Question

Starting from the expression for the total differential of enthalpy, H, express (δH/δP)_T in terms of C_p and the Joule-Thompson coefficient.

Answer 1

Starting with the total differential of the enthalpy:
dH = (∂H/∂T)_p dT + (∂H/∂p)_T dp
Note that one definition of the constant-pressure molar heat capacity is:
n x Cp = (∂H/∂T)_p
so:
dH = n x Cp dT + (∂H/∂p)_T dp
To express the second partial derivative in terms of other thermodynamic function, use the mathematical relationship among partial derivatives known as the reciprosity theorem:

(∂x/∂y)z x (∂y/∂z)_x x (∂z/∂x)_y = -1
Applying this relationship in this case gives:
(∂H/∂p)_T x (∂p/∂T)_H x (∂T/∂H)_P = -1
(∂H/∂p)_T = -(∂T/∂p)_H x (∂H/∂T)_P
The definition of the Joule-THompson coefficient is:
μ = (∂T/∂p)_H,

(∂H/∂p)_T = -n x μ xCp

Plugging this into the total differential:
dH = n x Cp dT - n x μ xCp dp
Assuming Cp and μ are independent of temperature and pressure, we can integrate this to get:

ΔH = n x Cp x (T_final - T_initial) - n x μ x Cp x (P_final - P_initial) = n x Cp x ΔT - n xμ x Cp x ΔP

This is the general expression for the change in enthalpy for a change in temperature and pressure. The expression you are trying to get to is obviously for a constant-temperature process (i.e., dT = ΔT = 0), with P1 = P_initial and P2 = P_final:

ΔH = n x μ x Cp x P_initial - P_final)