Thermodynamic Proof: Calculating (∂H/∂T)V

[Themanslayer]

Homework Statement

(∂H/∂T)_V = C_P( 1 - α_Pμ_JT / κ_T)

Homework Equations

1. dH=TdS + VdP
2. dU= Tds - PdV
3. all maxwell relations
4. every equation here: http://en.wikipedia.org/wiki/Bridgman's_thermodynamic_equations

The Attempt at a Solution

- divided equation 1 by (dT)_v
-used cyclic rule
- used maxwells relations

The closest answer I have obtained is the following:
(∂H/∂T)_V = T(∂S/∂T)_V + Vα_P / κ_T

Any help is appreciated.

 

[nate808]

Thank you for sharing your equation and the attempt at a solution. It seems like you have made some progress in using the given equations and relations to simplify the original equation.

However, I noticed that in your attempt, you have used the cyclic rule and Maxwell's relations, but you have not used the fact that H = U + PV. This relation can also be helpful in simplifying the equation further.

Here is a suggestion for a possible solution:

(∂H/∂T)V = (∂/∂T)(U + PV)V = (∂U/∂T)V + V(∂P/∂T)V

Using the equations 2 and 3, we can rewrite this as:

(∂H/∂T)V = T(∂S/∂T)V - P(∂V/∂T)V + V(∂P/∂T)V

Now, using the cyclic rule and Maxwell's relations, we can simplify this further to:

(∂H/∂T)V = T(∂S/∂T)V + VαP / κT

This is the same result that you have obtained, so it seems like you are on the right track. I hope this helps and good luck with your further work on this equation!