Van't Hoff equation

The Van-'t-Hoff equation or Van-'t-Hoff'sche or van-'t-Hoffsche reaction isobaric (after Jacobus Henricus van 't Hoff ) describes in thermodynamics the connection between the position of the equilibrium of a chemical Reaction and temperature (at constant pressure):

${\displaystyle {\Bigl (}{\frac {\partial \ln K}{\partial T}}{\Bigr )}_{p}={\frac {\Delta _{r}H^{0}(T)}{RT^{2}}}}$

in which

• ${\displaystyle K}$the equilibrium constant ,
• ${\displaystyle T}$the temperature ,
• ${\displaystyle \Delta _{r}H^{0}(T)}$the standard molar enthalpy of reaction as a function of temperature (the standard condition pressure is fulfilled) and ${\displaystyle T}$ ${\displaystyle p_{0}=1\,{\text{bar}}}$
• ${\displaystyle R}$is the general gas constant .

The index p stands for the constant pressure. Another formulation of the Van't Hoff equation for the inverse temperature with the Boltzmann constant is: ${\displaystyle \beta =(k_{\mathrm {B} }T)^{-1}}$ ${\displaystyle k_{\mathrm {B} }}$

${\displaystyle \left({\frac {\partial \ln K}{\partial \beta }}\right)_{p}=-{\frac {\Delta _{r}H^{0}(\beta )}{N_{\mathrm {A} }}},}$

where is Avogadro's constant . ${\displaystyle N_{\mathrm {A} }}$

Derivation

The following generally applies to the equilibrium constant : ${\displaystyle K}$

${\displaystyle \ln K=-{\frac {\Delta _{r}G^{0}(T)}{RT}}}$

Their partial derivative according to the temperature at constant pressure results in:

${\displaystyle {\Bigl (}{\frac {\partial \ln K}{\partial T}}{\Bigr )}_{p}=+{\frac {\Delta _{r}G^{0}(T)}{RT^{2}}}-{\frac {{\Bigl (}{\frac {\partial \Delta _{r}G^{0}(T)}{\partial T}}{\Bigr )}_{p}}{RT}}}$

The derivation of the molar, free enthalpy of reaction according to the temperature at constant pressure is calculated as follows:

${\displaystyle {\Bigl (}{\frac {\partial G_{m}^{0}}{\partial T}}{\Bigr )}_{p}=-S_{m}^{0}(T)}$

${\displaystyle \rightarrow {\Bigl (}{\frac {\partial \ln K}{\partial T}}{\Bigr )}_{p}=+{\frac {\Delta _{r}G^{0}(T)}{RT^{2}}}+{\frac {T\Delta _{r}S^{0}(T)}{RT^{2}}}}$

With the Gibbs-Helmholtz equation

${\displaystyle \Delta _{r}G^{0}(T)=\Delta _{r}H^{0}(T)-T\Delta _{r}S^{0}(T)}$

surrendered:

${\displaystyle {\Bigl (}{\frac {\partial \ln K}{\partial T}}{\Bigr )}_{p}={\frac {\Delta _{r}H^{0}(T)-T\Delta _{r}S^{0}(T)+T\Delta _{r}S^{0}(T)}{RT^{2}}}={\frac {\Delta _{r}H^{0}(T)}{RT^{2}}}}$

Van't Hoff reaction isochore

If the volume is kept constant during a reaction, the reaction is described by the change in the standard free energy . The result is the van-'t-Hoff'sche reaction isochore : ${\displaystyle \ln K=-\Delta _{r}F(N_{i},V,T)^{0}/(RT)}$

${\displaystyle {\Bigl (}{\frac {\partial \ln K}{\partial T}}{\Bigr )}_{V}={\frac {\Delta _{r}U^{0}}{RT^{2}}}}$

solution

The formal solution of the Van't Hoff equation is

${\displaystyle K(T)=K(T_{0})\exp \left(\int _{T_{0}}^{T}\mathrm {d} T{\frac {\Delta _{r}H^{0}(T)}{RT^{2}}}\right)}$

In Ulich's approximation , one assumes a standard reaction enthalpy that is constant - at least in a certain temperature interval.

This results in:

${\displaystyle K(T)=K(T_{0})\exp \left(-{\frac {\Delta _{r}H^{0}}{RT}}+{\frac {\Delta _{r}H^{0}}{RT_{0}}}\right)}$

See also

• RGT rule

Web links

• Video: VAN'T HOFF reaction isobars - how does the temperature change the position of the equilibrium? . Jakob Günter Lauth (SciFox) 2013, made available by the Technical Information Library (TIB), doi : 10.5446 / 15667 .

Individual evidence

1. ↑ Peter W. Atkins, Julio de Paula: Physical chemistry . 4th, completely revised edition. Wiley-VCH, 2006, ISBN 3-527-31546-2 , pp. 237 .
2. ↑ Derivation of Van't Hoff's reaction isochors - Chemgapedia. Retrieved February 8, 2019 . 
3. ↑ Ulich approximations . In: Lexicon of Chemistry. Retrieved July 25, 2014.