**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 = nC_{v}ΔT

Where,

C_{v} is a constant,
depending upon the nature of gas.

Hence,

\(C=\frac{Q}{n\Delta T}\),

If n = 1 and ΔT = 1 then C_{v}
= 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 = nC_{v}ΔT

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

U = nC_{v}ΔT

In particular, for an isochoric process:

Q = ΔU = nC_{v}ΔT [∵ W = 0]

Q = nC_{v}ΔT.