Editing Gibbs–Helmholtz equation (section)

{{Short description|A thermodynamic equation}}
The '''Gibbs–Helmholtz equation''' is a [[thermodynamics|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 "[[Thermodynamik chemischer Vorgänge|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>{{cite journal |last1=von Helmholtz |first1=Hermann |title=Die Thermodynamik chemischer Vorgange |journal=Ber. KGL. Preuss. Akad. Wiss. Berlin |date=1882 |volume=I |pages=22–39}}</ref> It was derived by [[Hermann von Helmholtz|Helmholtz]] first, and Gibbs derived it only 6 years later.<ref>{{Cite journal |last=Jensen |first=William B. |date=2016-01-27 |title=Vignettes in the history of chemistry. 1. What is the origin of the Gibbs–Helmholtz equation? |url=https://doi.org/10.1007/s40828-015-0019-8 |journal=ChemTexts |language=en |volume=2 |issue=1 |pages=1 |doi=10.1007/s40828-015-0019-8 |issn=2199-3793|doi-access=free }}</ref> The attribution to Gibbs goes back to [[Wilhelm Ostwald]], who first translated [[On the Equilibrium of Heterogeneous Substances|Gibbs' monograph]] into German and promoted it in Europe.<ref>At the last paragraph on page 638, of 

Bancroft, W. D. (1927). ''[https://books.google.com/books?id=umdGAQAAIAAJ&dq=%22Helmholtz+did+deduce+and+which+Gibbs+could+have%22&pg=PA638 Review of: Thermodynamics for Students of Chemistry. By C. N. Hinshelwood]''. The Journal of Physical Chemistry, 31, 635-638.</ref><ref>{{Cite journal |last=Daub |first=Edward E. |date=December 1976 |title=Gibbs phase rule: A centenary retrospect |url=https://pubs.acs.org/doi/abs/10.1021/ed053p747 |journal=Journal of Chemical Education |language=en |volume=53 |issue=12 |pages=747 |doi=10.1021/ed053p747 |issn=0021-9584|url-access=subscription }}</ref>

The equation is:<ref name="P">Physical chemistry, [[P. W. Atkins]], Oxford University Press, 1978, {{ISBN|0-19-855148-7}}</ref>

{{Equation box 1
|indent =:
|equation = <math>\left( \frac{\partial \left( \frac{G} {T} \right) } {\partial T} \right)_p = - \frac {H} {T^2},</math>
|border colour = #50C878
|background colour = #ECFCF4
}}

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/T''<sup>2</sup>.

Similar equations include<ref>{{Cite book |last=Pippard |first=Alfred B. |title=Elements of classical thermodynamics: for advanced students of physics |date=1981 |publisher=Univ. Pr |isbn=978-0-521-09101-5 |edition=Repr |location=Cambridge |chapter=5: Useful ideas}}</ref>
{| class="wikitable"
|+
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|<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>
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|<math>\updownarrow G-F = PV</math>
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|-
|<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>
|-
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|<math display="inline">H = -T^2\left(\frac{\partial}{\partial T}\frac GT\right)_P</math>
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|<math display="inline">G = -S^2\left(\frac{\partial}{\partial S}\frac{H}{S}\right)_p</math>
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|}