Show that the heat capacity at constant pressure equals the

Show that the heat capacity at constant pressure equals the temperature derivative of the enthalpy at constant pressure and number: C_P = (delta H/delta T)_p, N

Solution

We know that enthalpy of a system is defined as: H = E + pV

Also, from the first law of thermodynamics, we know that: q = E + pV

Now, we heat capacity can be defined as the rate of change of q with respect to temperature at constant pressure and number of particles.

That is C_p = (E/T)_P,N + p(V/T)_P,N

We subsitute the expression for E in terms of enthalpy to get:

C_p = ((H-pV)/T)_P,N + p(V/T)_P,N = (H/T)_P,N

Therefore we get thee expression as required. The heat capacity at constant pressure equals the temperature derivative of enthalpy at constant pressure and number.