Problem from Fermi's Thermodynamics

[Chasing_Time]

Hi all, I am getting stuck on this problem from chapter 4 of Fermi's Thermodynamics:

Homework Statement

"A body obeys the equation of state:

pV$$^{1.2}$$ = 10$$^{9}$$T$$^{1.1}$$

A measurement of its thermal capacity inside a container having the constant volume of 100 L shows that under these conditions, the thermal capacity (heat) capacity is constant and equal to 0.1 calories / K. Express the energy of the system as a function of T and V."

Homework Equations

The first law: dQ = dU + dW.

Since we want U=U(T,V), we can express the first law as:

$$\left(\frac{\partial Q}{\partial T}\right)_V dT + \left(\frac{\partial Q}{\partial V}\right)_T dV = \left(\frac{\partial U}{\partial T}\right)_V dT + [(\left(\frac{\partial U}{\partial V}\right)_T + (\frac{10^9*T^{1.1}}{V^{1.2}})] dV$$

We also know:

$$\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$$

where again p can be substituted from the defined equation of state.

The Attempt at a Solution

I'm tempted to say that $$\left(\frac{\partial Q}{\partial T}\right)_V = 0.1 cal / K$$ with the corresponding V = 100 L. If I do this and assume all dV = 0 for constant volume I get what appears to be a trivial result. I feel like I need to get a perfect differential to solve for U(T,V), but I don't see how. Other than that I haven't done much more than toy with the above equations. Thanks for your time.

 

[Chasing_Time]

I have looked at the problem again recently but still don't have much more insight. I see that, by the first law, since no work is being done (constant volume):

$$\left(\frac{\partial Q}{\partial T}\right)_V = C_v = \left(\frac{\partial U}{\partial T}\right)_V$$

Using $$\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p$$, setting up a perfect differential and taking the derivatives, we can get:

$$dU = (0.1 cal / K) dT + \frac{10^{8}T^{1.1}}{V^{1.2}}dV$$, but I don't see how to go further. Any ideas?

 

[Mapes]

Try expressing the constant-volume heat capacity as

$$C_V=T\left(\frac{\partial S}{\partial T}\right)_V$$

Then determine S by integration (note that a function of V appears that needs to be evaluated via a Maxwell relation), and use

$$U=TS-PV+\mu N$$

where $\mu$ needs to be evaluated by using the heat capacity condition.

This was pretty difficult to work through. There may be an easier way. I eventually got $U=(0.1\,\mathrm{cal}/\mathrm{K})T$ plus a function of T and V that vanished when $V=100\,\mathrm{L}$.