Gibbs-Helmholtz equation

From Wikipedia, the free encyclopedia

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} \right)_{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 infinite 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} \right)_{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 the 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.


The Gibbs free energy for an open system


dG = - SdT + VdP \,


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


dG_{p\,} = - SdT \,


or

\left(\frac{\partial G}{\partial T}\right)_{p\,} = - S \,


As the Gibbs energy for a closed system is


G = H - TS \,


it follows that


\left(\frac{\partial G}{\partial T}\right)_{P\,} = \frac{G - H}{T} \,


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} \right)_{p\,} = \frac{\frac{\partial g}{\partial T}}{\frac{\partial}{\partial T}} = \frac{T\left(\frac{\partial G}{\partial T}\right)_{P\,} - G\left(\frac{\partial T}{\partial T}\right)_{P\,} }{T^2} = \frac{1}{T}\left(\frac{\partial G}{\partial T}\right)_{P\,} - \frac {G} {T^2}


and with substitution:


\left( \frac{\partial ( \frac{G} {T} ) } {\partial T} \right)_{p\,} = \frac{1}{T}\frac{G - H}{T}- \frac {G} {T^2}


the result is


\left( \frac{\partial ( \frac{G} {T} ) } {\partial T} \right)_{p\,} = - \frac {H} {T^2}

  • Gibbs-Helmholtz equation @ www.chem.arizona.edu Link
  • Gibbs-Helmholtz equation @ www.owlnet.rice.edu Link
Retrieved from "http://en.wikipedia.org../../../g/i/b/Gibbs-Helmholtz_equation_a059.html"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.