Non-random two-liquid model

The non-random two-liquid model ( NRTL equation for short , dt. Non-random, two liquids ) is a thermodynamic model that correlates the activity coefficient of a chemical substance mixture with its composition, expressed by mole fractions . ${\displaystyle \gamma }$ ${\displaystyle x}$

The term “non random” refers to the structure of the liquid and the arrangement of the molecules. While the Porter, van Laar and Margules models do not take into account the structured arrangement of the molecules, this is introduced in the Wilson, Uniquac and NRTL models.

The NRTL model is considered to be the best VLE model. The minimum number of binary parameters is 2. The model has now been expanded to include up to 9 parameters. With the NRTL model, LLE, i.e. H. Liquid-liquid and SLE, i.e. H. Simulate solid-liquid equilibria very well.

NRTL models belong to the class of g ^E models , as they also use the excess free enthalpy ( excess amount of the free enthalpy ). ${\displaystyle G^{E}}$ ${\displaystyle G}$

Equations

The following equations apply to a binary mixture :

${\displaystyle \ln \gamma _{1}=x_{2}^{2}\left[\tau _{21}\left({\frac {G_{21}}{x_{1}+x_{2}G_{21}}}\right)^{2}+{\frac {\tau _{12}G_{12}}{(x_{2}+x_{1}G_{12})^{2}}}\right]}$

${\displaystyle \ln \gamma _{2}=x_{1}^{2}\left[\tau _{12}\left({\frac {G_{12}}{x_{2}+x_{1}G_{12}}}\right)^{2}+{\frac {\tau _{21}G_{21}}{(x_{1}+x_{2}G_{21})^{2}}}\right]}$

With

${\displaystyle \ln G_{12}=-\alpha _{12}\tau _{12}}$

${\displaystyle \ln G_{21}=-\alpha _{21}\tau _{21}}$

${\displaystyle \tau _{12}}$ and and are parameters that are adapted to the activity coefficients . ${\displaystyle \tau _{21}}$ ${\displaystyle \alpha _{12}}$

In most cases, however, the parameters are still based on the relationships ${\displaystyle \tau }$

${\displaystyle \tau _{12}={\frac {\Delta g_{12}}{RT}}}$

${\displaystyle \tau _{21}={\frac {\Delta g_{21}}{RT}}}$

scaled with the gas constant and the temperature and then adjusted the parameters and . ${\displaystyle R}$ ${\displaystyle T}$ ${\displaystyle \Delta g_{12}}$ ${\displaystyle \Delta g_{21}}$

Temperature dependent parameters

If activity coefficients are available over a larger temperature range (e.g. from vapor-liquid and also from solid-liquid equilibria), temperature-dependent parameters can be introduced.

Two approaches are common:

${\displaystyle {\begin{aligned}\tau _{ij}&=f(T)=a_{ij}+{\frac {b_{ij}}{T}}+c_{ij}\ln T+d_{ij}T\\\Delta g_{ij}&=f(T)=a_{ij}+b_{ij}\cdot T+c_{ij}T^{2}\end{aligned}}}$

Individual terms can be omitted. E.g. the logarithmic term is mostly only used when liquid-liquid equilibria ( miscibility gaps ) have to be modeled.

Origin of the activity coefficients

The required activity coefficients are mostly derived from experimentally determined phase equilibria (vapor-liquid, liquid-liquid, solid-liquid) and from the heat of mixing . The source of these experimental data are fact databases such as the Dortmund database . Alternatively, the activity coefficients are determined directly experimentally or with predictive models such as UNIFAC .

See also

• Margules equation

literature

1. ↑ Renon H., Prausnitz JM: Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures , AIChE J., 14 (1), pp. 135-144, 1968
2. ^ Reid RC, Prausnitz JM, Poling BE: The Properties of Gases & Liquids , 4th edition, McGraw-Hill, 1988