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Question: The ratio of the velocity of sound in hydrogen \(\left( \gamma = \frac{7}{5} \right)\) to that in he...

The ratio of the velocity of sound in hydrogen (γ=75)\left( \gamma = \frac{7}{5} \right) to that in helium(γ=53)\left( \gamma = \frac{5}{3} \right) at the same temperature is

A

542\sqrt{\frac{5}{42}}

B

521\sqrt{\frac{5}{21}}

C

425\frac{\sqrt{42}}{5}

D

215\frac{\sqrt{21}}{5}

Answer

425\frac{\sqrt{42}}{5}

Explanation

Solution

Velocity of sound in gas

v=γRTMv = \sqrt{\frac{\gamma RT}{M}}

Where the symbols have their usual meanings, At the same temperature.

v∝γMv \propto \sqrt{\frac{\gamma}{M}}

∴vH2vH2=γH2γH2×MHeMH2=75×35×42=425\therefore\frac{vH_{2}}{vH_{2}} = \sqrt{\frac{\gamma H_{2}}{\gamma H_{2}} \times \frac{M_{He}}{M_{H_{2}}}} = \sqrt{\frac{7}{5} \times \frac{3}{5} \times \frac{4}{2}} = \frac{\sqrt{42}}{5}