Question

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

Answer 1

The energy is proportional to the absolute temperature.

Explanation:

An immediate deduction of kinetic theory is the pressure expression,

`p = (mnv^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 = N/V` where `N` is number of molecules.

Making this substitution,

`pV = (mNv^2)/3`

But, `(mv^2)/2 = E` is kinetic energy of a molecule.

Therefore, `pV = (2NE)/3`

But, from ideal gas equation,

`pV = muRT` where, `mu = N/N_A` is the number of moles.

Using these results,

`muRT = (2NE)/3`

`implies (NRT)/N_A = (2NE)/3`

Which finally gives,
`E = (3kT)/2`

Where, `k = R/N_A` is the Boltzmann constant.

Thus the energy is proportional to the temperature.

This the the kinetic interpretation of temperature.