# Specific Heat of a Gas at Constant Volume

## Specific Heat of a Gas at Constant Volume

Consider,  moles of an ideal gas, confined in a cylinder, filled with a fixed piston. If  be the heat supplied to the gas and as expected, the increase in temperature be, then experimentally:

Q α n; ΔT = Constant … (1)

Also,

Q α ΔT; n = Constant … (1)

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

Q α nΔT

Q = nCvΔT

Where,

Cv is a constant, depending upon the nature of gas.

Hence,

$$C=\frac{Q}{n\Delta T}$$,

If n = 1 and ΔT = 1 then Cv = Q.

Thus,  is the amount of heat required to raise the temperature of unit mole of a gas by unit degree at constant volume and is known as the molar heat capacity of the gas at constant volume.

Thus, for any gas, ΔU = nCvΔT

Now, we can write inter energy (U) of a gas at any temperature T as:

U = nCvΔT

In particular, for an isochoric process:

Q = ΔU = nCvΔT [∵ W = 0]

Q = nCvΔT.