referred to as the Vukančić–Vuković equation.[2][3][4]

The van 't Hoff equation has been widely utilized to explore the changes in state functions in a thermodynamic system. The **van 't Hoff plot**, which is derived from this equation, is especially effective in estimating the change in enthalpy, or total energy, and entropy, or amount of disorder, of a chemical reaction.

## Equation

### Under standard conditions

Under standard conditions, the van 't Hoff equation is[5][6]

where *ln* denotes natural logarithm and *R* is the ideal gas constant. This equation is exact at any one temperature. In practice, the equation is often integrated between two temperatures under the assumption that the reaction enthalpy Δ*H* is constant. Since in reality Δ*H* and the reaction entropy Δ*S* do vary with temperature for most processes,[7] the integrated equation is only approximate.

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[5]

The definite integral between temperatures *T*_{1} and *T*_{2} is then

In this equation *K*_{1} is the equilibrium constant at absolute temperature *T*_{1}, and *K*_{2} is the equilibrium constant at absolute temperature *T*_{2}.

### Development from thermodynamics

Combining the definition of Gibbs free energy:

where *S* is the entropy of the system, and the Gibbs free energy isotherm equation:[8]

we obtain

Differentiation of this expression with respect to the variable T yields the van 't Hoff equation.

Provided that Δ*H*^{⊖} and Δ*S*^{⊖} 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 enthalpy and entropy changes 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.

### van 't Hoff isotherm

The Gibbs free energy changes with the temperature and pressure of the thermodynamic system. The **van 't Hoff isotherm** can be used to determine the Gibbs free energy for non-standard state reactions at a constant temperature:[9]

where Δ_{r}*G* is the Gibbs free energy for the reaction, and *Q*_{r} is the reaction quotient. When a reaction is at equilibrium, *Q*_{r} = *K*_{eq}. The van 't Hoff isotherm can help estimate the equilibrium reaction shift. When Δ_{r}*G* < 0, the reaction moves in the forward direction. When Δ_{r}*G* > 0, the reaction moves in the backwards directions. See Chemical equilibrium.

## 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

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*H*/*R* is the slope, and Δ*S*/*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.[10][11] Knowing the slope and intercept from the van 't Hoff plot, the enthalpy and entropy of a reaction can be easily obtained using

The van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. 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:

for an endothermic reaction, Δ*H* > 0 (and the gas constant *R* > 0), so

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:

from an exothermic reaction, Δ*H* < 0, so

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

### Error propagation

Using the fact that Δ*G*^{⊖} = −*RT* ln *K* = Δ*H*^{⊖} − *T*Δ*S*^{⊖}, it would appear that two measurements of *K* would suffice to be able to obtain a value of Δ*H*^{⊖}:

where *K*_{1} and *K*_{2} are the equilibrium constant values obtained at temperatures *T*_{1} and *T*_{2} respectively. The precision of Δ*H*^{⊖} values obtained in this way is highly dependent on the precision of the equilibrium constant values. A typical pair of temperatures might be 25 and 35 °C (298 and 308 K). For these temperatures

Inserting this value into the expression for Δ*H*^{⊖}:

Now, error propagation shows that the error on Δ*H*^{⊖} will be about 76 kJ/mol times the error on (ln *K*_{1} − ln *K*_{2}), or about 110 kJ/mol times the error on the ln *K* values. Assume for example that the error on each ln *K* is *σ* ≈ 0.05, a small but reasonable value. The error on Δ*H*^{⊖} will be about 5 kJ/mol. So, even though the individual stability constants were determined with good precision, the enthalpy calculated in this way is subject to a significant error.

The entropy will then be obtained from Δ*S*^{⊖} = 1/*T*(Δ*H*^{⊖} + *RT* ln *K*). In this expression the error on the second term is negligible compared to the error on the first term. The magnifying factor is then 76 kJ/mol ÷ 298 K, so for an error of 0.05 in the logarithms the error on Δ*S*^{⊖} will be of the order of 17 J/(K mol).

When equilibrium constants are measured at three or more temperatures, values of Δ*H*^{⊖} will be obtained by straight-line fitting. In this case the error on the standard enthalpy will be magnified to a somewhat lesser, but still substantial, extent.

## 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.[12] It is most effective in determining the favored product in a reaction.

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.[13] 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.[14]

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.

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

### Temperature dependence

The van 't Hoff plot is linear based on the tacit assumption that the enthalpy and entropy are constant with temperature changes. 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*/*T*^{2} 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:[15]

where

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 Δ*H*^{⊖}_{m} of surfactants from the temperature dependence of the critical micelle concentration (CMC):

However, the relation loses its validity when the aggregation number is also temperature-dependent, and the following relation should be used instead:[16]

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[17] or polyoxypropylene–polyoxyethylene block copolymers (Poloxamers, Pluronics, Synperonics).[18] The extended equation can be exploited for the extraction of aggregation numbers of self-assembled micelles from differential scanning calorimetric thermograms.[19]

## See also

## References

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