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| The '''Gibbs–Helmholtz equation''' is a [[thermodynamics|thermodynamic]] [[equation]] useful for calculating changes in the [[Gibbs energy]] of a system as a function of [[temperature]]. It is named after [[Josiah Willard Gibbs]] and [[Hermann von Helmholtz]].
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| ==Equation==
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| The equation is:<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7</ref>
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| {{Equation box 1
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| |equation = <math>\left( \frac{\partial (G/T) } {\partial T} \right)_p = - \frac {H} {T^2}</math>
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| 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>.
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| ==Chemical reactions==
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| {{Main|Thermochemistry}}
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| The typical applications 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>
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| :<math>\left( \frac{\partial ( \Delta G^\ominus/T ) } {\partial T} \right)_p = - \frac {\Delta H} {T^2}</math>
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| with Δ''G'' as the change in Gibbs energy and Δ''H'' as the enthalpy change (considered independent of temperature). The <s>o</s> denotes [[Standard conditions for temperature and pressure]] (approximately temperature 298.15 [[Kelvin]] and pressure 101,325 [[pascal (unit)|pascal]]s).
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| Integrating with respect to ''T'' (again ''p'' is constant) it becomes:
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| :<math> \frac{\Delta G^\ominus(T_2)}{T_2} - \frac{\Delta G^\ominus(T_1)}{T_1} = \Delta H^\ominus(p)\left(\frac{1}{T_2} - \frac{1}{T_1}\right) </math>
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| 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.
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| 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 | |
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| :<math>\frac{\Delta G^\ominus}{T} = -R \ln K </math>
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| 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>
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| ==Derivation==
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| ===Background===
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| {{main|Defining equation (physical chemistry)|Enthalpy|Thermodynamic potential}}
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| The definition of the Gibbs function is
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| :<math>H= G + ST \,\!</math>
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| where ''H'' is the enthalpy defined by:
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| :<math>H= U + pV \,\!</math>
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| Taking [[differential of a function|differentials]] of each definition to find ''dH'' and ''dG'', then using the [[fundamental thermodynamic relation]], aka "master equation" (always true for [[Reversible process (thermodynamics)|reversible]] or [[Irreversible process|irreversible]] [[thermodynamic process|processes]]):
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| :<math>dU= TdS - pdV \,\!</math>
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| where ''S'' is the [[entropy]], ''V'' is [[volume]], (minus sign due to reversibility, in which ''dU'' = 0: work other than pressure-volume may be done and is equal to −''pV'') leads to the "reversed" form of the initial fundamental relation into a new master equation:
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| :<math>dG= - SdT + Vdp \,\!</math>
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| 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>physical chemistry, p.W. Atkins, Oxford University press, 1978, ISBN 0-19-855148-7</ref>
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Derivation
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| |-
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| |Starting from the equation
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| :<math>dG = - SdT + Vdp = \frac{\partial G}{\partial T}dT + \frac{\partial G}{\partial p}dp \,\!</math>
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| at constant [[pressure]] meaning ''dp'' = 0. Then ''dG'' reduces to
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| :<math>dG_p = - SdT = \left(\frac{\partial G}{\partial T}\right)_p dT \quad \rightarrow \quad -S = \left(\frac{\partial G}{\partial T}\right)_p. \,\!</math>
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| The dependence of the ratio ''G/T'' on ''T'' is found by the [[product rule]] of [[Differentiation (mathematics)|differentiation]]:
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| :<math>\left( \frac{\partial(G/T)} {\partial T} \right)_p = \frac{1}{T}\left(\frac{\partial G}{\partial T}\right)_p + G\frac{\partial (T^{-1})}{\partial T} = \dfrac{T\left ( \dfrac{\partial G}{\partial T} \right)_p- G}{T^2} = \frac{-ST - G}{T^2} = -\frac{H}{T^2} \,\!</math>
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| |}
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| ==Sources==
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| {{reflist}}
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| ==External links==
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| * [http://www.chem.arizona.edu/~salzmanr/480a/480ants/gibshelm/gibshelm.html Link] - Gibbs–Helmholtz equation
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| * [http://www.owlnet.rice.edu/~chem312/Handouts%20Folder/Gibbs_Helmholtz.pdf Link] - Gibbs–Helmholtz equation
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| {{DEFAULTSORT:Gibbs-Helmholtz equation}}
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| [[Category:Thermodynamic equations]]
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| [[Category:Equations]]
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My name is Gaston Frye. I life in Sao Paulo (Brazil).
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