Deriving the relationship of Cp - Cv in terms of P, T, V, and H

Question

We're learning the relationship between C_p and C_v. We seem to be doing it a little uniquely because we're looking for equations of C_p - C_v in terms of P, T, V, and U, and in terms of P, T, V, and H

For the first one, which was also isobaric, this was the process:

H = U + PV

dH = dU + d(PV)

dH = dU + PdV + VdP ⁰

dH = dU + PdV

(∂H/∂T)_P = (∂U/∂T)_P + P(∂V/∂T)_P

C_p = (∂U/∂T)_P + P(∂V/∂T)_P [ EQN 1 ]

dU = C_vdT + (∂U/∂V)_TdV

(∂U/∂T)_P = C_v (∂T/∂T)_P ¹ + (∂U/∂V)_T(∂V/∂T)_P

(∂U/∂T)_P = C_v + (∂U/∂V)_T(∂V/∂T)_P [ EQN 2 ]

Plug EQN 2 into EQN 1 to get:

C_p = C_v + (∂U/∂V)_T(∂V/∂T)_P + P(∂V/∂T)_P

C_p - C_v = (∂U/∂V)_T(∂V/∂T)_P + P(∂V/∂T)_P

C_p - C_v = (∂V/∂T)_P[ (∂U/∂V)_T + P]

Now I'm supposed to derive an equation of C_p - C_v at isochoric conditions in terms of P, T, V, and H but I'm just completely lost in a lot of parts.

Supposedly the end equation is as follows:

C_p - C_v = (∂P/∂T)_v [V-(∂H/∂P)_T]

I tried starting from H = U + PV and got C_v = (∂H/∂T)_V - V(∂P/∂T)_V before realizing that's a dead-end.

I'm sure I'll need to start off from H = U + PV but that's as far as I can imagine. Even just where to start off in will be helpful to me. I'm just super stuck.

Answer 1

I don't know about the isochoric constraint, but in general, $$dH=C_pdT+\left(\frac{\partial H}{\partial P}\right)_TdP$$so,$$dU=dH-VdP-PdV$$$$dU=C_pdT+\left[\left(\frac{\partial H}{\partial P}\right)_T-V\right]dP-PdV$$ So, $$C_v=C_p+\left[\left(\frac{\partial H}{\partial P}\right)_T-V\right]\left(\frac{\partial P}{\partial T}\right)_V$$