For adiabatic processes in an ideal gas, show that (a) the bulk modulus is given bywhere(See Eq. 17-2.) (b) Then show that the speed of sound in the gas is${\mathbf{v}}_{\mathbf{s}}\mathbf{=}\sqrt{\frac{\mathbf{\gamma p}}{\mathbf{\rho }}}\mathbf{=}\sqrt{\frac{\mathbf{\gamma RT}}{\mathbf{M}}}\mathbf{,}$where is the density, T is the temperature, and M is the molar mass. (See Eq. 17-3.)

1. $\gamma =C_p/C_V$
2. The bulk modulus, according to Equation 17-2, is-

$B=-\frac{\Delta p}{(\Delta V/V)}$

$v_s=\sqrt{\frac{B}{\rho }}$

Where, $\rho$is the density, T is the temperature, and M is the molar mass.

If for a process, no exchange of heat takes place between the system and surroundings and all the work done during the process appears as a change in the internal energy of the gas, then the process is known as an adiabatic process. For the adiabatic process, the following equation must be satisfied.

$p{V}^{\gamma }=a\left(constant\right)$

The density is given as-

$\rho =\frac{m}{V}$

Themolar mass is given as-

$M=\frac{m}{n}$

where,$\rho$is the density, Tis the temperature, nis no. of moles, Vis volume, p is pressure and Mis the mass of gas.

The bulk modulus is given as-

$B=-\frac{\Delta p}{(\Delta V/V)}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\dots \dots \dots \dots \dots \dots .(17-2)$

Where $\Delta p$is the change in the pressure, $\Delta V$is the change in the volume and V is the volume.

For infinitesimally small changes,the above equation can be written as-

$B=-V\frac{dp}{dV}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots ..\left(1\right)$

For the adiabatic processes,

$p{V}^{\gamma }=C\hspace{0.33em}\left(constant\right)\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .(19-53)$

Therefore, the pressureis given by,

$\begin{array}{c}p=\frac{C}{V^{\gamma }}\\ =C{V}^{-\gamma }\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(2\right)\end{array}$

Putting this in Equation (1),

$B=-V\frac{d\left(C{V}^{-\gamma }\right)}{dV}$

role="math" localid="1662696912597" $B=-CV\frac{d\left(V^{-\gamma }\right)}{dV}$

$B=-(-\gamma )CV.V^{-{\gamma }^{-1}}$

$=\left(\gamma \right)C{V}^{-\gamma }$

using Equation (2), we have-

$B=\gamma p\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(3\right)$

Hence, proved

The speed of sound in air is,

$v_s=\surd (B/\rho )\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .(17-3)$

using Equation (3),

$v_s=\surd (\gamma p/\rho )\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(4\right)$

But, the density is given by,

$\rho =m/V\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\dots (14-2)$

And molar mass is given by,

$M=\frac{m}{n}$

Where n is the number of moles.

Therefore,

$\rho =\frac{nM}{V}$

Putting this in Equation (4),

$v_s=\sqrt{\frac{\gamma pV}{nM}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(5\right)$

But, from Equation 19-5,

$pV=nRT\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots ..(19-5)$

Putting this in Equation (5),

$$\begin{gathered} v_s=\sqrt{\frac{n\gamma RT}{nM}} \\ {v}_s=\sqrt{\frac{\gamma RT}{M}} \end{gathered}$$

Hence, proved.

Therefore, by using the equations for bulk modulus, adiabatic processes, ideal gas law, and the formula for density, the above relations are proved.