Adiabatic equation

For an adiabatic process, $\db Q = 0$ so $\d U = -p \d V$ .

For an ideal gas ( $PV = N k_B T$ ), the energy $U = \frac f2 N k_B T$ where $f$ is the number of active degrees of freedom. Thus $\d U = \frac f2 N k_B \d T$ .

We differentiate the ideal gas law to find

\begin{align*} N k_B \d T &= p \d V + V \d p \\ -p \d V &= \frac f2 \big(V \d P + p \d V\big)\\ 0 &= \left(\frac f2 + 1 \right) p\d V + \frac f2 V \d p \end{align*}

Now define $\gamma = 1 + \frac 2f$ . Since we are considering an ideal gas, $\gamma = C_p / C_v$ is the heat capacity ratio. This gives us

\gamma p \d V + V \d p = 0.

The solution is

\begin{align*} \d (pV^\gamma ) &= 0 \\ pV^\gamma &= \text{constant}. \end{align*}

This is the adiabatic equation. It only holds for adiabatic processes.