The molecules of a given mass of a gas have r.m.s. velocity of 200 at and pressure. When the temperature and pressure of the gas are respectively, and , the r.m.s. velocity of its molecules in is [2016]

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Solution

Q.
The molecules of a given mass of a gas have r.m.s. velocity of 200 ${ms}^{- 1}$ at $27 ° C$ and $1.0 \times 10^{5}$ ${Nm}^{- 2}$ pressure. When the temperature and pressure of the gas are respectively, $127 ° C$ and $0.05 \times 10^{5}$ ${Nm}^{- 2}$ , the r.m.s. velocity of its molecules in ${ms}^{- 1}$ is [2016]

1 $\frac{100 \sqrt{2}}{3}$

2 $\frac{100}{3}$

3 $100 \sqrt{2}$

4 $\frac{400}{\sqrt{3}}$

Ans.
(4)

As, $v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$

$∴$ $\frac{v_{27}}{v_{127}} = \sqrt{\frac{27 + 273}{127 + 273}} = \sqrt{\frac{300}{400}} = \frac{\sqrt{3}}{2}$

or $v_{127} = \frac{2}{\sqrt{3}} \times v_{27} = \frac{2}{\sqrt{3}} \times 200 {ms}^{- 1} = \frac{400}{\sqrt{3}} {ms}^{- 1}$

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