We have

dQ= dU + dW

= dU+ PdV

If the volume is constant i.e., V = C or dV = 0

So dW = PdV = 0

(dQ)_{v} = (dU)_{v} … (4.10)

Specific heat is defined as the amount of heat required to raise the temperature of unit mass by one degree.

So the specific heat (c_{v}) is

So we can properly redefine the specific heat at constant volume as “It is the rate of change of specific internal energy with respect to temperature when the volume of system is kept constant”.

We also have

Equation (4.10} states that heat transfer at constant volume increases the internal energy of the system.

Equation (4.11) is a proper definition of C_{v} rather than the previous one.

Heat capacity at constant volume is the product of mass to the specific heat at constant volume.

mc_{v} = C_{v}. The unit of C_{v} = J/K.