[Doubt] Chemical Potential of an Ideal Gas

[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 beggining 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 excercise 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.

 

[paranormal]

Your approach to solving for the chemical potential of an ideal gas using thermodynamic equations is correct. However, it is important to note that the chemical potential is not a constant and can vary depending on the temperature, pressure, and number of moles of the gas. Therefore, it is not necessary to include a constant term \mu_0 in your calculation.

Additionally, it is important to consider the physical interpretation of the chemical potential in this context. The chemical potential of an ideal gas is a measure of the change in the Gibbs free energy (G = U - TS + PV) when adding one mole of the gas to a system at constant temperature and pressure. It represents the energy required to add one mole of the gas to the system while keeping the temperature and pressure constant.

In summary, your approach to solving for the chemical potential is correct, but it is not necessary to include a constant term and it is important to consider the physical interpretation of the chemical potential in this context.