JEE Main · 2008 · 27 Apr 2008 · Offline · Waves & Sound

For an ideal gas, the speed of sound is $v=\sqrt{\frac{\gamma RT}{M}}$.

At the same temperature $T$, comparing helium with oxygen:

$\frac{v_{\mathrm{He}}}{v_{{\mathrm{O}}_2}}=\sqrt{\frac{{\gamma }_{\mathrm{He}}}{{\gamma }_{{\mathrm{O}}_2}}\times \frac{M_{{\mathrm{O}}_2}}{M_{\mathrm{He}}}}$

Helium is monatomic: ${\gamma }_{\mathrm{He}}=\frac{5}{3}$, $M_{\mathrm{He}}=4$ g/mol
Oxygen is diatomic: ${\gamma }_{{\mathrm{O}}_2}=\frac{7}{5}$, $M_{{\mathrm{O}}_2}=32$ g/mol

$\frac{v_{\mathrm{He}}}{v_{{\mathrm{O}}_2}}=\sqrt{\frac{5/3}{7/5}\times \frac{32}{4}}=\sqrt{\frac{25}{21}\times 8}=\sqrt{\frac{200}{21}}=\sqrt{9.524}=3.086$

$v_{\mathrm{He}}=460\times 3.086\approx 1421\mathrm{m}{\mathrm{s}}^{-1}$

Note: the mass ratio dominates. Helium is 8 times lighter per mole, which alone would give a factor $\sqrt{8}=2.83$; the larger $\gamma$ of a monatomic gas raises this only slightly, to 3.09.