Chapter 21, Problem 21.56AP

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°C and state how it compares with the value in Table 16.1. Take M = 28.9 g/mol. (c) Show that the speed of sound in an ideal gas is

$v=\sqrt{\frac{\gamma k_{\text{B}}T}{m_0}}$

where m₀ is the mass of one molecule. (d) Slate how the result in part (c) compares with the most probable, average, and rms molecular speeds.