Editing Gibbs–Helmholtz equation (section)

===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>
}}