Chemical potential of an ideal gas from heat capacities

[Sabian]

Homework Statement

Basically, find the chemical potential of an ideal gas knowing its heat capacities.

Homework Equations

$P V = n R T \ \ \ \ (1)$
$U = n c_V T + U_0 \ \ \ \ (2)$
$S = S_0 + n c_V ln (T) + nR ln (V) = S_0 + n c_V ln (T) + nR ln \left ( \frac{nRT}{P} \right ) \ \ \ \ (3)$
$\mu = \left ( \frac {\partial G}{\partial n} \right )|_{T,P} \ \ \ \ (4)$
$G = U - TS + PV \ \ \ \ (5)$

The Attempt at a Solution

Mixing (1), (2) and (3) into (5) I get

$G = n c_V T + U_0 - T \left (S_0 + n c_V ln (T) + nR ln \left ( \frac{nRT}{P} \right ) \right ) + nRT$

Then differentiating with n, while treating P and T as constants

$\mu (P, T, n) = c_V T - T \left (c_V ln (T) + R ln \left ( \frac{nRT}{P} \right ) + R \right ) + RT$

Which has no constants, but I suppouse that the chemical potential, as every good classical potential, must be defined beginning at some constant $\mu_0$.

What I am doing wrong?

Thank you for your time.

 

[Chestermiller]

In the third term for S, the V should just be RT/P. Then, re-express μ as

μ = μ₀(T) + RT ln P

μ is an intensive property and cannot depend on the number of moles.

Determine what μ₀ is as a function of T.

 

[Sabian]

Good catch on the molar dependance, hadn't looked at that. The exercise I was looking at says $\mu (P,T,n)$. I have some doubts on that being on purpose, but anyway...
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Then I have

$G = n c_v T + U_0 - T \left (S_0 + n c_v ln (T) + nR ln \left ( \frac{RT}{P} \right ) \right ) + nRT$

Then the differentiation is$\mu (P, T,) = c_v T - T \left (c_v ln (T) + R ln \left ( \frac{RT}{P} \right ) \right ) + RT$

Which is the same as

$\mu (P,T) = c_v T - T c_v ln(T) + TR ln (RT) - TR ln (P) + RT$

On a side note, I understand your approach, but I'm concerned about what's wrong with my derivation rather than getting the actual answer. I don't mean to be rude and I greatly appreciate your help on that other way to make the derivation, but I'm afraid I might have some wrong concepts and that's why what I've done is wrong.

Thanks :)

 

[Chestermiller]

There are some sign errors in your derivation. Watch the algebra. Also, you can use C_p = C_v + R

to make the final result more concise.

I might also mention that the chemical potential is usually regarded as the Gibbs free energy per mole. You also need to understand that U₀ and S₀ are extensive properties, and thus are proportional to the number of moles. Thus, they can be expressed as U₀ = n u₀ and S₀ = n s₀, where u₀ is the internal energy per mole in some reference state. With these substitutions, you can divide G by n and get μ.