Skip main navigation

New offer! Get 30% off one whole year of Unlimited learning. Subscribe for just £249.99 £174.99. New subscribers only T&Cs apply

Find out more

Cp vs. P

Cp vs. P
0.7
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.
60.3
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.

In week 1 article(‘T_ P dependence of heat capacity’), pressure dependence of heat capacity is ignorable in most cases while the temperature dependence is significant. However, heat capacity strongly depends on pressure sometimes.

In this video, the variation of heat capacity with pressure is mathematically derived. The derivation starts from the enthalpy function. Through mathematical manipulation and a property relation, the variation of heat capacity with pressure is estimated.

This article is from the free online

Thermodynamics in Energy Engineering

Created by
FutureLearn - Learning For Life

Reach your personal and professional goals

Unlock access to hundreds of expert online courses and degrees from top universities and educators to gain accredited qualifications and professional CV-building certificates.

Join over 18 million learners to launch, switch or build upon your career, all at your own pace, across a wide range of topic areas.

Start Learning now