Gibbs–Helmholtz equation

The Gibbs–Helmholtz equation is a 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:

:$left(\; frac\{partial\; (\; frac\{G\}\; \{T\}\; )\; \}\; \{partial\; T\}\; ight)\_\{p,\}\; =\; -\; frac\; \{H\}\; \{T^2\}$

With:

:$H,$ the enthalpy:$T,$ the absolute temperature:$G,$ the Gibbs free energy

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).

For a chemical reaction the equation reads:

:$left(\; frac\{partial\; (\; frac\{Delta\; G\}\; \{T\}\; )\; \}\; \{partial\; T\}\; ight)\_\{p,\}\; =\; -\; frac\; \{Delta\; H\}\; \{T^2\}$

with $Delta\; G,$ as the change in Gibbs energy and $Delta\; H,$ as the enthalpy change (which is considered independent of temperature).

which can rearrange to:

:$frac\{Delta\; G,T\_2\}\{T\_2\}\; -\; frac\{Delta\; G^circ,T\_1\}\{T\_1\}\; =\; Delta\; H^circ(P)*(frac\{1\}\{T\_2\}\; -\; frac\{1\}\{T\_1\})$

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 at 25°C and 1 bar.

Through:

:$Delta\; G^circ\; =\; -RT\; ln\; K$

which relates Gibbs energy to an equilibrium constant, the van 't Hoff equation is derived.

Proof

The Gibbs free energy for a closed system

:$dG\; =\; -\; SdT\; +\; VdP\; ,$

at constant pressure $P,$ (dP = 0) reduces to

:$dG\_\{p,\}\; =\; -\; SdT\; ,$

or

:$left(frac\{partial\; G\}\{partial\; T\}\; ight)\_\{p,\}\; =\; -\; S\; ,$

The dependence of the G/T ratio on T is found with the aid of the quotient rule:

:$left(\; frac\{partial\; (\; frac\{G\}\; \{T\}\; )\; \}\; \{partial\; T\}\; ight)\_\{p,\}=\; frac\{left(\; frac\{partial\; G\}\{partial\; T\}\; ight)\_\{p,\{T\}\; -\; frac\{G\}\{T^2\}\; =\; frac\{Tleft\; (\; frac\{partial\; G\}\{partial\; T\}\; ight)\_\{p,\}-\; G\}\{T^2\}\; =\; frac\{-ST\; -\; G\}\{T^2\}\; =\; frac\{-H\}\{T^2\}$

Sometimes it can be found like this:

:$left(\; frac\{partial\; (\; frac\{G\}\; \{T\}\; )\; \}\; \{partialleft(frac\{1\}\{T\}\; ight)\}\; ight)\_\{p,\}=\; H$

Here, the chain rule has been used with a bit of rearrangement. If $y$ and $u$ are functions of $x$, the chain rule says that $frac\{dy\}\{dx\}\; =\; frac\{dy\}\{du\}\; frac\{du\}\{dx\}$. Dividing both sides through by $frac\{du\}\{dx\}$ gives $frac\{dy\}\{du\}\; =\; frac\{frac\{dy\}\{dx\{frac\{du\}\{dx$. In the equation above, $y$ is $frac\{G\}\{T\}$, $x$ is $T$, and u is $frac\{1\}\{T\}$. Thus, $frac\{du\}\{dx\}$ is $frac\{-1\}\{T^2\}$.

External links

* Gibbs–Helmholtz equation @ www.chem.arizona.edu [http://www.chem.arizona.edu/~salzmanr/480a/480ants/gibshelm/gibshelm.html Link]
* Gibbs–Helmholtz equation @ www.owlnet.rice.edu [http://www.owlnet.rice.edu/~chem312/Handouts%20Folder/Gibbs_Helmholtz.pdf Link]