Helmholtz free energy for mixing?

In summary, there is a discussion about excess Helmholtz energy in the paper by Wong and Sandler. They use a mixing rule from Smith and Van Ness' book to derive equations for a Van der Waals gas mixture. These equations can be found in Chapters 10 and 11 of the book.
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maistral
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TL;DR Summary
Derivation of formula.
Hi, I have been reading a few literature regarding excess Helmholtz energy and I encountered this definition from the paper of Wong and Sandler (apparently, from the mixing rule used in a EOS):
wong.png


In particular, the ones in the red boxes.

How did these equations come into being? I tried to look for it in books like Tester and SVNA and I still am unable to find where on Earth did these come from. Can someone point me to a reliable resource? Thanks.
 
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Check out Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics, Chapters 10 and 11. They work mainly with Gibbs free energy, but you'll get the idea.

It looks like Sanders derived these equations specifically for an Van der Waals gas mixture.
 
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1. What is Helmholtz free energy for mixing?

Helmholtz free energy for mixing is a thermodynamic concept that describes the amount of energy required to mix two substances together. It takes into account the changes in both enthalpy and entropy during the mixing process.

2. How is Helmholtz free energy for mixing calculated?

Helmholtz free energy for mixing can be calculated using the equation ΔFmix = ΔHmix - TΔSmix, where ΔHmix is the change in enthalpy, ΔSmix is the change in entropy, and T is the temperature.

3. What is the significance of Helmholtz free energy for mixing?

Helmholtz free energy for mixing is important in understanding the spontaneity of a mixing process. If the ΔFmix is negative, the process is spontaneous and the substances will mix on their own. If the ΔFmix is positive, energy must be added to the system for the substances to mix.

4. How does Helmholtz free energy for mixing relate to other thermodynamic quantities?

Helmholtz free energy for mixing is closely related to Gibbs free energy, which takes into account both enthalpy and entropy changes at constant pressure. It is also related to the chemical potential, which is a measure of the energy required to add a molecule to a system.

5. Can Helmholtz free energy for mixing be negative?

Yes, Helmholtz free energy for mixing can be negative, indicating that the mixing process is spontaneous. This means that the substances will mix on their own without any external energy input. However, it is important to note that the overall ΔFmix may still be positive if the substances have different concentrations or temperatures before mixing.

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