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?

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  • $\begingroup$ What is the parameter $\beta$ ? And what is $\Delta E$ ? $\endgroup$
    – Maurice
    Nov 24, 2021 at 19:44
  • $\begingroup$ As can be seen from the previous post, @porpyrin defined in his derivations $\beta = N_0 z w/RT = zw/k_B T = \Delta E/k_B T$. $\endgroup$
    – Josiah_H
    Nov 24, 2021 at 20:23
  • $\begingroup$ 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 $ln\gamma_B=\beta x_A^2$. $\endgroup$
    – Josiah_H
    Dec 12, 2021 at 4:13


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