# Derivatives - Examples and Exercises

Thermodynamics | |
---|---|

Introduction | What is this thing called Thermodynamics??? | Definitions | Thermal Equilibrium and Zeroth Law | Limitations |

First Law | Work, Heat, Energy, and the First Law | Work, Heat, Energy, and the First Law (simplied) | Derivatives | Derivatives Exercise | Reversibility, Enthalpy, and Heat Capacity |

Second Law | Things to Think About | Observations and Second Law of Thermodynamics | Alternative Approach - the Clausis Inequality | Consequences of the Second Law | Consequences of the Second Law (simplified) | Carnot Principle - motivation and examples | Equivalence of Second Law Statements* |

Third Law | Third Law of Thermodynamics | Consequences of Third Law* |

Development of Thermodynamics | The Thermodynamic Network | Network Exercise | Equations of State (EOS) | EOS Example, Reading Tables, and Numerical Analysis | EOS Exercises | Thermochemistry |

* Optional Section |

# 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 C_{V}

[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].