Heat Capacity - Relationship between Cp and Cv

What is Heat Capacity?

When heat is absorbed by a body, the temperature of the body increases and when heat is lost, the temperature decreases. The temperature of an object is the measure of the total kinetic energy of the particles that make up that object. So when heat is absorbed by an object this heat gets translated into the kinetic energy of the particles and as a result the temperature increases. Thus, the change in temperature is proportional to the heat transfer.

The formula q = n C ∆T represents the heat q required to bring about a ∆T difference in temperature of one mole of any matter. The constant C here is called the molar heat capacity of the body. Thus, the molar heat capacity of any substance is defined as the amount of heat energy required to change the temperature of 1 mole of that substance by 1 unit. It depends on the nature, size, and composition of the system.

In this article, we will discuss two types of molar heat capacity – C_P and C_V and derive a relationship between Cp and Cv.

What are Heat Capacity C, C_P, and C_V?

• The molar heat capacity C, at constant pressure, is represented by C_P.
• At constant volume, the molar heat capacity C is represented by C_V.

In the following section, we will find how C_P and C_V are related, to an ideal gas.

The relationship between C_P and C_V for an Ideal Gas

From the equation q = n C ∆T, we can say:

At constant pressure P, we have

q_P = n C_P∆T

This value is equal to the change in enthalpy, that is,

q_P = n C_P∆T = ∆H

Similarly, at constant volume V, we have

q_V = n C_V∆T

This value is equal to the change in internal energy, that is,

q_V = n C_V∆T = ∆U

We know that for one mole (n=1) of an ideal gas,

∆H = ∆U + ∆(pV ) = ∆U + ∆(RT) = ∆U + R ∆T

Therefore, ∆H = ∆U + R ∆T

Substituting the values of ∆H and ∆U from above in the former equation,

C_P∆T = C_V∆T + R ∆T
C_P = C_V + R
C_P – C_V = R

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