# Van 't Hoff equation
The van 't Hoff equation is a relation in chemical thermodynamics that describes how the equilibrium constant of a chemical reaction changes with temperature, connecting that change to the reaction's standard enthalpy change. In its differential form it reads
\[ \left( \frac{\partial \ln K}{\partial T} \right)_P = \frac{\Delta H^{\circ}}{R T^{2}} \]
where \( K \) is the thermodynamic equilibrium constant, \( T \) the absolute temperature, \( R \) the gas constant (8.314 J mol⁻¹ K⁻¹), and \( \Delta H^{\circ} \) the standard reaction enthalpy[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation) • [2](https://media.iupac.org/didac/Didac%20Eng/Didac02/Content/E11.htm). It is named after Jacobus Henricus "Henry" van 't Hoff Jr. (1852–1911)[3](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/10%3A_Chemical_Equilibrium/10.02%3A_Temperature_Dependence_of_Keq).
| Key fact | Detail |
|---|---|
| Differential form | \( (\partial \ln K / \partial T)_P = \Delta H^{\circ}/(RT^{2}) \), derived from the Gibbs-Helmholtz equation and \( \Delta G^{\circ} = -RT \ln K \)[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation) |
| Integrated form | \( \ln(K_2/K_1) = -(\Delta H^{\circ}/R)(1/T_2 - 1/T_1) \), valid when \( \Delta H^{\circ} \) is treated as constant over the temperature range[4](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation) |
| Linear form | \( \ln K = -(\Delta H^{\circ}/R)(1/T) + \Delta S^{\circ}/R \); a plot of \( \ln K \) against \( 1/T \) has slope \( -\Delta H^{\circ}/R \) and intercept \( \Delta S^{\circ}/R \)[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation) |
| Direction of shift | Endothermic reactions (\( \Delta H^{\circ} > 0 \)) have \( K \) increasing with temperature; exothermic reactions (\( \Delta H^{\circ} < 0 \)) have \( K \) decreasing[3](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/10%3A_Chemical_Equilibrium/10.02%3A_Temperature_Dependence_of_Keq) |
| Practical use | Translates a \( K \) value measured at one temperature to another, and lets \( \Delta H^{\circ} \) and \( \Delta S^{\circ} \) be determined from equilibrium data without calorimetry[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation) |
| Interpretation | Described as the mathematical expression of Le Chatelier's principle[3](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/10%3A_Chemical_Equilibrium/10.02%3A_Temperature_Dependence_of_Keq) |
## Derivation
The equation follows from the Gibbs-[Helmholtz equation](https://www.edgepedia.org/helmholtz-equation), which at constant pressure gives \( (\partial [\Delta G^{\circ}/T]/\partial T)_P = -\Delta H^{\circ}/T^{2} \). Substituting the equilibrium relation \( \Delta G^{\circ} = -RT \ln K \) and dividing by \( R \) yields the differential form above[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation). An equivalent route starts from \( \Delta G^{\circ}_{T_2}/T_2 - \Delta G^{\circ}_{T_1}/T_1 = \Delta H^{\circ}(1/T_2 - 1/T_1) \) and substitutes \( \Delta G^{\circ} = -RT \ln K \) at the two temperatures[4](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation).
**Integrated form.** If the temperature change is small enough that \( \Delta H^{\circ} \) can be considered constant, integration between two temperatures gives
\[ \ln \frac{K(T_2)}{K(T_1)} = -\frac{\Delta H^{\circ}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
This is the working form used to translate a \( K \) value at one temperature into a value at another[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation) • [4](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation).
## The van 't Hoff plot
Combining \( K = e^{-\Delta_r G^{\circ}/RT} \) with \( \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \) gives the linear relation
\[ \ln K = -\frac{\Delta H^{\circ}}{R} \cdot \frac{1}{T} + \frac{\Delta S^{\circ}}{R} \]
A plot of \( \ln K \) against \( 1/T \), called a van 't Hoff plot, should therefore be a straight line with slope \( -\Delta_r H^{\circ}/R \) and intercept \( \Delta_r S^{\circ}/R \). Endothermic reactions give plots with negative slope (rising \( K \) with \( T \)) and exothermic reactions give positive slope[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation). The enthalpy and entropy can thus be determined from equilibrium measurements alone, without calorimetry; the main assumption is that \( \Delta_r H^{\circ} \) and \( \Delta_r S^{\circ} \) depend only weakly on temperature, which is usually valid[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation).
