11. (JEE Main 2022 (Online) 28th July Morning Shift )

Given below are two statements :

Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature.

Statement II : The rms speed of oxygen molecules in a gas is $v$ . If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2 v$ .

In the light of the above statements, choose the correct answer from the options given below :

(A) Both Statement I and Statement II are true

(B) Both Statement I and Statement II are false

(C) Statement I is true but Statement II is false

(D) Statement I is false but Statement II is true

Correct answer is (D)

Average momentum $= ⟨ \vec{P} ⟩ = 0$

$v_{r m s} = \sqrt{\frac{3 R T}{M}}$

If temperature is doubled and oxygen atoms are used then

$v_{r m s}^{'} = \sqrt{\frac{3 R (2 T)}{M / 2}} = 4 v_{r m s}$

12. (JEE Main 2022 (Online) 27th July Morning Shift )

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $1 : 4$ , then

A. The r.m.s. velocity of gas molecules in two vessels will be the same.

B. The ratio of pressure in these vessels will be $1 : 4$ .

C. The ratio of pressure will be $1 : 1$ .

D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $1 : 4$ .

Choose the correct answer from the options given below :

(A) A and C only

(B) B and D only

(C) A and B only

(D) C and D only

Correct answer is (C)

$v_{r m s} = \sqrt{\frac{3 R T}{M_{0}}}$ because T is same

v_rms will be same so, A is correct D is incorrect

$\frac{P_{1}}{P_{2}} = \frac{n_{1} R T_{1} / V_{1}}{n_{2} R T_{2} / V_{2}} = \frac{n_{1}}{n_{2}} = \frac{1}{4}$

B is correct

C is incorrect

13. (JEE Main 2022 (Online) 25th July Evening Shift )

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is $\sqrt{2}$ times the speed of sound, then the value of n will be :

(A) 1

(B) 2

(C) 3

(D) 4

Correct answer is (B)

Molar mass $M - \frac{2 \times 4 + n \times 1}{2 + n}$ ..... (i)

Also, $γ = \frac{n_{1} C_{P_{1}} + n_{2} C_{P_{2}}}{n_{1} C_{V_{1}} + n_{2} C_{V_{2}}} = \frac{2 \times 5 R + n \times 7 R}{2 \times 3 R + n \times 5 R}$

$\Rightarrow γ = \frac{10 + 7 n}{6 + 5 n}$ ...... (ii)

Given that $V_{r m s} = \sqrt{2} V_{s o u n d}$

$\Rightarrow \sqrt{\frac{3 R T}{M}} = \sqrt{2} \sqrt{\frac{γ R T}{M}}$

$\Rightarrow γ = \frac{3}{2}$

$\Rightarrow n = 2$

14. (JEE Main 2022 (Online) 28th June Evening Shift )

What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?

(A) The velocity of atomic oxygen remains same

(B) The velocity of atomic oxygen doubles

(C) The velocity of atomic oxygen becomes half

(D) The velocity of atomic oxygen becomes four times

Correct answer is (B)

As $v_{r m s} = \sqrt{\frac{3 R T}{M_{0}}}$

T is doubled and oxygen molecule is dissociated into atomic oxygen molar mass is halved.

So, $v_{r m s}^{'} = \sqrt{\frac{3 R T \times 2 T_{0}}{M_{0} / 2}} = 2 v_{r m s}$

So velocity of atomic oxygen is doubled.

15. (JEE Main 2022 (Online) 25th June Morning Shift )

The relation between root mean square speed (v_rms) and most probable sped (v_p) for the molar mass M of oxygen gas molecule at the temperature of 300 K will be :

(A) $v_{r m s} = \sqrt{\frac{2}{3}} v_{p}$

(B) $v_{r m s} = \sqrt{\frac{3}{2}} v_{p}$

(C) $v_{r m s} = v_{p}$

(D) $v_{r m s} = \sqrt{\frac{1}{3}} v_{p}$

Correct answer is (B)

$v_{r m s} = \sqrt{\frac{3 R T}{M}}$

$v_{p} = \sqrt{\frac{2 R T}{M}}$

$\Rightarrow v_{r m s} = \sqrt{\frac{3}{2}} v_{p}$

Kinetic Theory of Gases