# Gibbs–Helmholtz equation

The Gibbs–Helmholtz equation is a thermodynamic relation that gives the temperature dependence of the Gibbs energy at constant pressure, linking the change in Gibbs energy \( \Delta G \) to the enthalpy change \( \Delta H \) and the temperature derivative of \( \Delta G \). In its common form for a process or reaction,

\[ \Delta G = \Delta H + T \left( \frac{\partial \Delta G}{\partial T} \right)_{p} \]

where \( \Delta G \), \( \Delta H \), and \( \Delta S \) are the Gibbs energy, enthalpy, and entropy changes of the reaction, \( T \) is the temperature, and \( p \) is the pressure. Since \( -\Delta S = \left( \partial \Delta G / \partial T \right)_{p} \), the equation is equivalently a statement that \( \Delta G = \Delta H - T \Delta S \) expressed in differential form.<sup>[1](https://doi.org/10.1007/s40828-014-0002-9)</sup>

| Key fact | Detail |
|---|---|
| Differential form | \( \Delta G = \Delta H + T (\partial \Delta G / \partial T)_p \), with \( -\Delta S = (\partial \Delta G / \partial T)_p \)<sup>[1](https://doi.org/10.1007/s40828-014-0002-9)</sup> |
| \( G/T \) form | The temperature dependence of \( G/T \) at constant pressure equals \( -H/T^2 \)<sup>[2](https://www.youtube.com/watch?v=7EuBgXpekv4)</sup> |
| Integrated form | If \( \Delta H \) is independent of \( T \): \( \Delta G_2/T_2 - \Delta G_1/T_1 = \Delta H (1/T_2 - 1/T_1) \)<sup>[3](https://archive.nptel.ac.in/content/storage2/courses/122101001/Slide/lect21/ch21_28.htm)</sup> |
| Electrochemical form | \( E_{\text{cell}} = -\Delta H/(nF) + T (\partial E_{\text{cell}} / \partial T)_p \)<sup>[1](https://doi.org/10.1007/s40828-014-0002-9)</sup> |
| Typical temperature coefficient | Zero-current cell potential coefficients usually lie in the range \( 10^{-4} \) to \( 10^{-5} \) V K<sup>−1</sup><sup> • </sup><sup>[4](https://chemistry.tcd.ie/assets/pdf/jf/2011-2012/MEGL%20JF%20Lecture%20Notes/JF%20Chemistry%201101%202011%20%20Lectures%2016-18.pdf)</sup> |
| Practical use | Allows \( \Delta G \) to be obtained from a measurement of \( \Delta H \); most useful for phase changes and chemical reactions at constant pressure<sup>[5](https://gioretikto.github.io/chemistry/thermodynamics/Gibbs_Helmholtz.html)</sup> |

## Forms of the equation

The equation is often written for the ratio \( G/T \) rather than for \( G \) itself. In that form it expresses the temperature dependence of \( G/T \) at constant pressure, where \( G/T \) is a composite function of temperature because \( G \) itself also depends on temperature. The derivative is the negative enthalpy divided by the temperature squared,<sup>[2](https://www.youtube.com/watch?v=7EuBgXpekv4)</sup><sup> • </sup><sup>[5](https://gioretikto.github.io/chemistry/thermodynamics/Gibbs_Helmholtz.html)</sup>

\[ \left( \frac{\partial (G/T)}{\partial T} \right)_{p} = -\frac{H}{T^{2}} \]

Applied to a change, such as a phase change or a chemical reaction at constant pressure, the equation holds for both the final and initial states. Writing \( \Delta G = G_{f} - G_{i} \) and applying the relation to each state gives the corresponding equation for \( \Delta G \) and \( \Delta H \).<sup>[6](https://staff.um.edu.mt/jgri1/teaching/che2372/notes/04/g_props/g_props.html)</sup>

**Integrated form.** If the enthalpy change is independent of temperature, the \( G/T \) form can be integrated between two temperatures \( T_1 \) and \( T_2 \) to give<sup>[3](https://archive.nptel.ac.in/content/storage2/courses/122101001/Slide/lect21/ch21_28.htm)</sup>

\[ \frac{\Delta G_{2}}{T_{2}} - \frac{\Delta G_{1}}{T_{1}} = \Delta H \left( \frac{1}{T_{2}} - \frac{1}{T_{1}} \right) \]

The same formula applies to standard quantities \( \Delta G^{\circ} \). This integrated version is the route by which the equation leads to the temperature dependence of equilibrium constants and of freezing point depression constants in dilute solutions.<sup>[2](https://www.youtube.com/watch?v=7EuBgXpekv4)</sup>

## Behavior it describes

Because the entropy \( S \) is positive, the Gibbs energy of a system decreases as temperature increases at constant pressure. The steepness of that decrease is set by the entropy: the slope of a \( G \) versus \( T \) plot is greatest for a gas, then a liquid, then a solid.<sup>[6](https://staff.um.edu.mt/jgri1/teaching/che2372/notes/04/g_props/g_props.html)</sup> This is why the equation is most useful for changes, including changes of phase and chemical reactions, at constant pressure.<sup>[6](https://staff.um.edu.mt/jgri1/teaching/che2372/notes/04/g_props/g_props.html)</sup>

