Editing Gibbs–Helmholtz equation

{{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"
|+
|
|<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==

{{Main|Thermochemistry}}

The typical applications of this equation are to [[chemical reaction]]s. The equation reads:<ref>Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, {{ISBN|0-356-03736-3}}</ref>

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

with Δ''G'' as the change in Gibbs energy due to reaction, and Δ''H'' as the [[enthalpy of reaction]] (often, but not necessarily, assumed to be independent of temperature). The <s>o</s> denotes the use of [[Standard state|standard states]], and particularly the choice of a particular standard pressure (1 bar), to calculate Δ''G'' and Δ''H''.

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 ''T''<sub>2</sub> 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, {{ISBN|0-19-855148-7}}</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, {{ISBN|0-356-03736-3}}</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">{{cite book |last1=Gerasimov |first1=Ya |title=Physical Chemistry Volume 1 |date=1978 |publisher=MIR Publishers |location=Moscow  |page=118 |edition=1st}}</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">{{cite book |last1=Gerasimov |first1=Ya |title=Physical Chemistry Volume 2 |date=1978 |publisher=MIR Publishers |location=Moscow  |page=497 |edition=1st}}</ref>

==Derivation==

===Background===

{{main|Defining equation (physical chemistry)|Enthalpy|Thermodynamic potential}}

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

Taking [[differential of a function|differentials]] of each definition to find {{math|''dH''}} and {{math|''dG''}}, then using the [[fundamental thermodynamic relation]] (always true for [[Reversible process (thermodynamics)|reversible]] or [[Irreversible process|irreversible]] [[thermodynamic process|processes]]):
<math display="block">dU = T\,dS - p\,dV </math>
where {{mvar|S}} is the [[entropy]], {{mvar|V}} is [[volume]], (minus sign due to reversibility, in which {{math|1=''dU'' = 0}}: work other than pressure-volume may be done and is equal to {{math|−''pV''}}) 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" />

{{math proof|title=Derivation
|proof=
Starting from the equation
<math display="block">dG =  - S\,dT + V\,dp</math>
for the differential of ''G'', and remembering <math display="block">H = G + T\,S ,</math> one computes the differential of the ratio {{math|''G''/''T''}} by applying the [[product rule]] of [[Differentiation (mathematics)|differentiation]] in the version for differentials:
<math display="block">\begin{align}
  d\left(\frac{G}{T}\right)
    &= \frac{T\, dG - G\, dT}{T^2} 
     = \frac{T\, (-S \,dT + V \, dp) - G\, dT}{T^2} \\
    &= \frac{-T\, S \,dT -G\,dT + T\, V \, dp}{T^2} 
     = \frac{-(G + T\,S)\,dT + T\, V \, dp}{T^2} \\
    &= \frac{-H \,dT + T\,V\,dp}{T^2} 
\end{align}\,\!</math>

Therefore, 
<math display="block">d\left(\frac{G}{T}\right) = -\frac{H}{T^2}\, dT + \frac{V}{T}\, dp \,\!</math>

A comparison with the general expression for a total differential
<math display="block">d\left(\frac{G}{T}\right) =
  \left( \frac{\partial(G/T)}{\partial T} \right)_p \, dT
  + \left( \frac{\partial(G/T)}{\partial p} \right)_T \, dp</math>
gives the change of {{math|''G''/''T''}} with respect to {{mvar|T}} at constant [[pressure]] (i.e. when {{math|1=''dp'' = 0}}), the Gibbs–Helmholtz equation:
<math display="block">\left( \frac{\partial(G/T)}{\partial T} \right)_p = -\frac{H}{T^2} </math>
}}

==Sources==

{{reflist}}

==External links==
* [https://web.archive.org/web/20151007133812/http://www.chem.arizona.edu/~salzmanr/480a/480ants/gibshelm/gibshelm.html Gibbs–Helmholtz equation, by W. R. Salzman (2004)].
* [https://carnotcycle.wordpress.com/2013/12/01/the-gibbs-helmholtz-equation/#:~:text=The%20Gibbs%2DHelmholtz%20equation%20was,the%20Thermodynamics%20of%20Chemical%20Processes Gibbs-Helmholtz Equation, by P. Mander (2013)]
{{DEFAULTSORT:Gibbs-Helmholtz equation}}
[[Category:Thermodynamic equations]]
[[Category:Hermann von Helmholtz]]