- #1

Potatochip911

- 318

- 3

## Homework Statement

Prove that ##TdS = C_vdT + \alpha T / \kappa dV##

## Homework Equations

##T dS = dU - pdV##

##\alpha = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right )_P##

##\kappa = -\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T##

## The Attempt at a Solution

The ##C_vdT## part is quite easy since for a constant volume process ##dU = C_vdT## but I can't seem to figure out how to get the second part of the expression. After multiplying by forms of 1 I end up with $$-pdV = \frac{\alpha\left(\frac{\partial v}{\partial P}\right)_T}{\kappa \left(\frac{\partial v}{\partial T}\right)_P}PdV$$, now using the cyclical rule here doesn't seem logical since that would introduce a negative so it seems like I need to replace the pressure P with something although I'm not sure what relation I can use to do that.