Recall from the previous post that the First Law of Thermodynamics can be expressed

\[ \alpha+\beta = -dU \]

where \(U\) is the total energy of a thermodynamic system and \(\alpha,\beta\) are one-forms whose integrals along a transformation express the work done by the system on its environment, and the heat supplied by the system to its environment, in the course of the transformation. (In Fermi’s notation \(\alpha=dL\), \(\beta=-dQ\), where \(L,Q\) should be thought of as functions not on the state space but on the *path space* of the state space – their values depend on how you got there). If the state space is a 2-manifold parametrized by volume \(V\) and pressure \(p\), then \(\alpha=pdV\). The **thermal capacity** of the system is the derivative \(dQ/dT\), that is the marginal amount of heat absorbed for an increase in temperature. There are two versions of the heat capacity, \(C_V\) (heat capacity at constant volume) and \(C_p\) (heat capacity at constant pressure). For the situation we’re considering, \(\alpha=pdV = 0\) at constant volume, so

\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V,\qquad C_p = \left(\frac{\partial U}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p \]

by standard calculations with partial derivatives. Continue reading →