Question

Consider a rigid, adiabatic, impermeable cylinder (i.e., a closed isolated system) that consists of two chambers (i.e., simple systems) that are separated by a wall that is permeable, immovable, and diathermal. Suppose that initially, the fluids in the two chambers are at the same temperature, T, but have different chemical potentials with µ2 > µ1. Following our earlier arguments considering entropy changes for heat transfer, demonstrate that

a. the fluid will flow from chamber 2 into chamber 1, and

b. this flow will continue until the two fluids are at the same chemical potential.

Answer 1

Let’s have two systems S₁ and S₂ in contact with a thermal reservoir R at temperature T. The systems themselves can exchange particles and energy but their respective volumes and the total particle distribution N = N₁ + N₂ remains constant. The Helmholtz free energy of S₁ + S₂ should be minimized at equilibrium i.e.

but since , we have at equilibrium

We define chemical potential as follows:

Chambers S₁ and S₂ will be in diffusive equilibrium if . If particles will flow from S₂ to S₁ since this leads to a decrease in Gibbs free energy by first equation. Two systems are said to be in thermal and diffusive equilibrium if both their temperatures and chemical potentials are equal.