Derive a relation between the two specific heats of a gas on the basis of the first law of thermodynamics.

Relation between C_P and C_V :  Suppose one mole of a gas is heated so that its temperature rises by dT.

Heat supplied = 1 x C_V x dT = C_V dT................................(i)

Since the volume is constant, the gas will not perform external work in accordance with the first law of thermodynamics and the heat supplied will be just equal to the increase in the internal energy of the gas.

Therefore, dU = C_V dT ...........................................(ii)

Let the gas be heated at a constant pressure to again increase its temperature by dT and dQ be the amount of  heat supplied, therefore,

dQ = 1 x C_P x dT = C_P dT .................................(iii)

The heat supplied at a constant pressure increases the temperature by dT, hence increases its internal energy by dU as well as enables the gas to perform work dW.

dW = PdV...........................................(iv)

From the first law of thermodynamics, we have, dQ = dU -dW

Substituting the values we get, C_P dT = C_V dT + P dV

But PV =RT (For one mole of the gas) or P dV = R dT

or C_P - C_V = R .........................................(v)

This is the relation between two principle specific heats of the gas, when C_P, C_V, and R are measured in the units of either heat or work.