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 | |
| 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].
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