Specific Heat Capacity

Suppose that we add an amount of heat $dQ$ to an ideal gas causing its temperature to rise by $dT$. The specific heat capacity of the gas is defined

$$C = \frac{dQ}{dT}.$$ (5.103)

In fact, an ideal gas possesses a number of different specific heat capacities depending on what is held constant as heat is added to the system. Suppose that the volume is held constant. It follows from Equation (5.102) that $dQ=dU$. Hence, the specific heat capacity at constant volume of the gas is

$$C_V=\left(\frac{\partial Q}{\partial T}\right)_V = \left(\frac{\partial U}{\partial T}\right)_V.$$ (5.104)

However, according to Joule's second law, which was established experimentally by James Joule in 1843, the internal energy of an ideal gas depends on its temperature alone, and is independent of the volume or pressure. We also expect $C_V$ to be an extensive quantity. It follows that

$$dU = \nu\,c_V(T)\,dT,$$ (5.105)

where $c_V = C_V/\nu$ is termed the molar specific heat capacity at constant volume, and is an intensive quantity. In fact, $c_V$ is constant for an idea gas. For a monatomic gas,

$$c_V = \frac{3}{2}\,R,$$ (5.106)

whereas for a diatomic gas,

$$c_V = \frac{5}{2}\,R.$$ (5.107)

(See Sections 5.3.6 and 5.5.8.) In both cases, Equation (5.105) can be integrated to give

$$U(T) = \nu\,c_V\,T.$$ (5.108)

Consider the specific heat capacity of an ideal gas at constant pressure. Making use of Equations (5.102) and (5.105),

$$dQ = \nu\,c_V\,dT + p\,dV.$$ (5.109)

However, at constant pressure, the ideal gas law, (5.97), yields

$$p\,dV = \nu\,R\,dT.$$ (5.110)

The previous two equations give

$$dQ=\nu\,(c_V+R)\,dT.$$ (5.111)

Now, the molar specific heat capacity at constant pressure of an ideal gas is

$$c_p = \frac{1}{\nu} \left(\frac{\partial Q}{\partial T}\right)_p.$$ (5.112)

Hence, we deduce that

$$c_p = c_V + R.$$ (5.113)

Note that the specific heat capacity at constant pressure is greater than that at constant volume, because some of the heat energy added to a gas held at constant pressure is consumed by the work that the gas does on its surroundings, in order to expand its volume slightly, and, therefore, does not lead to an increase in the internal energy (i.e., temperature) of the gas. On the other hand, for a gas held at constant volume, all of the added heat energy goes to increase its internal energy.