How is Work Calculated from Chemical Potential and ΔG in a Two-Chamber System?

[tahaha]

Homework Statement

Consider a container with two chambers of the same size separated by a fixed membrane in the middle (permeable only to the ideal solute, but not the solvent). One chamber initially contains 1M of solute, and the other contains 0.5M of solute.

Write an equation for the amount of work that could potentially be harnessed when the system goes from the initial condition to equilibrium.

Homework Equations

μ=∂G/∂n at constant temperature and pressure

The Attempt at a Solution

Some confusions that I have:

1. So the definition of the chemical potential of a species at constant temperature, pressure is ∂G/∂n.
When we try to find the total free energy G of the species, we integrate ∫dG=∫μdn. There is then an equation in my textbook saying that G=μn. It kinda makes sense, because it's like summing up the chemical potential (defined as G per molecule or per mole) in the system. But does this imply that μ does not depend on n? How can it depend on concentration while not depending on the number of moles?

2. The answer says Work=∫dG= ∫[RTln(C1-n)/(C2+n)]dn integrated from n=0 to n=0.25, where C1=1M and C2=0.5M.
Apparently, the chemical potential difference between the two chambers changes as the reaction proceeds and is dependent on n. But I don't quite get why we have to integrate it.

Most importantly, can someone please explain in detail the mathematical relationship between chemical potential and free energy (and how to get from one to the other)?

 

[DrDu]

1. So the definition of the chemical potential of a species at constant temperature, pressure is ∂G/∂n.
When we try to find the total free energy G of the species, we integrate ∫dG=∫μdn. There is then an equation in my textbook saying that G=μn. It kinda makes sense, because it's like summing up the chemical potential (defined as G per molecule or per mole) in the system. But does this imply that μ does not depend on n? How can it depend on concentration while not depending on the number of moles?

∫μdn=μn if the integration path is for constant composition (or concentration c) pressure p and temperature T as μ(c, p,T) does not depend on n.
However, this is not the integration path taken in the second part of your answer.

 

[Chestermiller]

An ideal solution is defined as one in which the chemical potential of each species, by analogy to the chemical potentials of species in an ideal gas, is given by $μ_i=G_i+RT\ln x_i$, where μ_i is the chemical potential of species i, G_i is the free energy of species i at the same temperature and pressure as the system, and x_i is the mole fraction of species i in the solution. Apparently, there are a significant number of liquid solutions that approach this type of behavior. At final steady state equilibrium, there will be 0.75M solution on both sides of the membrane. Therefore, Δn=0.25 moles per liter will be removed from the high concentration chamber and transferred to the lower concentration chamber. The change in free energy for each chamber is the concentration-dependent chemical potential for that chamber integrated over the change in the moles of solute in that chamber.