Question

If the adiabatic constant from helium and hydrogen gases at the same temperature are $\frac{5}{3}$ and $\frac{7}{5}$ respectively, then the ratio of velocity of sound in these gases will be

• A. $42:5$
• B. $5:\sqrt{42}$
• C. $\sqrt{42}:5$
• D. $5:42$

Answer 1

The correct option is B $5:\sqrt{42}$

The velocity of sound in gas is

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

where, $\gamma$ is the adiabatic constant$,R$ is gas constant$,T$ is the temperature of gas and $M$ is molecular weight of gas.

Hence, for helium $(M=4),$

$v_{he}=\sqrt{\frac{{\gamma }_{he}RT}{M_{he}}}$
$v_{he}=\sqrt{\frac{\left(\frac{5}{3}\right)RT}{4}}$
$v_{he}=\sqrt{\frac{5RT}{12}}...............\left(1\right)$
for hydrogen $(M=2),$
$v_{hy}=\sqrt{\frac{{\gamma }_{hy}RT}{M_{hy}}}$
$v_{hy}=\sqrt{\frac{\left(\frac{7}{5}\right)RT}{2}}$
$v_{hy}=\sqrt{\frac{7RT}{10}}...............\left(2\right)$

Dividing $\left(1\right)$ by $\left(2\right),$ we get,

$\frac{v_{he}}{v_{hy}}=\sqrt{\frac{5RT}{12}}\times \sqrt{\frac{10}{7RT}}$
$\frac{v_{he}}{v_{hy}}=\sqrt{\frac{50}{84}}$
$\frac{v_{he}}{v_{hy}}=\sqrt{\frac{25}{42}}$
$\frac{v_{he}}{v_{hy}}=\frac{5}{\sqrt{42}}$