# The average kinetic energy of the molecules of an ideal gas is directly proportional to what?

Jun 6, 2017

The energy is proportional to the absolute temperature.

#### Explanation:

An immediate deduction of kinetic theory is the pressure expression,

$p = \frac{m n {v}^{2}}{3}$

Where $m$ is mass of a gas molecule, $n$ is no. of molecules per unit volume and $v$ is the rms speed.

Thus, $n = \frac{N}{V}$ where $N$ is number of molecules.

Making this substitution,

$p V = \frac{m N {v}^{2}}{3}$

But, $\frac{m {v}^{2}}{2} = E$ is kinetic energy of a molecule.

Therefore, $p V = \frac{2 N E}{3}$

But, from ideal gas equation,

$p V = \mu R T$ where, $\mu = \frac{N}{N} _ A$ is the number of moles.

Using these results,

$\mu R T = \frac{2 N E}{3}$

$\implies \frac{N R T}{N} _ A = \frac{2 N E}{3}$

Which finally gives,
$E = \frac{3 k T}{2}$

Where, $k = \frac{R}{N} _ A$ is the Boltzmann constant.

Thus the energy is proportional to the temperature.

This the the kinetic interpretation of temperature.