Question

Why is the specific heat of a gas at constant pressure always greater than the specific heat of a gas at constant volume?

Answer 1

Hint: The specific heat of a gas at constant pressure is the amount of heat required to raise the temperature of one mole of a gas by unit temperature at constant pressure. On the other hand, the specific heat of a gas at constant volume is the amount of heat required to raise the temperature of one mole of the gas by unit temperature at constant volume.

Complete step by step answer:
When heat is given to a gas at constant pressure, the heat is absorbed by the gas and two things happen. Part of the thermal energy (heat) is used by the gas to increase its internal energy. This internal energy also represents the temperature change of the gas. The rest of the heat is utilized by the gas to do mechanical work (the gas expands). Mechanical work done by a gas is defined as the product of the pressure of the gas and the change in volume of the gas.
Hence, the magnitude of the work done $W$ by a gas at constant pressure $P$ is
$W=P\mathrm{\Delta }V$ --(1)
where $\mathrm{\Delta }V$ is the change in volume of the gas.
Hence, a part of the heat results in a rise in temperature and the rest for doing mechanical work at constant pressure.

Now, when heat is given to a gas at constant volume, all of the heat is used to raise the internal energy and the temperature of the gas and no mechanical work is done. This is because in a constant volume process, the change in volume $\mathrm{\Delta }V$ is zero and hence, from (1), we can see that zero work is done by the gas. Hence, all of the heat is used up for raising the temperature of the gas.
The specific heat of a gas at constant pressure is the amount of heat required to raise the temperature of one mole of a gas by unit temperature at constant pressure. On the other hand, the specific heat of a gas at constant volume is the amount of heat required to raise the temperature of one mole of the gas by unit temperature at constant volume.
Therefore, since a lesser amount of heat will be enough for attaining the same rise in temperature in a constant volume process than in a constant pressure process because no heat will be wasted for mechanical work, the specific heat of a gas at constant volume is less than the specific heat of the gas at constant pressure.

Note: The specific heat of a gas at constant pressure and volume are specific constants and parameters of a gas that define a lot of its properties. For example, the ratio of the specific heat at constant pressure to that at constant volume is called the adiabatic constant of the gas.
The fact that the specific heat of a gas at constant pressure $\left(C_P\right)$ is more than that at constant volume $\left(C_V\right)$ is also evident from their mathematical relation
$C_P=C_V+R$
Where $R=8.314J.mo{l}^{-1}{K}^{-1}$ is a constant greater than zero known as the universal gas constant.