N molecules each of mass m of gas A and 2N molecules each of mass 2m of gas B are contained in the vessel which is maintained at a temperature T. The mean square of velocity of the molecules of B type is denoted by v2 and the mean square of the x-component of the velocity of A type is denoted by w2. The ratio of w2: v2 is

Question

N molecules each of mass m of gas A and 2N molecules each of mass 2m of gas B are contained in the vessel which is maintained at a temperature T. The mean square of velocity of the molecules of B type is denoted by v2 and the mean square of the x-component of the velocity of A type is denoted by w2. The ratio of w2: v2 is (A) 3: 2 (B) 1: 3 (C) 2: 3 (D) 1: 1

Tardigrade Admin · Accepted Answer

The mean square velocity of gas molecules is given by

v^{2} = \frac{3 k T}{m}

.
For gas

A

,

v_{A}^{2} = \frac{3 k T}{m} ... (i)

For a gas molecule,

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2} = 3 v_{x}^{2} (∵ v_{x}^{2} = v_{y}^{2} = v_{z}^{2})

or

v_{x}^{2} = \frac{v ^{2}}{3}

From eqn.

(i)

, we get

w^{2} = v_{x}^{2} = [\frac{\frac{3 k T}{m}}{3}] = \frac{k T}{m} ... (ii)

For gas

B

,

v_{B}^{2} = v^{2} = \frac{3 k T}{m} ... (iii)

Dividing eqn.

(ii)

by eqn.

(iii)

, we get

\frac{w ^{2}}{v ^{2}} = \frac{\frac{k T}{m}}{\frac{3 k T}{2 m}} = \frac{2}{3}

$3 : 2$

$1 : 3$

$2 : 3$

$1 : 1$

3:2

1:3

2:3

1:1

$3 : 2$

$1 : 3$

$2 : 3$

$1 : 1$