# Interpretation of azeotropes from statistical mechanics derivation

In an earlier posted question titled Reason for the formation of azeotropes, @porphyrin presented an elegant derivation from statistical mechanics to interpret partial pressures for the two components of a binary mixture and the azeotropic composition.

The figure of $$p_A/p_A^0$$, which is normalized partial pressure or activity $$\alpha_A$$, shows several curves calculated with different $$\beta$$ values or equivalently $$\Delta E/k_B T$$ as a function of $$x_A$$. Most curves in that figure show monotonic changes with mole fraction except for the curves with $$\beta$$=2 and 3. From realistic experimental data, however, partial pressure curves appear to only show monotonic changes as a function of mole fraction, i.e, the derivative of partial pressure curves won't change sign (it does change sign for the total pressure thereby yielding one point with a derivative of zero at the azeotropic composition). Therefore, the partial pressure curve with $$\beta$$=2 and at least curve with $$\beta$$=3 may not be realistic, even though math does produce such curves if large $$\Delta E$$ is used. By taking the derivative of partial pressure $$p_A$$ w.r.t. $$x_A$$, one can find the condition for $$p_A$$ to show only monotonic changes as a function of mole fraction. It turns out that when $$\Delta E/k_B T$$ = 2, $$x_A$$ is located exactly at $$1/2 \pm 1/2 \sqrt{1-2/2}$$ = 0.5 giving only one derivative value of zero, and when $$\Delta E/k_B T$$ = 3, $$x_A$$ is located at $$1/2 \pm 1/2 \sqrt{1-2/3}$$ = 0.211 and 0.789 giving two derivative values of zero. Therefore, as long as $$\Delta E/k_B T$$ is less than 2, the partial pressure curves will be showing monotonic changes as a function of mole fraction. An educated guess is that the binary mixtures are miscible if $$\Delta E/k_B T$$ < 2 and not completely miscible if $$\Delta E/k_B T$$ > 2, which form homogeneous and heterogeneous azeotropes, respectively.

I am wondering if the derivations given by @porphyrin based on statistical mechanics for the partial pressures are available somewhere in the literature (papers or books) so I can read more to understand and possibly cite the reference instead of the website link in my work? • What is the parameter $\beta$ ? And what is $\Delta E$ ? Nov 24, 2021 at 19:44
• As can be seen from the previous post, @porpyrin defined in his derivations $\beta = N_0 z w/RT = zw/k_B T = \Delta E/k_B T$. Nov 24, 2021 at 20:23
• Found in Atkins' Physical Chemistry that the critical parameter $\beta=2$ has to do phase separation, and $p_A= p_A^0x_Ae^{\beta x_B^2}$ and $p_B=p_B^0x_Be^{\beta x_A^2}$ are related to the Margules equations $ln\gamma_A=\beta x_B^2$ and $ln\gamma_B=\beta x_A^2$. Dec 12, 2021 at 4:13