Gibbs-Helmholtz equation

The Gibbs-Helmholtz equation (also Gibbs-Helmholtz equation ) is an equation of thermodynamics . It is named after the US physicist Josiah Willard Gibbs and the German physiologist and physicist Hermann von Helmholtz . It describes the relationship between the Gibbs energy and the enthalpy as a function of the temperature . The Gibbs-Helmholtz equation is generally: ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle T}$

${\displaystyle \left.{\frac {\partial }{\partial T}}{\frac {G}{T}}\right|_{p}=-{\frac {H}{T^{2}}}}$

The symbol stands for a partial derivative according to the temperature and the notation means that the pressure in this equation is kept constant in all sizes. This is not explicitly mentioned in the notation below. ${\displaystyle \partial /(\partial T)}$ ${\displaystyle |_{p}}$ ${\displaystyle p}$

The relationship , actually just a Legendre transformation , which describes the relationship between the Gibbs energy, the enthalpy, the temperature and the entropy , is referred to in some literature as the Gibbs-Helmholtz equation. Both forms are equivalent, since they can be converted into one another through mathematical operations of differential calculus (and the use of the physical relationship between temperature, entropy and Gibbs energy). This form plays a central role when considering the chemical equilibrium , as it allows the influence of enthalpy and entropy on the Gibbs energy to be compared directly with one another (see exergonic and endergonic reaction ). It can be used to estimate which side of the equilibrium is thermodynamically preferred. ${\displaystyle \Delta G=\Delta H-T\Delta S}$ ${\displaystyle S}$

Derivation

The enthalpy and the Gibbs energy can be transformed into one another via a variable transformation , more precisely via the Legendre transformation:

${\displaystyle H(S,p)=G(T,p)+TS}$

The total differential of the Gibbs energy is when the number of particles is fixed

${\displaystyle \mathrm {d} G=-S\mathrm {d} T+V\mathrm {d} p}$,

so . It follows: ${\displaystyle S=-{\tfrac {\partial G}{\partial T}}}$

${\displaystyle {\frac {\partial }{\partial T}}{\frac {G}{T}}=-{\frac {G}{T^{2}}}+{\frac {1}{T}}{\frac {\partial G}{\partial T}}=-{\frac {H}{T^{2}}}+{\frac {S}{T}}-{\frac {S}{T}}=-{\frac {H}{T^{2}}}}$

Relation to the Van't Hoff equation

In thermodynamics, the Van-'t-Hoff equation describes the relationship between the position of the equilibrium of a chemical reaction and the temperature at constant pressure. The following generally applies to the equilibrium constant of a chemical reaction and the change in the free enthalpy during the reaction under standard conditions ${\displaystyle K}$ ${\displaystyle \Delta _{R}G^{0}}$

${\displaystyle \ln K=-{\frac {\Delta _{R}G^{0}}{RT}}}$

with the general gas constant . The Gibbs-Helmholtz equation leads directly to the Van't Hoff equation : ${\displaystyle R}$

${\displaystyle {\frac {\partial \ln K}{\partial T}}=-{\frac {1}{R}}{\frac {\partial }{\partial T}}{\frac {\Delta _{R}G^{0}}{T}}={\frac {\Delta _{R}H^{0}}{RT^{2}}}}$

Other spellings

The Gibbs-Helmholtz equation can be expressed as a function of the inverse temperature with the Boltzmann constant as ${\displaystyle \beta =(k_{\mathrm {B} }T)^{-1}}$ ${\displaystyle k_{\mathrm {B} }}$

${\displaystyle {\frac {\partial (\beta G)}{\partial \beta }}=H(\beta )}$

represent. This follows from the chain rule of differential calculus:

${\displaystyle {\frac {\partial (\beta G)}{\partial \beta }}=G+\beta {\frac {\partial G}{\partial \beta }}=G-(k_{\mathrm {B} }T)^{-1}(k_{\mathrm {B} }T^{2}){\frac {\partial G}{\partial T}}=G+TS=H}$

Individual evidence

1. ^ ^{A b} J. A. Campbell: Allgemeine Chemie . Verlag Chemie, Weinheim 1975, p. 774-775 .
2. ^ JA Campbell: General Chemistry . Verlag Chemie, Weinheim 1975, p. 812 .