Derivatives - Examples and Exercises

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Preknowledge

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


Example

Prove the following:


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


Solution


Note the rules mentioned are given HERE


[math]C_P = \left (\frac{\partial H}{\partial T} \right )_P[/math] by definition


Using rule #2


[math]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[/math]


since [math]H=U+PV[/math]


[math]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[/math]


The first term on the right is CV


[math]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[/math]


Using rule #1


[math]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[/math]


Using rule #4 (Euler's relation)


[math]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 ][/math]


rearranging,


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

Exercise

If

[math]\kappa_T = -\frac{1}{V} \left (\frac{\partial V}{\partial P} \right )_T[/math]

and

[math]\kappa_S = -\frac{1}{V} \left (\frac{\partial V}{\partial P} \right )_S[/math]


Prove the following:


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