1. ⇒ (MHT CET 2023 11th May Evening Shift )

If temperature of gas molecules is raised from $127^{\circ} C$ to $527^{\circ} C$ , the ratio of r.m.s. speed of the molecules is respectively

A. $1 : 2$

B. $2 : 1$

C. $1 : \sqrt{2}$

D. $2 : \sqrt{2}$

Correct Option is (C)

We know

$V_{rms} = \sqrt{\frac{3 RT}{m}}$

The temperature of the same gas molecule is raised.

$V_{ms} \propto \sqrt{T}$

$∴$ The ratio of the velocities is

$\begin{aligned} \frac{V_{1}}{V_{2}} = \frac{\sqrt{T_{1}}}{\sqrt{T_{2}}} \\ \frac{V_{1}}{V_{2}} = \frac{\sqrt{400}}{\sqrt{800}} \\ \frac{V_{1}}{V_{2}} = \frac{1}{\sqrt{2}} \end{aligned}$

Convert temperatures given in centigrade to Kelvin before calculation.

2. ⇒ (MHT CET 2023 11th May Morning Shift )

If a gas is compressed isothermally then the r.m.s. velocity of the molecules

A. decreases.

B. increases.

C. remains the same.

D. first decreases and then increases.

Correct Option is (C)

The root-mean-square (rms) velocity of gas molecules is,

$v_{r, m . s} = \sqrt{\frac{3 kT}{m}}$

$\Rightarrow v_{rms} \propto \sqrt{T}$ for a given gas.

As the temperature is constant in isothermal process, the r.m.s velocity remains the same.

3. ⇒ (MHT CET 2023 9th May Evening Shift )

The pressure exerted by an ideal gas at a particular temperature is directly proportional to

A. the mean speed of the gas molecules.

B. mean of the square of the speed of the gas molecules.

C. the square of the mean speed of the gas molecules.

D. the root mean square speed of the gas molecules.

Correct Option is (B)

Currently no explanation available

4. ⇒ (MHT CET 2023 9th May Evening Shift )

The temperature of an ideal gas is increased from $27^{\circ} C$ to $927^{\circ} C$ . The r.m.s. speed of its molecules becomes

A. twice

B. four times.

C. half.

D. one-fourth.

Correct Option is (A)

$\begin{aligned} We know v_{rms} = \sqrt{\frac{3 R T}{M}} \\ \Rightarrow v_{rms} \propto \sqrt{T} \\ ∴ \frac{v_{mm}}{v_{{mm}_{2}}} = \sqrt{\frac{T_{1}}{T_{2}}} = \sqrt{\frac{300}{1200}} = \sqrt{\frac{1}{4}} \\ \frac{v_{ms}}{v_{{ms}_{2}}} - \frac{1}{2} \\ ∴ v_{{mss}_{2}} = 2 \cdot v_{rms 1} \end{aligned}$

5. ⇒ (MHT CET 2021 21th September Evening Shift )

What is the ratio of the velocity of sound in hydrogen $(γ = \frac{7}{5})$ to that in helium $(γ = \frac{5}{3})$ at the same temperature? (Molecular weight of hydrogen and helium is 2 and 4 respectively.)

A. $\frac{\sqrt{42}}{5}$

B. $\frac{5}{\sqrt{42}}$

C. $\frac{\sqrt{21}}{5}$

D. $\frac{5}{\sqrt{21}}$

Correct Option is (A)

Velocity of sound is given by

$V = \sqrt{\frac{γ R T}{M}}$ $∴$ $\frac{V_{H}}{V_{H e}} = \sqrt{\frac{γ_{H}}{γ_{H e}} . \frac{M_{H e}}{M_{H}}}$ (T is the same for both)

$= \sqrt{\frac{7}{5} \times \frac{3}{5} \times \frac{4}{2}} = \frac{\sqrt{42}}{5}$

⇒Kinetic Theory of Gases