An acoustic gas sensor is constructed using a resonance chamber consisting of a long cylindrical tube that is closed at one end and open at the other. A stationary sound wave is set up in the tube, creating a fundamental resonant tone. The lowest frequency of resonance that this chamber produces in air is 260 Hz260\text{ Hz}260 Hz. The speed of sound in air at a temperature of 15 ∘C15\,^\circ\text{C}15∘C is 340 m s−1340\text{ m s}^{-1}340 m s−1.
Show that the resonance tube has an approximate length of 0.33 m0.33\text{ m}0.33 m.
In an ideal gas, the speed vvv of sound is given by
v=(γRTM)1/2 v = \left(\frac{\gamma RT}{M}\right)^{1/2} v=(MγRT)1/2where γ\gammaγ is a dimensionless constant that depends on the gas, RRR is the molar gas constant (8.31 J mol−1K−18.31\text{ J mol}^{-1}\text{K}^{-1}8.31 J mol−1K−1), TTT is the absolute temperature, and MMM is the molar mass of the gas.
The table below shows values of γ\gammaγ and MMM for both air and neon:
| Gas | γ\gammaγ | M/g mol−1M / \text{g mol}^{-1}M/g mol−1 |
|---|---|---|
| Air | 1.40 | 29.0 |
| Neon | 1.67 | 20.2 |
The kinetic model of an ideal gas assumes that there are a large number of particles in rapid, random motion. State two further assumptions for the kinetic model of an ideal gas.
The resonance chamber is filled with neon gas at a temperature of 60 ∘C60\,^\circ\text{C}60∘C. Calculate the lowest frequency of resonance that the tube produces under these conditions. (Assume the length of the tube remains constant.)
