Gibbs–Helmholtz equation

Template:Short description The Gibbs–Helmholtz equation is a thermodynamic equation used to calculate changes in the Gibbs free energy of a system as a function of temperature. It was originally presented in an 1882 paper entitled "Die Thermodynamik chemischer Vorgänge" by Hermann von Helmholtz. It describes how the Gibbs free energy, which was presented originally by Josiah Willard Gibbs, varies with temperature.<ref>Template:Cite journal</ref> It was derived by Helmholtz first, and Gibbs derived it only 6 years later.<ref>Template:Cite journal</ref> The attribution to Gibbs goes back to Wilhelm Ostwald, who first translated Gibbs' monograph into German and promoted it in Europe.<ref>At the last paragraph on page 638, of

Bancroft, W. D. (1927). Review of: Thermodynamics for Students of Chemistry. By C. N. Hinshelwood. The Journal of Physical Chemistry, 31, 635-638.</ref><ref>Template:Cite journal</ref>

The equation is:<ref name="P">Physical chemistry, P. W. Atkins, Oxford University Press, 1978, Template:ISBN</ref>

where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all at constant pressure p. The equation states that the change in the G/T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor H/T².

Similar equations include<ref>Template:Cite book</ref>

	<math display="inline">U = -T^2\left(\frac{\partial}{\partial T}\frac FT\right)_V</math>		<math display="inline">F = -S^2\left(\frac{\partial}{\partial S}\frac US\right)_V</math>
<math>U = -P^2\left(\frac{\partial}{\partial P}\frac{H}{P}\right)_S</math>	<math>U</math>	<math>\leftrightarrow U-F = TS</math>	<math>F</math>	<math display="inline">F = -P^2\left(\frac{\partial}{\partial P}\frac GP\right)_T</math>
	<math>\updownarrow U-H = -PV</math>		<math>\updownarrow G-F = PV</math>
<math display="inline">H = -V^2\left(\frac{\partial}{\partial V}\frac UV\right)_S</math>	<math>H</math>	<math>\leftrightarrow G-H = -TS</math>	<math>G</math>	<math display="inline">G = -V^2\left(\frac{\partial}{\partial V}\frac{F}{V}\right)_T</math>
	<math display="inline">H = -T^2\left(\frac{\partial}{\partial T}\frac GT\right)_P</math>		<math display="inline">G = -S^2\left(\frac{\partial}{\partial S}\frac{H}{S}\right)_p</math>

Chemical reactions and work

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The typical applications of this equation are to chemical reactions. The equation reads:<ref>Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, Template:ISBN</ref>

<math>\left( \frac{\partial ( \Delta G^\ominus/T ) } {\partial T} \right)_p = - \frac {\Delta H^\ominus} {T^2}</math>

with ΔG as the change in Gibbs energy due to reaction, and ΔH as the enthalpy of reaction (often, but not necessarily, assumed to be independent of temperature). The o denotes the use of standard states, and particularly the choice of a particular standard pressure (1 bar), to calculate ΔG and ΔH.

Integrating with respect to T (again p is constant) yields:

<math> \frac{\Delta G^\ominus(T_2)}{T_2} - \frac{\Delta G^\ominus(T_1)}{T_1} = \Delta H^\ominus \left(\frac{1}{T_2} - \frac{1}{T_1}\right) </math>

This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T₂ with knowledge of just the standard Gibbs free energy change of formation and the standard enthalpy change of formation for the individual components.

Also, using the reaction isotherm equation,<ref>Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, Template:ISBN</ref> that is

<math>\frac{\Delta G^\ominus}{T} = -R \ln K </math>

which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.<ref>Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, Template:ISBN</ref>

Since the change in a system's Gibbs energy is equal to the maximum amount of non-expansion work that the system can do in a process, the Gibbs–Helmholtz equation may be used to estimate how much non-expansion work can be done by a chemical process as a function of temperature.<ref name="P Chem 1">Template:Cite book</ref> For example, the capacity of rechargeable electric batteries can be estimated as a function of temperature using the Gibbs–Helmholtz equation.<ref name="P Chem 2">Template:Cite book</ref>

Derivation

Background

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The definition of the Gibbs function is <math display="block">H = G + ST </math> where Template:Mvar is the enthalpy defined by: <math display="block">H = U + pV </math>

Taking differentials of each definition to find Template:Math and Template:Math, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes): <math display="block">dU = T\,dS - p\,dV </math> where Template:Mvar is the entropy, Template:Mvar is volume, (minus sign due to reversibility, in which Template:Math: work other than pressure-volume may be done and is equal to Template:Math) leads to the "reversed" form of the initial fundamental relation into a new master equation: <math display="block">dG = - S\,dT + V\,dp </math>

This is the Gibbs free energy for a closed system. The Gibbs–Helmholtz equation can be derived by this second master equation, and the chain rule for partial derivatives.<ref name="P" />

External links

• Gibbs–Helmholtz equation, by W. R. Salzman (2004).
• Gibbs-Helmholtz Equation, by P. Mander (2013)