#### Question

Prove that root mean square velocity of gas molecule is directly proportional to the square root of its absolute temperature.

#### Solution

Expression for r.m.s velocity:

**a. ** Let, P = pressure exerted by one mole of an ideal gas

V = volume of one mole of the gas

T = absolute temperature**b**. Pressure exerted by gas is given by,

`P=1/3(Mc^2)/V`

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

`PV=1/3(Mc^2)` .....................(1)

**c.** But for one mole of an ideal gas, PV = RT

`therefore RT=1/3 MC^2` [From Equation 1]

`therefore c^2=(3RT)/M`

`therefore c=sqrt((3RT)/M)`........................(2)

Equation (2) represents expression for r.m.s velocity of gas molecules.

**d.** As R and M in equation (2) are constant,

`therefore c prop sqrtT`

`therefore c_1/c_2=sqrt(T_1/T_2)`