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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.