## Physical meaning
The sign of \( \Delta H^{\circ} \) fixes the direction of the temperature shift. For an [exothermic reaction](https://www.edgepedia.org/exothermic-reaction) \( K \) decreases as temperature increases, and for an endothermic reaction \( K \) increases; the equation is accordingly described as the mathematical expression of [Le Chatelier's principle](https://www.edgepedia.org/le-chateliers-principle)[3](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/10%3A_Chemical_Equilibrium/10.02%3A_Temperature_Dependence_of_Keq). A temperature change differs from a pressure change in this respect: pressure alters the equilibrium composition but not the value of \( K \), whereas temperature changes the value of \( K \) itself[4](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation).
The equation resembles the Clausius-Clapeyron equation, which governs phase equilibrium between a gas and a condensed phase, but it applies to chemical equilibria generally[5](https://faculty.washington.edu/jcv2/CHEM452_WI2023/html/chemical_equilibrium.html).
## Worked example
A reaction has \( K_p = 0.0260 \) at 25 °C and \( \Delta H^{\circ}_{rxn} = 32.4 \) kJ/mol. With \( T_1 = 298 \) K and \( T_2 = 310 \) K (37 °C), the integrated equation gives \( \ln(K_2/0.0260) = -(32400 \text{ J/mol})/(8.314 \text{ J/(mol K)}) \times (1/310 - 1/298) \), so \( K_2 = 0.0431 \). The constant rises with temperature, as expected for an endothermic reaction[4](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation).
## Limitations and extensions
The integrated form assumes \( \Delta H^{\circ} \) is independent of temperature over the range considered, an assumption inherited from the derivation of the Gibbs-Helmholtz equation[4](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation). When more precision is required, the temperature variation of \( \Delta H^{\circ} \) can be corrected using heat capacity data[1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation).
The same relation appears in other settings under the same name. In protein biochemistry, equilibrium constants extracted from thermal denaturation curves yield an enthalpy termed the van 't Hoff enthalpy, \( \Delta H^{0}_{vHoff} \)[6](https://bio.libretexts.org/Bookshelves/Biochemistry/Fundamentals_of_Biochemistry_%28Jakubowski_and_Flatt%29/01%3A_Unit_I-_Structure_and_Catalysis/04%3A_The_Three-Dimensional_Structure_of_Proteins/4.12%3A_Laboratory_Determination_of_the_Thermodynamic_Parameters_for_Protein_Denaturation). In solubility thermodynamics, the temperature dependence of the saturated mole fraction of a solute is likewise expressed by a van 't Hoff-type equation derived by equating the chemical potential of a pure solid solute to that of the solute in its saturated solution[7](https://stax.strath.ac.uk/downloads/gf06g3139?locale=en).
## References
1. [26.7: The van 't Hoff Equation, Chemistry LibreTexts (Mark E. Tuckerman)](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%253A_Chemical_Equilibrium/26.07%253A_The_van_%27t_Hoff_Equation)
2. [The van 't Hoff equation, IUPAC Didactica](https://media.iupac.org/didac/Didac%20Eng/Didac02/Content/E11.htm)
3. [10.2: Temperature Dependence of Keq, Chemistry LibreTexts (Roberto Peverati)](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_%28Peverati%29/10%3A_Chemical_Equilibrium/10.02%3A_Temperature_Dependence_of_Keq)
4. [9.6: Temperature Dependence of Equilibrium Constants - the van 't Hoff Equation, Chemistry LibreTexts (Patrick Fleming)](https://chem.libretexts.org/Courses/Millersville_University/CHEM_341-_Physical_Chemistry_I/09%253A_Chemical_Equilibria/9.06%253A_Temperature_Dependence_of_Equilibrium_Constants_-_the_van_t_Hoff_Equation)
5. [6. Chemical Equilibrium, Thermodynamics and Kinetics, University of Washington (CHEM 452)](https://faculty.washington.edu/jcv2/CHEM452_WI2023/html/chemical_equilibrium.html)
6. [4.12: Laboratory Determination of the Thermodynamic Parameters for Protein Denaturation, Biology LibreTexts](https://bio.libretexts.org/Bookshelves/Biochemistry/Fundamentals_of_Biochemistry_%28Jakubowski_and_Flatt%29/01%3A_Unit_I-_Structure_and_Catalysis/04%3A_The_Three-Dimensional_Structure_of_Proteins/4.12%3A_Laboratory_Determination_of_the_Thermodynamic_Parameters_for_Protein_Denaturation)
7. [3.2 The van't Hoff Equation, University of Strathclyde thesis](https://stax.strath.ac.uk/downloads/gf06g3139?locale=en)
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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Thermodynamics and equilibrium › Chemical equilibrium › Equilibrium constant*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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