Thermodynamic Identity: Chemical Potential

[WWCY]

Homework Statement

Homework Equations

Thermodynamic Identity

The Attempt at a Solution

While I was able to work out the problem with the help of the hint, I couldn't completely understand the implication of said hint. The hint suggests that the equations for Chemical Potential in a process where $U, V$ are constant and in another where $S,V$ are both constant, are both identical. Why is this? Should we not expect that the Chemical Potential varies differently with different processes (that have different state variables held constant)?

Also, what does this imply about other state variables such as $S, U, V$? Do we expect their equations to be the same under varying conditions as well?

 

[Chestermiller]

Homework Statement

View attachment 213602

Homework Equations

Thermodynamic Identity

The Attempt at a Solution

While I was able to work out the problem with the help of the hint, I couldn't completely understand the implication of said hint. The hint suggests that the equations for Chemical Potential in a process where $U, V$ are constant and in another where $S,V$ are both constant, are both identical. Why is this? Should we not expect that the Chemical Potential varies differently with different processes (that have different state variables held constant)?

Also, what does this imply about other state variables such as $S, U, V$? Do we expect their equations to be the same under varying conditions as well?

Most thermo books have a derivation to show that all the various expressions for the chemical potential in terms of the thermodynamic functions are equivalent.

 

[DrClaude]

The answer is in the name: state variables can only depend on the state of the system, not on any process that resulted in that state.

I think you may be confused the fact that the chemical potential is obtained through a derivative. The equation you get for μ is not equation for a change of chemical potential.

 

[WWCY]

Thanks for the responses.

Is it right to say that because we have already assumed that the changes were quasistatic by deriving $\mu$ from the thermodynamic identity, the changes are always governed by the same equation of state, and therefore both derivations lead to the same expression?