The non-random two-liquid model (abbreviated NRTL model) is an activity coefficient model introduced by Renon and Prausnitz in 1968 that correlates the activity coefficients γ i {\displaystyle \gamma _{i}} of a compound with its mole fractions x i {\displaystyle x_{i}} in the liquid phase concerned. It is frequently applied in the field of chemical engineering to calculate phase equilibria. The concept of NRTL is based on the hypothesis of Wilson, who stated that the local concentration around a molecule in most mixtures is different from the bulk concentration. This difference is due to a difference between the interaction energy of the central molecule with the molecules of its own kind U i i {\displaystyle U_{ii}} and that with the molecules of the other kind U i j {\displaystyle U_{ij}} . The energy difference also introduces a non-randomness at the local molecular level. The NRTL model belongs to the so-called local-composition models. Other models of this type are the Wilson model, the UNIQUAC model, and the group contribution model UNIFAC. These local-composition models are not thermodynamically consistent for a one-fluid model for a real mixture due to the assumption that the local composition around molecule i is independent of the local composition around molecule j. This assumption is not true, as was shown by Flemr in 1976. However, they are consistent if a hypothetical two-liquid model is used. Models, which have consistency between bulk and the local molecular concentrations around different types of molecules are COSMO-RS, and COSMOSPACE.
Derivation Like Wilson (1964), Renon & Prausnitz (1968) began with local composition theory, but instead of using the Flory–Huggins volumetric expression as Wilson did, they assumed local compositions followed
x 21 x 11 = x 2 x 1 exp ( − α 21 g 21 / R T ) exp ( − α 11 g 11 / R T ) {\displaystyle {\frac {x_{21}}{x_{11}}}={\frac {x_{2}}{x_{1}}}{\frac {\exp(-\alpha _{21}g_{21}/RT)}{\exp(-\alpha _{11}g_{11}/RT)}}}
with a new "non-randomness" parameter α. The excess Gibbs free energy was then determined to be
G e x R T = ∑ i N x i ∑ j N τ j i G j i x j ∑ k N G k i x k {\displaystyle {\frac {G^{ex}}{RT}}=\sum _{i}^{N}x_{i}{\frac {\sum _{j}^{N}\tau _{ji}G_{ji}x_{j}}{\sum _{k}^{N}G_{ki}x_{k}}}} . Unlike Wilson's equation, this can predict partially miscible mixtures. However, the cross term, like Wohl's expansion, is more suitable for H ex {\displaystyle H^{\text{ex}}} than G ex {\displaystyle G^{\text{ex}}} , and experimental data is not always sufficiently plentiful to yield three meaningful values, so later attempts to extend Wilson's equation to partial miscibility (or to extend Guggenheim's quasichemical theory for nonrandom mixtures to Wilson's different-sized molecules) eventually yielded variants like UNIQUAC.
Equations for a binary mixture For a binary mixture the following functions are used:
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