# 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? Jun 14, 2019 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) Jun 15, 2019 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 $PV=nRT$ Jun 15, 2019 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.