## Specific Heats of Gases

The First Law of thermodynamics does not state how the energy is to be supplied to a system. It simply states that the total energy remains conserved. But, in reality, the quantity of heat required to be supplied to the system to raise its temperature through 1 °C depends on whether the heat is added at constant volume or constant pressure. In the case of a liquid or solid, the specific heat does not vary appreciably with pressure or volume. So, the manner in which heat is added does not matter much. But it is not quite so with a gas where it can vary considerably depending upon whether heat is added at constant volume or constant pressure. It is obvious that when heat is added at constant pressure, some additional heat is required to do work against external pressure. So, prima facie, specific heat at constant pressure should be greater than that at constant volume. This can be shown as follows:

Let us take 1 mol of a gas and add a quantity of heat §Q to it. Also, let us denote the specific heat at constant volume by cv and that at constant pressure by cp. Then, using (3.2.2), we derive thermodynamic expressions for them as follows:

(a) Specific heat at constant volume, cv

Since volume is kept constant, the second term in (3.2.2) vanishes and we get, cv =(5Q/dT)v = (dU/dT)v (3.3.1)

(b) Specific heat at constant pressure, cp

In this case, the pressure is kept constant but the volume changes, so that the substitution for dU from (3.2.1) in (3.2.2) gives

Expressing dV as a function of p and T, we obtain, for constant pressure, dV = (dV/dT)p dT (3.3.3)

Using this value of dV in (3.3.2) and dividing by dT, we get

In the case of an ideal gas, it was shown by the experiments of Gay-Lussac that the internal energy U does not vary with volume at constant temperature. So, in (3.3.4), we put (dU/dV)T = 0 and get cp - cv = p(dV/dT)p (3.3.5)

On account of the equation of state for dry air, (3.3.5) simplifies to cp - cv = p(dV/dT)p = R (3.3.6)

This is known as Mayer's equation for an ideal gas.

Thus, the difference between the values of the specific heats of an ideal gas at constant pressure and constant volume is equal to value of the gas constant of that particular gas. Since dry air is composed largely of diatomic molecules, it can be shown that the values of its specific heats are:

For dry air, R = R*/Md, where Md is the molecular weight of dry air. Substitution of the value of R(= 287 J Kg-1 K-1) gives cp = 1004.5J Kg-1 K-1 cv = 717.5JKg-1 K-1

Integration of (3.3.1) gives the energy function U of an ideal gas as

o where Uo is the internal energy at T = 0.