**Specific Heat
of a Gas at Constant Pressure**

Consider n moles of a gas contained in a cylinder fitted with a frictionless movable piston bearing some load. Let heat be supplied to the gas slowly so that the pressure of the gas never exceeds the external pressure more than by an infinitesimal amount. Assuming, no loss of heat to the surrounding, if Q be the total heat supplied to the gas and ΔT the total rise in temperature. Then, it is found that:

Q α n; ΔT = Constant … (1)

Also,

Q α ΔT; n = Constant … (2)

Combining equations (1) and (2) we get:

ΔQ α nΔT

Q = C_{P}nΔT … (3)

Where, C_{P} is a constant, depending upon the nature of the gas and is known as the molar heat capacity of the gas at constant pressure.

Now, from equation (3):

\({{C}_{p}}=\frac{Q}{n\Delta T}\).

If n = 1 and ΔT = 1, then C_{P}
= Q. Thus, C_{P} is the amount of heat required to raise the
temperature of unit mole of a gas by unit degree at constant pressure, and is
known as the molar heat capacity of the gas at constant pressure.

From, first law of thermodynamics:

Q = ΔU + W

nC_{P}ΔT = nC_{V}ΔT +
PΔV

From equation of state for an ideal gas: PV = nRT

PΔV = nRΔT

nC_{P}ΔT = nC_{V}ΔT +
nRΔT … (3)

C_{P} = C_{V} + R …
(4)

From equation (4), we can write:

C_{P} – C_{V} = R is
known as Mayer’s Relation.