For some time now I've been confused about heat capacity. The way I understand it, if I put in an amount of heat energy into the system, $dQ$, its temperature will change by $CdT$. But now, everywhere I look I only find $C_p$ and $C_V$, the heat capacities in isobaric/isochoric processes. But for an arbitrary process, how do I calculate its heat capacity, in other words how do I find $$\frac{\partial Q}{ \partial T}, \mbox{ with } p=p(V)$$
where $p=p(V)$ is the equation of this process.
My question: I wish to calculate the heat capacity in a polytropic process with exponent $n$, so the equation is $pV^n=const$, and so $p=\frac{p_1 V_1 ^n}{V^n}$.
How do I get from here to actually calculating the heat capacity, that is the partial derivative?
P.S. I apologize if I'm mixing or misunderstanding notions here, but the way I was taught thermodynamics was horrible, just learning how to solve particular problem types without understanding the physics of it at all, and I hadn't yet had time to look for some source to correct all these misconceptions. Would be greatful if someone could recommend a mathematically rigorous text on thermodynamics that doesn't make unexplained simplifications and helps intuitively grasp the physics of it.
Best Answer
Let's start by computing heat capacities in a context that is a bit more general than polytropic processes; those defined by the constancy of some state variable $X$.
For concreteness, let's assume we are considering a thermodynamic system, like an ideal gas, whose state can be characterized by its temperature, pressure, and volume $(T,P,V)$ and for which the first law of thermodynamics reads \begin{align} dE = \delta Q - PdV \end{align} We further assume that there exists some equation of state which relates $T$, $V$, and $P$ so that the state of the system can in fact be specified by any two of these variables. Suppose that we want to determine the heat capacity of the system for a quasistatic process (curve in thermodynamic state space) for which some quantity $X=X(P,V)$ is kept constant. The trick is to first note that every state variable can be written (at least locally in sufficiently non-pathological cases), as a function of $T$ and $X$. Then the first law can be written as follows: \begin{align} \delta Q &= dE + PdV \\ &= \left[\left(\frac{\partial E}{\partial T}\right)_{X}+P\left(\frac{\partial V}{\partial T}\right)_{X}\right]dT + \left[\left(\frac{\partial E}{\partial X}\right)_{T}+P\left(\frac{\partial V}{\partial X}\right)_{T}\right]dX \end{align} Now, we see that if we keep the quantity $X$ constant along the path, then $\delta Q$ is proportional to $dT$, and the proportionality function is (by definition) the heat capacity for a process at constant $X$; \begin{align} C_X = \left(\frac{\partial E}{\partial T}\right)_{X}+P\left(\frac{\partial V}{\partial T}\right)_{X} \end{align} Now, if you want an explicit expression for this heat capacity, then you simply need to determine the energy, pressure, and volume functions of $T$ and $X$ and then take the appropriate derivatives.
Consider, for example, a polytropic process like you originally described, and further, consider a monatomic ideal gas for which the energy and equation of state can be written as follows: \begin{align} E = \frac{3}{2} NkT, \qquad PV = NkT \end{align} For this process, we have \begin{align} X = PV^n \end{align} Using the equation of state and the definition of $X$, we obtain \begin{align} V = (NkT)^{1/(1-n)}X^{1/(n-1)}, \qquad P = (NkT)^{n/(n-1)}X^{1/(1-n)} \end{align} and now you can take the required derivatives in to obtain $C_X$ where $X$ is appropriate for an arbitrary polytropic process.
Moral of the story. If the system you care about can be written as a function of only two state variables, write all quantities in terms of $T$ and $X$, the variable you want to keep constant. Then, the first law takes the form $\delta Q = \mathrm{stuff}\,dT + \mathrm{stuff}\,dX$ and the $\mathrm{stuff}$ in front of $dT$ is, by definition, the desired heat capacity.