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

AI Thread Summary
In a two-chamber system with different solute concentrations, the work that can be harnessed as the system reaches equilibrium is derived from the change in free energy, G, which is related to the chemical potential, μ. The chemical potential is defined as μ = ∂G/∂n, and the total free energy can be expressed as G = μn under constant temperature and pressure. The integration of the chemical potential over the change in moles, from n=0 to n=0.25, captures the dependency of μ on concentration. As the system progresses towards equilibrium, the chemical potential difference between the chambers changes, necessitating integration to accurately calculate work. Ultimately, at equilibrium, both chambers will contain a 0.75M solution, reflecting the transfer of solute from the higher concentration chamber.
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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)?
 
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tahaha said:
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.
 
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 μi is the chemical potential of species i, Gi is the free energy of species i at the same temperature and pressure as the system, and xi 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.
 
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