Question

Show that $R.M.S$ velocity of gas molecules is directly proportional to square root of its absolute temperature.

Answer 1

Expression for r.m.s velocity:

Let $P=$pressure exerted by one mole of an ideal gas

$V=$volume of one mole of the gas.

$T=$absolute temperature.

Pressure exerted by gas is given by, $P=\frac{1}{3}\frac{M{c}^2}{V}$

where $M=$ mass of one mole(molecular weight) of the gas.

$PV=\frac{1}{3}M{c}^2$ ......$\left(1\right)$

But for one mole of an ideal gas,$PV=RT$

$\therefore RT=\frac{1}{3}M{c}^2$ from eqn$\left(1\right)$

$\therefore c^2=\frac{3RT}{M}$

$\therefore c=\sqrt{\frac{3RT}{M}}$ ......$\left(2\right)$

represents expression for r.m.s velocity of gas molecules.

As $R$ and $M$ in eqn$\left(2\right)$ are constant,

$\therefore c\propto \sqrt{T}$

$\therefore \frac{c_1}{c_2}=\sqrt{\frac{T_1}{T_2}}$