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A student is performing an experiment using a resonance column and a tuning fork of frequency $244 s ^{-1}$. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is $(0.350 \pm 0.005) m$, the gas in the tube is
(Useful information) : $\sqrt{167 R T}=640 j^{1 / 2} mole ^{-1 / 2} ; \sqrt{140 RT }=590 j ^{1 / 2} mole ^{-1 / 2}$. The molar masses $M$ in grams are given in the options. Take the value of $\sqrt{\frac{10}{ M }}$ for each gas as given there.)
Neon $\left(M=20, \sqrt{\frac{10}{20}}=\frac{7}{10}\right)$
Nitrogen $\left(M=28, \sqrt{\frac{10}{28}}=\frac{3}{5}\right)$
Oxygen $\left(M=32, \sqrt{\frac{10}{32}}=\frac{9}{16}\right)$
Argon (M=36, $\left.\sqrt{\frac{10}{36}}=\frac{17}{32}\right)$
Solution
$f =\frac{1}{4 \ell} \sqrt{\frac{\gamma RT }{ M }} \& \frac{\Delta f }{ f }=\frac{\Delta \ell}{\ell}$
(A) $M=20 \times 10^{-3} \quad f =320 Hz$ $\quad\quad$$\Delta f = \pm 4.5 Hz$ Not possible
(B) $M=20 \times 10^{-3} \quad f =253 Hz$ $\quad\quad$$\Delta f = \pm 3.6 Hz$ Not possible
(C) $M=32 \times 10^{-3} \quad f =237 Hz$ $\quad\quad$$\Delta f = \pm 3.4 Hz$ Not possible
(D) $M =36 \times 10^{-3}$ $\quad$$f =242.8 Hz$ $\quad\quad$$\Delta f = \pm 3.5 Hz$ possible
