JEE MainPhysicsToughKinetic Theory of Gases

A container of fixed volume has a mixture of one mole of hydrogen
and one mole of helium in equilibrium at temperature T. Assuming the
gases are ideal, the correct statement(s) is (are)

Show answer and solution

\

For hydrogen,

\ n_{1} = 1,C_{v,} = \frac{5}{2}R

For mixture of gases,

C_{v} = \frac{n_{1}C_{V,} + n_{2}C_{V_{2}}}{n_{1} + n_{2}}

= \frac{1 \times \frac{5}{2}R + 1 \times \frac{3}{2}R}{1 + 1} = 2R

C_{P} = C_{v} + R = 3R,\gamma_{mix} = \frac{C_{P}}{C_{v}} = \frac{3}{2}

Also,

\gamma_{mx} = 1 + \frac{2}{f} \Rightarrow \frac{3}{2} = 1 + \frac{2}{f}\therefore f = 4

\therefore

Average energy per mole

= \frac{1}{2}fRT = 2RT

M_{\max} = \frac{n_{1}M_{1} + n_{2}M_{2}}{n_{1} + n_{2}} = \frac{1 \times 2 + 1 \times 4}{1 + 1} = 3gm{ol}^{- 1}

Speed of sound in a gas,

v = \sqrt{\frac{\gamma RT}{M}}

For a given value of T,v

\propto \sqrt{\frac{\gamma}{M}}

Rms speed of a gas molecule at temperature T is given by

T,v \propto \sqrt{\frac{\gamma}{M}}

For a given value of T,

\ v_{rms} \propto \frac{1}{\sqrt{M}}

\frac{\left( v_{rms\ } \right)He}{\left( v_{rms\ } \right)H_{2}} = \sqrt{\frac{M_{H_{2}}}{M_{He}}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}}

So, options (a), (b) and (c) are correct

Source: Grass-G / P-08 Physics paper 23 October.docx

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