# Preknowledge

 Read the section on Derivatives. You will also need the definitions from the section on Enthalpy and Heat Capacity.

## Example

Prove the following:

$C_P = C_V + \left [V - \left (\frac{\partial H}{\partial P} \right )_T \right ] \left (\frac{\partial P}{\partial T} \right )_V$

Solution

Note the rules mentioned are given HERE

$C_P = \left (\frac{\partial H}{\partial T} \right )_P$ by definition

Using rule #2

$C_P = \left (\frac{\partial H}{\partial T} \right )_V + \left (\frac{\partial H}{\partial V} \right )_T \, \left (\frac{\partial V}{\partial T} \right )_P$

since $H=U+PV$

$C_P = \left (\frac{\partial U}{\partial T} \right )_V + V\, \left (\frac{\partial P}{\partial T} \right )_V + \left (\frac{\partial H}{\partial V} \right )_T \, \left (\frac{\partial V}{\partial T} \right )_P$

The first term on the right is CV

$C_P = C_V + V\, \left (\frac{\partial P}{\partial T} \right )_V + \left (\frac{\partial H}{\partial V} \right )_T \, \left (\frac{\partial V}{\partial T} \right )_P$

Using rule #1

$C_P = C_V + V\, \left (\frac{\partial P}{\partial T} \right )_V + \left (\frac{\partial H}{\partial P} \right )_T \, \left (\frac{\partial P}{\partial V} \right )_T \, \left (\frac{\partial V}{\partial T} \right )_P$

Using rule #4 (Euler's relation)

$C_P = C_V + V \, \left (\frac{\partial P}{\partial T} \right )_V - \left (\frac{\partial H}{\partial P} \right )_T \, \left [ \left (\frac{\partial P}{\partial T} \right )_V \right ]$

rearranging,

$C_P = C_V + \left [ V - \left (\frac{\partial H}{\partial P} \right )_T \right ] \, \left (\frac{\partial P}{\partial T} \right )_V$ ##

## Exercise

If

$\kappa_T = -\frac{1}{V} \left (\frac{\partial V}{\partial P} \right )_T$

and

$\kappa_S = -\frac{1}{V} \left (\frac{\partial V}{\partial P} \right )_S$

Prove the following:

$\kappa_S - \kappa_T = \frac{1}{V} \left (\frac{\partial V}{\partial S} \right )_P \left (\frac{\partial S}{\partial P} \right )_T$.