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

Explanation

Let, one mole of an ideal gas enclosed in volume (V) with pressure (P) and temperature (T) then, pressure exerted by gas is given by
`P=(1)/(3)(m)/(V)C_(r.m.s)^(2) " " ` …(i)
Where `C_(r.m.s)` is the RMS speed of the gas molecules and m is the mass of the gas.
If there are n moles of the gas and M is the molar mass then,
m = nM
For one mole, n = 1
m = M
Then, equation (i) becomes
`P=(1)/(3) (M)/(V) C_(r.m.s)^(2) `
`PV=(1)/(3)MC_(r.m.s)^(2)" " ` ....(ii)
For an ideal gas : PV = RT
`RT=(1)/(3) MC_(r.m.s)^(2) " " `[From equation (ii)]
`C_(r.m.s)^(2) =(3RT)/(M)`
`C_(r.m.s)=sqrt((3RT)/(M))`
`C_(r.m.s) prop sqrt(T) (because (3R)/(M)" is constant")`
Thus, RMS velocity of molecules of gas is directly proportional to square root of its absolute temperature.