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)
Q α ΔT; n = Constant … (1)
Combining equations (1) and (2), we get:
Q α nΔT
Q = nCvΔT
Cv is a constant, depending upon the nature of gas.
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.