Question

What is the relation between pressure and kinetic energy of gas?

Answer 1

1. From the kinetic theory of gas, it is known that a gas is made of tiny particles which move in random directions and undergo elastic collisions with themselves and the walls of the container.
2. The particles thus have kinetic energy and the continuous collisions on the walls are what impart pressure.
3. The pressure exerted by a gas is given by the expression, $P=\frac{Nm\overline{v^2}}{3V}$. Here, $P$ is the pressure, $N$ is the total number of molecules, $m$ is the mass of each molecule, $\overline{v^2}$ is the mean of the square of the speed and $V$ is the volume.
4. The kinetic energy of the gas can be expressed as $K=N\frac{1}{2}m\overline{v^2}$, where $K$ denotes the kinetic energy.
5. By comparing the two expressions above, we can get a relation between the pressure and the kinetic energy.

$\begin{array}{rcl}P& =& \frac{Nm\overline{v^2}}{3V}\\ P& =& \frac{2}{3V}N\frac{1}{2}m\overline{v^2}\\ P& =& \frac{2}{3}\frac{K}{V}\\ PV& =& \frac{2}{3}K\end{array}$

Hence, the relation between the pressure and the kinetic energy of a gas is $PV=\frac{2}{3}K$.