An organ pipe is a musical instrument made from a long tube that is open at both ends. A stationary sound wave in the tube produces a musical note. The lowest frequency note that this organ pipe produces in air is 320 Hz320\text{ Hz}320 Hz. The speed of sound in air at a temperature of 20 ∘C20\,^\circ\text{C}20∘C is 340 m s−1340\text{ m s}^{-1}340 m s−1.
Show that the organ pipe has an approximate length of 0.53 m0.53\text{ m}0.53 m.
In an ideal gas, the speed vvv of sound is given by v=(γRTM)1/2v = \left(\frac{\gamma RT}{M}\right)^{1/2}v=(MγRT)1/2 where 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 helium: | Gas | γ\gammaγ | M/g mol−1M / \text{g mol}^{-1}M/g mol−1 | | :--- | :---: | :---: | | Air | 1.40 | 29.0 | | Helium | 1.67 | 4.00 | 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 organ pipe is placed inside a sealed chamber. The chamber is filled with helium at a temperature of 35 ∘C35\,^\circ\text{C}35∘C. Calculate the lowest frequency that the organ pipe could produce inside the chamber. (Assume the length of the organ pipe remains constant.)