Show that for an ideal gas, Cp - Cv= R

From the definitions , it is clear that two heat capacities are not equal and C_P is greater than C_V by a factor which is related to the work done. At a constant pressure part of heat absorbed by the system is used up in increasing the internal energy of the system and the other for doing work by the system. While at constant volume the whole of heat absorbed is utilized in increasing the temperature of the system as there is no work done by the system. Thus increase in temperature of the system would be lesser at constant pressure than at constant volume. Thus C_P is greater than C_v.

We know             C_p = dH/dT                               …(1)

And                      C_v = dE/dT                               ….(2)

By definition,  H = E + PV for 1 mole of an ideal gas

Or                           H = E + RT                        (PV= RT)

Differentiating w.r.t. temperature, T, we get

                                dH/dT = dE/dT + R

or                            C_p = C_v+ R                            By using (1) and (2)

or                            C_p – C_v = R

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