# Why specific heat at constant pressure is greater than specific heat at constant volume?

I know the relation between specific heat at constant volume and pressure and I also know how to calculate it. Thing is, I don't understand its concept I want to know why at constant pressure, specific heat is always greater than at constant volume

At constant volume, all the heat that goes into the system goes into raising the temperature of the system, and no external work is done.

At constant pressure, some of the heat goes into expanding the system, which does external work, and therefore leaves less energy available for raising the temperature.

• This explanation can't be correct in general, as the constant-pressure heat capacity always exceeds the constant-volume heat capacity, but some materials shrink when you heat them, meaning that they can't possibly do any "external work"—in fact, the atmosphere does work on them. (The explanation is correct for the ideal gas, though.) Jul 15 '21 at 23:24

Specific heat at constant volume represents the heat supplied to a unit mass of the system to raise its temperature through 1K, keeping the volume constant. Since, V= Constant, dV = 0 and the work done by the system W = PdV = 0. The first law of thermodynamics says: Q = (dU+W) = (dU+PdV) = dU.

Specific heat at constant pressure represents the heat supplied to a unit mass of the system to raise its temperature through 1K, keeping the pressure constant. Since, P= Constant, dV > 0 and the work done by the system, W = PdV > 0. The first law of thermodynamics says: Q = (dU+W) = (dU+PdV) > dU.

As can be seen from the above, we need a quantity equal to dU units of heat to raise the temperature by 1K under constant volume conditions, where as we need a greater quantity, (dU + W)>dU units of heat to raise the temperature by 1K under constant pressure conditions.

Thus we find that we need more heat to raise the temperature of unit mass of the system through 1K under constant pressure conditions, compared to the heat required to raise the temperature of the same unit mass of the system through the same 1K, under constant volume conditions.

When the gas is heated in constant volume,the volume is constant so their will not be further expansion of gas only the internal energy increase but in case of constant pressure the volume changes therefore an extra heat should be applied to increase the temperature of system as some heat is lost during expansion this is the reason why cp>cv