Expressions for constant volume and constant pressure heat capacities My lecture handout says:

$$C_V = \frac{\mathrm{d}q_V}{\mathrm{d}T} = \left(\frac{\partial U}{\partial T}\right)_{\!V}$$

I understand that $C_V = \mathrm{d}q_V/\mathrm{d}T$ and that $\mathrm{d}U = \mathrm{d}q_V$, but what's the need for the partial derivative? Why can't you just substitute and get:

$$C_V = \frac{\mathrm{d}U}{\mathrm{d}T}$$

Likewise for the constant pressure heat capacity:

$$C_p = \frac{\mathrm{d}q_p}{\mathrm{d}T} = \left(\frac{\partial H}{\partial T}\right)_{\!p}$$
The difference in internal energy ($\mathrm{d}U$) is a function of many different variables:

$$\mathrm{d}U = T\,\mathrm{d}S - p\,\mathrm{d}V \!$$

Therefore you must specify – via a partial derivative – with respect to which of the variables you are doing the derivative, otherwise you would need a total derivative, which is not what the equation requires.