Thread by @zhigangsuo

#threadsofpolymers
Entropy of mixing in condensed phases

1/5 When a pure substance of molecule A mixes with a pure substance of molecule B, the entropy of the mixture exceeds the sum of the entropies of the two pure substances.

The excess is called the entropy of mixing.

2/5 Molecules A and B can have different volumes. A molecule can jiggle into many shapes. For example, A can be a macromolecule, and B a solvent molecule.

Mixing is taken to change neither the volume of each molecule, not the number of shapes that each molecule can jiggle into.

3/5 What does mixing do? It lets each molecule explore a larger volume!

Each molecule A explores volume (n_A)(V_A) in the pure substance, but explores volume (n_A)(V_A) + (n_B)(V_B) in the mixture. Similarly for B.

Mixing increases the number of configurations by a factor:

4/5 Recall the definition:
entropy = log (number of microstates)

Here the Boltzmann constant is set to be k = 1.

The entropy of mixing is

5/5 This derivation simplifies the original derivation by Huggins and Flory, and removes extraneous assumptions.

I placed this derivation in a 2016 paper (Section 5). Let me know if you know an earlier derivation published elsewhere.
suo.seas.harvard.edu/files/suo/file…