I have a question pertaining to the following derivation:
For the heat capacity at constant volume part, we apparently have:
$$dQ = C_v dT + P dV$$
But I find this confusing, as if we are to assume volume is constant, then $dV =0 $ so I would say that
$$dQ = C_v dT$$
Secondly, I don't understand the part how the previous thing I addressed implies
$$C_p = \frac{d Q_p}{dT}$$
As $$C_V = \left(\frac{\partial U}{\partial T}\right)_V = \frac{d U}{dT}$$
So I'd assume
$$C_P = \left(\frac{\partial U}{\partial T}\right)_P = \frac{d U}{dT}$$
but this doesn't look to be the case. What's going on here?
Best Answer
I call this equation ($C_V=(\partial U/\partial T)_V$) the cruelest equation in introductory thermodynamics because of how often it trips people up.
It looks like the misconception here is thinking that the heat capacity is how much the internal energy $U$ increases for a given increase in temperature $T$. This is not the case. I recommend thinking of the heat capacity as how much you have to heat the system to obtain a given increase in $T$. At constant pressure, this heating corresponds to the increase in enthalpy $H$, not the increase in internal energy $U$. Thus, $C_P=(\partial H/\partial T)_P$.