Ulichian approximation

The Ulich approximations , also written Ulich approximations , denote three different possibilities in thermodynamics to approximate the temperature dependence of the molar, isobaric heat capacity difference between two chemical states. These allow a significantly simplified calculation of the temperature dependency z. B. the molar enthalpy of reaction or the molar entropy of reaction . ${\displaystyle \Delta _{R}c_{p}}$ ${\displaystyle \Delta _{R}H_{m}}$ ${\displaystyle \Delta _{R}S_{m}}$

They were introduced by the German chemist Hermann Ulich (born January 13, 1895 in Dresden , † April 14, 1945 in Karlsruhe ).

Basics

From the total differentials of the molar enthalpy and the molar entropy, it can be deduced that their isobaric temperature change depends on the molar, isobaric heat capacity:

${\displaystyle dH=\delta Q+Vdp\Rightarrow \left({\frac {\partial H_{m}}{\partial T}}\right)_{p}=\left({\frac {\delta Q}{\partial T}}\right)_{p}=c_{p}}$

${\displaystyle dS={\frac {\delta Q}{T}}\Rightarrow \left({\frac {\partial S_{m}}{\partial T}}\right)_{p}={\frac {1}{T}}\left({\frac {\delta Q}{\partial T}}\right)_{p}={\frac {c_{p}}{T}}}$

This results in the standard molar enthalpy at temperature : ${\displaystyle H_{m}^{0}}$ ${\displaystyle T}$

${\displaystyle H_{m}^{0}(T)=H_{m}^{0}(T_{0})+\int _{T_{0}}^{T}c_{p}(T')dT'}$

where the standard enthalpy is at 298 K. For the standard molar entropy it results analogously: ${\displaystyle H_{m}^{0}(T_{0})}$

${\displaystyle S_{m}^{0}(T)=S_{m}^{0}(T_{0})+\int _{T_{0}}^{T}{\frac {c_{p}(T')}{T'}}dT'}$

In the case of chemical reactions or physical changes of state, one calculates with the reaction enthalpy , the difference between the product enthalpies and the educt enthalpies. ${\displaystyle \Delta _{R}H_{m}^{0}}$

${\displaystyle \Delta _{R}H_{m}^{0}(T)=H_{m}^{0}(T_{0})_{\text{Produkte}}+\int _{T_{0}}^{T}c_{p}(T')_{\text{Produkte}}dT'-H_{m}^{0}(T_{0})_{\text{Edukte}}-\int _{T_{0}}^{T}c_{p}(T')_{\text{Edukte}}}$

${\displaystyle =\Delta _{R}H_{m}^{0}(T_{0})+\int _{T_{0}}^{T}\Delta _{R}c_{p}(T')dT'}$

Likewise for entropy:

${\displaystyle \Delta _{R}S_{m}^{0}(T)=\Delta _{R}S_{m}^{0}(T_{0})+\int _{T_{0}}^{T}{\frac {\Delta _{R}c_{p}(T')}{T'}}dT'}$

Since the exact course of the function is mostly unknown, Ulich's approximations are often used in theoretical calculations to simplify the integrals . ${\displaystyle \Delta _{R}c_{p}(T')}$

Ulich approximations

First Ulich approximation

${\displaystyle \Delta _{R}c_{p}(T')=0}$

This would be the case if products and starting materials had the same molar heat capacity. It follows from this that the standard molar enthalpy or entropy of reaction is independent of temperature, i.e. has the same value for all temperatures . This simplifies, for example, the calculation of the free enthalpy of reaction to: ${\displaystyle T}$ ${\displaystyle \Delta _{R}G_{m}^{0}=\Delta _{R}H_{m}^{0}-T\Delta _{R}S_{m}^{0}}$

Second Ulich approximation

${\displaystyle \Delta _{R}c_{p}(T')=\alpha ={\text{konst.}}}$

This would result, for example, if the temperature change of the educts would just level out that of the products. In general, however, this is not the case. Typically, you can choose a mean value for , for a certain temperature range , so that the deviations due to the temperature dependency remain small. The standard reaction enthalpies or entropies are then calculated as follows. ${\displaystyle \Delta c_{p}}$

${\displaystyle \Delta _{R}H_{m}^{0}(T)=\Delta _{R}H_{m}^{0}(T_{0})+\alpha (T-T_{0})}$

${\displaystyle \Delta _{R}S_{m}^{0}(T)=\Delta _{R}S_{m}^{0}(T_{0})+\alpha \cdot \ln \left({\frac {T}{T_{0}}}\right)}$

Third Ulich approximation

Instead of choosing an average value for a temperature range as in the 2nd approximation, in the third approximation a larger area is divided into several intervals. For each one sets a mean value for , so that the course of a step function is obtained: ${\displaystyle \Delta _{R}c_{p}}$ ${\displaystyle \Delta _{R}c_{p}(T)}$

${\displaystyle \Delta _{R}c_{p}(T)={\begin{cases}\alpha _{1},&{\text{für }}T_{0}\leq T<T_{1}{\text{,}}\\\alpha _{1},&{\text{für }}T_{1}\leq T<T_{2}{\text{,}}\\\alpha _{2},&{\text{für }}T_{2}\leq T<T_{3}{\text{,}}\\\ldots \end{cases}}}$

The integral for calculating the standard reaction enthalpies or entropies can then be broken down into several integrals over the individual sub-intervals, which can be calculated separately from one another.

${\displaystyle \Delta _{R}H_{m}^{0}(T)=\Delta _{R}H_{m}^{0}(T_{0})+\int _{T_{0}}^{T_{1}}\alpha _{1}dT'+\int _{T_{1}}^{T_{2}}\alpha _{2}dT'+\int _{T_{2}}^{T_{3}}\alpha _{3}dT'+\dots }$

${\displaystyle \Delta _{R}S_{m}^{0}(T)=\Delta _{R}S_{m}^{0}(T_{0})+\int _{T_{0}}^{T_{1}}{\frac {\alpha _{1}}{T'}}dT'+\int _{T_{1}}^{T_{2}}{\frac {\alpha _{2}}{T'}}dT'+\int _{T_{2}}^{T_{3}}{\frac {\alpha _{3}}{T'}}dT'+\ldots }$

application

Generally it can be about the standard reaction and Standardreaktionsentropie the free standard reaction enthalpy ${\displaystyle \Delta _{R}H_{m}^{0}(T)}$ ${\displaystyle \Delta _{R}S_{m}^{0}(T)}$

${\displaystyle \Delta _{R}G_{m}^{0}(T)=\Delta _{R}H_{m}^{0}(T)-T\cdot \Delta _{R}S_{m}^{0}(T)}$

determine. In this context, Ulich's approximations are used, for example, for statements about the temperature dependence of the equilibrium constant of chemical equilibria .

Individual evidence

1. ^ Entry on Hermann Ulich in the Catalogus Professorum Rostochiensium

literature

• Gerd Wedler, Hans-Joachim Freund: Textbook of Physical Chemistry Sixth, completely revised and updated edition . Wiley-VCH, Weinheim 2012, ISBN 978-3-527-32909-0 , pp. 432-433.

Web links

• Ulichian approximations. In: Lexicon of Chemistry. Spectrum Academic Publishing House, Heidelberg 1998.