• This lecture should bring in points that were previously covered in a discussion of the Dulong and Petit Rule.
• It should be noted that although the heat capacity depends on many different factors, the values should all be reasonably close to the value given by Dulong and Petit

As we learned last week, heat capacity is amount of energy required to raise the temperature of an object by a small amount. $$C \sim \frac{đ Q}{\partial T}$$ $$đ Q = C dT \text{ At constant what?}$$ If we hold the volume constant, then we can see from the first law that $$dU = đQ - pdV$$ since $dV=0$ for a constant-volume process, $\newcommand\myderiv[3]{\left(\frac{\partial #1}{\partial #2}\right)_{#3}}$ $$C_V = \myderiv{U}{T}{V}$$ But we didn't measure $C_V$ on Monday, since we didn't hold the volume of the water constant. Instead we measured $C_p$, but what is that? To distinguish between different sorts of heat capacities, we need to specify the sort of path used. So, for instance, we could write $$đQ=Tds$$ $$đQ=C_αdT+?dα$$ $$TdS=C_αdT+?dα$$ $$dS= \frac{C_\alpha}{T} dT + \frac{?}{T}d\alpha$$ $$C_\alpha = T \myderiv{S}{T}{\alpha}$$

This may look like an overly-tricky derivative, so let's go through the first law and check that we got it right in a few cases. I'll do the $C_V$ case. We already know that $$dU=đQ-pdV$$ $$C_V= \myderiv{U}{T}{V}$$ $$= \myderiv{U}{S}{V} \myderiv{S}{T}{V}$$ $$= T \myderiv{S}{T}{V}$$ where the second step just uses the ordinary chain rule.