Discussion
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 He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas).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: \(\sqrt{167RT} = 640\ J^{1/2}mole^{-1/2}\); \(\sqrt{140RT} = 590\ J^{1/2}mole^{-1/2}\). The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{M}}\) for each gas as given there.)
Answer: D
The gas is Argon.
In a resonance column closed at one end, the shortest resonating length is a quarter wavelength, so the speed of sound in the gas is
\[ v = 4fL = 4 \times 244 \times 0.350 = 341.6\ \text{m/s} \]
With \(L\) between 0.345 and 0.355 m, v lies between about 336.7 and 346.5 m/s.
The speed of sound is \(v = \sqrt{\gamma RT/M}\), with M in kg, which can be written \(\sqrt{100\gamma RT}\sqrt{10/M}\) with M in grams. For a monatomic gas \(\gamma = 5/3\), so \(\sqrt{100\gamma RT} = \sqrt{167RT} = 640\); for a diatomic gas \(\gamma = 7/5\), giving \(\sqrt{140RT} = 590\).
- Neon (monatomic): \(640 \times 7/10 = 448\) m/s.
- Nitrogen (diatomic): \(590 \times 3/5 = 354\) m/s.
- Oxygen (diatomic): \(590 \times 9/16 \approx 332\) m/s.
- Argon (monatomic): \(640 \times 17/32 = 340\) m/s.
Only argon falls inside the measured range.
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