Serway & Jewett — The Kinetic Theory of Gases: Problem 32.

Review. As a sound wave passes through a gas, the compressions are either so rapid or so far apart that thermal conduction is prevented by a negligible time interval or by effective thickness of insulation. The compressions and rarefactions are adiabatic. (a) Show that the speed of sound in an ideal gas is \[ v = \sqrt{\frac{\gamma RT}{M}} \] where \( M \) is the molar mass. The speed of sound in a gas is given by Equation 16.35; use that equation and the definition of the bulk modulus from Section 12.4. (b) Compute the theoretical speed of sound in air at \( 20.0^\circ\text{C} \) and state how it compares with the value in Table 16.1. Take \( M = 28.9\text{ g/mol} \). (c) Show that the speed of sound in an ideal gas is \[ v = \sqrt{\frac{\gamma k_B T}{m_0}} \] where \( m_0 \) is the mass of one molecule. (d) State how the result in part (c) compares with the most probable, average, and rms molecular speeds.