# Mean kinetic energy of an ideal gas particle

The **mean kinetic energy of a gas particle $$(\langle e_{k} \rangle )$$** of mass $$m$$ is given by:$$$\langle e_{k} \rangle=\frac{1}{2}m\langle v^{2}\rangle=\frac{3}{2}kT$$$ $$\langle v^{2}\rangle$$ is the mean square speed of the particle.

The mean kinetic energy is **directly proportional** to the thermodynamic temperature $$T$$.

It is necessary to take the mean square speed (as opposed to the mean speed squared $$(\langle v\rangle)^{2}$$) because the mean speed ($$\langle v\rangle$$) of a particle is zero.

This is because the particles within a gas move in random directions and the probability that a particle moves in a particular direction is equal to that of any other direction.

This means that on average, each particle is considered to be stationary (i.e. $$\langle v\rangle=0$$). This can be inferred from the fact that a container filled with gas does not move even though the gas particles are in perpetual motion.

The mean kinetic energy equation is derived by equating the **macroscopic** (i.e. using the ideal gas equation) and **microscopic** (i.e. considering the force of a particle on a wall) pressure values of a gas particle.

$$p=$$pressure of the gas; $$V=$$volume of the gas; $$n=$$number of particles of the gas measured in moles; $$k=$$Boltzmann constant; $$T=$$thermodynamic temperature of the gas.