Show that for one mole of an ideal gas, Cp – C V = R. Use the fact that PV = nRT for an ideal gas and that the energy U of an ideal gas depends only on temperature T and not on volume/pressure.
Solution
Expand U as a function of T and V
dU = (U/T) V dT + (U/V) T dV
Take the derivative of this with respect to T at constant p. Note that (dT/dT) p = 1.
(U/T) p = (U/T) V + (U/V) T (V/T) p = C V + (U/V) T (V/T) p
Now Cp = (H/T)p
(H/ T)P = (U/T) p + ((pV)/T) p = (U/T) p + P(V/T) p
Substitute for ( U/ T) p from the earlier equation
Cp = C V + (V/T) p ((U/V) T + P)
But , (V/T)p=/T (nRT/p) = nR/P and (U/V)T = 0 for an ideal gas
