Process

A thermodynamic process is a change in the state of the system. Different types of thermodynamic processes have different properties, such as constant temperature, zero heat, constant pressure, etc.

Adiabatic process

An adiabatic process has zero heat transfer $\Delta Q = 0$ . For such a process, $\d U = -p \d V$ . From the adiabatic equation we know $pV^\gamma = \text{constant}$ .

The work done is

\begin{align*} \Delta W &= \int -p \d V = \int_{V_0}^{V_1} \frac{p_0 V_0^\gamma}{V^\gamma} \d V \\ &= -p_0 V_0^\gamma \frac{1}{1-\gamma} \left[V^{1-\gamma}\right]_{V_0}^{V_1} \\ &= \frac{p_0V_0}{\gamma - 1}\left[ V_1^{1-\gamma} - V_0^{1-\gamma} \right] \\ &= \frac{N k_B T}{\gamma - 1} \left[ \left( \frac{V_0}{V_1} \right)^{\gamma - 1} - 1 \right], \quad\begin{cases} <0 & \text{if } V_0 < V_1, \\ >0 & \text{if } V_0 > V_1. \end{cases} \end{align*}

The adiabat $pV^\gamma = \text{constant}$ is steeper than the isotherm $pV = \text{constant}$ since $\gamma > 1$ . Graphically, this means that the adiabatic work done is greater than the isothermal work, since the area in phase space is larger.

Isochoric/isometric process

For an isochoric process, $\Delta V = 0$ , so $\Delta W = 0$ and $\d U|_V = \db Q|_V$ . In this case

\begin{align*} \frac{\db Q}{\d T} &= \frac{\d U}{\d T} = C_V \\ \Delta U &= \int C_V(T,V) \d T. \end{align*}

For an ideal gas, the heat capacity $C_V = \frac f2 N k_B$ . We conclude

\Delta U = \frac f2 N k_B (T_1 - T_0).

Isobaric process

For an isobaric process, $\Delta p = 0$ .