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

Asked by Topperlearning User | 4th Jun, 2014, 01:23: PM

Expert Answer:

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

Heat supplied = 1 x CV x dT = CV 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 = CV 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 CP x dT = CP 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, CP dT = CV dT + P dV

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

or CP - CV = R .........................................(v)

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

Answered by  | 4th Jun, 2014, 03:23: PM