Chemical potential of an ideal gas problem

AI Thread Summary
The discussion focuses on deriving the chemical potential of an ideal gas in terms of temperature (T) and volume (V). The user attempts to express the chemical potential using the relationship between entropy, internal energy, and Gibbs free energy. They initially struggle with missing terms in their calculations and seek clarification on how pressure-volume (PV) relates to the chemical potential. The conversation highlights the relationships between specific heat capacities and the ideal gas law, leading to a clearer understanding of the chemical potential equation. Ultimately, the user gains insights into the derivation process and the connections between the variables involved.
arenaninja
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Hey everyone. I hope someone can help. I'm off on this by several factors so I'm wondering what I may be inferring incorrectly.

Homework Statement


Express the chemical potential of an ideal gas in termps of T and V:
\mu = c_{P}T - c_{V}T\ln T - RT\ln V - s_{0}T + const

Homework Equations


(Hint: Find the entropy S = S(T,V); use G = U - TS + PV and write \mu = G/n)

The Attempt at a Solution


For S = S(T,V) of an ideal gas we have:
S = nc_{V}\ln T + nR\ln V
Now we attempt to find G:
G = U - TS + PV
Recognize that U for an ideal gas is a constant (Nfk/2), and we have:
G = -nc_{V}T\ln T - nRT\ln V + \frac{nfk_{B}}{2} + PV

As you can see, I'm missing two terms. I'm not sure how PV would translate into those two terms. So overall I'm not faring very well in this problem.

Any hints? Insights? Corrections?
 
Last edited:
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What is PV for an ideal gas?
 
PV = nRT
Also, for ideal gases:
c_{P} = c_{V} + nR
Ohhh I see (I think). So the last term:
nk_{B}T = c_{P}T - c_{V}T

I'm guessing c_{V}T = s_{0}T. Great!

Thank you very much!
 
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