The equation is also valued in experimental thermodynamics because it allows \( \Delta G \) to be obtained from a measurement of \( \Delta H \).<sup>[5](https://gioretikto.github.io/chemistry/thermodynamics/Gibbs_Helmholtz.html)</sup>

## Application in electrochemistry

For a galvanic cell, the Gibbs energy change of the cell reaction is related to the measured cell voltage by \( \Delta G = -nFE \), where \( n \) is the number of electrons transferred and \( F \) is the [Faraday constant](https://www.edgepedia.org/faraday-constant). Substituting this into the Gibbs–Helmholtz equation gives<sup>[1](https://doi.org/10.1007/s40828-014-0002-9)</sup>

\[ E_{\text{cell}} = -\frac{\Delta H}{nF} + T \left( \frac{\partial E_{\text{cell}}}{\partial T} \right)_{p} \]

This form turns voltage measurements into a calorimeter. Measuring the zero-current cell potential \( E \) as a function of temperature gives the temperature coefficient \( (\partial E / \partial T)_p \) from experimental \( E = E(T) \) data, from which the reaction enthalpy \( \Delta H \) and, once \( \Delta H \) and \( \Delta G \) are known, the reaction entropy \( \Delta S \) can be evaluated for the cell reaction.<sup>[4](https://chemistry.tcd.ie/assets/pdf/jf/2011-2012/MEGL%20JF%20Lecture%20Notes/JF%20Chemistry%201101%202011%20%20Lectures%2016-18.pdf)</sup> Typical values of the temperature coefficient of the zero-current cell potential lie in the range \( 10^{-4} \) to \( 10^{-5} \) V K<sup>−1</sup>, and the measured \( E(T) \) curve is commonly fitted to a polynomial in \( T - T_0 \) about a reference temperature such as 298 K, with coefficients that may be positive or negative.<sup>[4](https://chemistry.tcd.ie/assets/pdf/jf/2011-2012/MEGL%20JF%20Lecture%20Notes/JF%20Chemistry%201101%202011%20%20Lectures%2016-18.pdf)</sup> The equation likewise links the temperature coefficient of the [Galvani potential](https://www.edgepedia.org/galvani-potential) of individual electrodes to the energy effects or entropy changes of the electrode reactions occurring at them.<sup>[7](https://users.encs.concordia.ca/~tmg/images/9/9d/Fundamentals_of_Electrochemistry_2nd_Ed.pdf)</sup>

For exothermic reactions \( \Delta H < 0 \), and for endothermic reactions \( \Delta H > 0 \).<sup>[1](https://doi.org/10.1007/s40828-014-0002-9)</sup>

## Historical note: the endothermic cell

Before 1897 there had been no experimental evidence that an endothermic reaction can proceed spontaneously. That changed when István Bugarszky (1868–1941) discovered a galvanic cell whose cell reaction was endothermic, built from the electrode sequence Hg | Hg₂Cl₂ | KCl | KOH | Hg₂O | Hg.<sup>[1](https://doi.org/10.1007/s40828-014-0002-9)</sup> In such a cell the electrical work delivered exceeds the magnitude of the enthalpy change, which is possible only because the entropy term \( T \Delta S \) contributes to the driving force, exactly the distinction the Gibbs–Helmholtz equation makes explicit.

## References

1. [Crossing the bridge between thermodynamics and electrochemistry. From the potential of the cell reaction to the electrode potential, ChemTexts (Springer)](https://doi.org/10.1007/s40828-014-0002-9)
2. [Chemical Thermodynamics 6.8 – Gibbs-Helmholtz Equation, TMP Chem lecture](https://www.youtube.com/watch?v=7EuBgXpekv4)
3. [Lecture 21: Review of Thermodynamics, NPTEL course notes](https://archive.nptel.ac.in/content/storage2/courses/122101001/Slide/lect21/ch21_28.htm)
4. [JF Chemistry 1101: Basic Electrochemistry, Lectures 16–18, Trinity College Dublin](https://chemistry.tcd.ie/assets/pdf/jf/2011-2012/MEGL%20JF%20Lecture%20Notes/JF%20Chemistry%201101%202011%20%20Lectures%2016-18.pdf)
5. [The Gibbs–Helmholtz equation, chemistry notes](https://gioretikto.github.io/chemistry/thermodynamics/Gibbs_Helmholtz.html)
6. [Properties of the Gibbs free energy, University of Malta teaching notes](https://staff.um.edu.mt/jgri1/teaching/che2372/notes/04/g_props/g_props.html)
7. [Fundamentals of Electrochemistry, 2nd edition](https://users.encs.concordia.ca/~tmg/images/9/9d/Fundamentals_of_Electrochemistry_2nd_Ed.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Laws, states, and potentials › Thermodynamic potentials and free energy › Gibbs free energy › Temperature and pressure dependence: Gibbs–Helmholtz relations*

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