Chemical potential of an ideal gas problem

[arenaninja]

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?

 

[Mapes]

What is PV for an ideal gas?

 

[arenaninja]

$$PV = nRT$$
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!