Chapter 19, Problem 91P

For adiabatic processes in an ideal gas, show that (a) the bulk modulus is given by

$B=-V\frac{dp}{dV}=\gamma p,$

where γ = C_p/C_V. (See Eq. 17-2.) (b) Then show that the speed of sound in the gas is

$v_s=\sqrt{\frac{\gamma p}{\rho }}=\sqrt{\frac{\gamma RT}{M},}$

where ρ is the density, T is the temperature, and M is the molar mass. (See Eq. 17-3.)