Cp vs. P

Now, let’s apply the property relation to see the pressure dependence of Cp. Cp is (dH over dT) at constant pressure. Let’s start from enthalpy as a function of temperature and pressure. Then, the total differential of enthalpy is like this. (dH over dT) at constant P times dT + (dH over dP) at constant T times dP. This is Cp by definition. Then from the exactness of a state function such as enthalpy, differentiation of Cp with respect to pressure at constant T is the same with differentiation of (dH over dP) at constant T with respect to temperature as shown in this equation. So the temperature dependence of this function, dH over dP, gives pressure dependence of Cp.

Let’s look at the temperature dependence of this function. Previously we have shown that (dH over dP) at constant T is - T times (dV over dP) at const P + V. Insert it into this equation, we just derived. Then, we need to differentiate (V - T times (dV over dT) at const P) with respect to temperature. Expand the equation. It becomes (dV over dT) - (dV over dT) - T times second derivative of V with respect to T. Cancel out the same things. Then, dCp over dP at constant T is - T times second derivative of V with respect to T at constant P.