A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at $STP$. Assuming the speed of sound in air at $STP$ is $300\; \mathrm{m} / \mathrm{s}$, the frequency difference between the fundamental and second harmonic of this pipe is ___ $\mathrm{Hz}$
JEE MAIN 2020, Diffcult
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$v_{s}=\sqrt{\frac{\gamma P}{\rho}}$
$\frac{\mathrm{v}_{\mathrm{gas}}}{\mathrm{v}_{\mathrm{air}}}=\sqrt{\frac{\rho_{\mathrm{air}}}{\rho_{\mathrm{gas}}}}$
$\Rightarrow \frac{\mathrm{v}_{\mathrm{gas}}}{300}=\frac{1}{\sqrt{2}}$
$\Rightarrow \mathrm{v}_{\mathrm{gas}}=\frac{300}{\sqrt{2}}$
$\therefore \mathrm{v}_{\mathrm{grs}}=150 \sqrt{2}$
Now $\mathrm{n}_{2}-\mathrm{n}_{1}=\frac{\mathrm{v}_{\mathrm{gss}}}{2 \ell}=\frac{150 \sqrt{2}}{2(1)}=75 \sqrt{2}$
$\Rightarrow \Delta \mathrm{n}=106.06 \mathrm{Hz}$
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