How to find equation of state of an ideal gas from heat capacity?

I'm having a problem where I have to derive the equation of state of an ideal gas from the formula for molar heat capacity $$C=C_V+ \beta V$$. Is this even ideal gas? Can someone help me, or at least give me a clue on how to solve this problem?

• What is the exact word-for-word statement of the problem? – Chet Miller Jun 14 at 11:43
• I forgot. I have to do it for an exam, and I went home remembering it like this. Just want to know the answer since the teacher is unlikely to release solutions until someone solves it (he reused exam question lol) – J. Doe Jun 15 at 8:59
• I'm assuming an ideal gas since the goal is find the equation from the heat capacities. $dQ=dU+dW, dU=C_{v}dT \rightarrow dQ=C_{V}dT+dW$. $dQ=dH-VdP, dH=C_{p}dT \rightarrow dQ=C_{P}dT-VdP$. Hence, $C_{V}dT+dW=C_{P}dT-VdP$ implies $C_{V}dT+PdV=C_{P}dT-VdP$ where $dW=PdV$. Hence $RdT=d(PV)$ where the molar heat capacities $C_{P}-C_{V}=nR$. Or $PV=nRT$ – Cinaed Simson Jun 15 at 19:36

You have to use the Maxwell thermodynamical relations to derive a potential. Most likely you need to start from $$\frac{\partial U}{\partial T} = C_{V}\\C=\frac{\partial Q}{\partial T}$$
Take a look at the thermodynamic square, maybe you will first need to derive a potential ($$U$$?). Once you have a potential it is trivial to get the equation of state.