Question: From the knowledge that C_v , the molar specific heat at constant volume, for a gas in a container is 5.0 R , calculate the ratio of the speed of sound in that gas to the rms speed of the molecules, for gas temperature T. (Hint: See Problem 91.)

1. The molar-specific heat of the gas, when the volume is kept constant, is $C_V=5.0R$.
2. Hint: See Problem 19-91.

By using the equation for the speed of sound in gas from problems 19-91 and using Equations 19-34 and 19-49, find the ratio of speed of sound in the gas to the rms speed of the molecules, for a given temperature .

Formulae are as follow:

${\mathbf{v}}_{\mathbf{s}}\mathbf{=}\sqrt{\frac{\gamma \mathbf{RT}}{\mathbf{M}}}$

${\mathbf{v}}_{\mathbf{rms}}\mathbf{=}\sqrt{\frac{\mathbf{3}\mathbf{RT}}{\mathbf{M}}}$

${\mathbf{C}}_{\mathbf{p}}\mathbf{=}{\mathbf{C}}_{\mathbf{V}}\mathbf{+}\mathbf{R}$

where,$\mathbf{\gamma }\mathbf{=}{\mathbf{C}}_{\mathbf{p}}\mathbf{/}{\mathbf{C}}_{\mathbf{V}}$, Ris the gas constant, Tis the temperature, vis velocity and M is the molar mass.

From problem 19-91, the speed of sound in the gas is,

$v_s=\sqrt{\frac{\gamma RT}{M}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(1\right)$

Where, $\gamma =C_p/C_V$, R is the gas constant, T is the temperature, and M is the molar mass.

From Equation 19-34, therms speed of the molecules is given by,

$v_{rms}=\sqrt{\frac{3RT}{M}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(19-34\right)$

From Eq. (1) and (19-34), the ratio of speed of sound in the gas to the rms speed of the molecules, for gas temperature T is,

$$\begin{gathered} \frac{v_s}{v_{rms}}=\frac{\sqrt{\frac{\gamma RT}{M}}}{\sqrt{\frac{3RT}{M}}} \\ \frac{v_s}{v_{rms}}=\sqrt{\frac{\gamma }{3}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(2\right) \end{gathered}$$

But,$\gamma ={\text{C}}_p/{\text{C}}_V$ ,

Putting in Eq. (2),

$$\begin{gathered} \frac{v_s}{v_{rms}}=\sqrt{\frac{\left(C_p/C_V\right)}{3}} \\ \frac{v_s}{v_{rms}}=\sqrt{\frac{C_p}{3C_V}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(3\right) \end{gathered}$$

But, from Eq. (19-49),

$C_p=C_V+R\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(19-49\right)$

Putting in Eq. (3),

$$\begin{gathered} \frac{v_s}{v_{rms}}=\sqrt{\frac{\left(C_V+R\right)}{3C_V}} \\ =\sqrt{\frac{\left(\left(5.0R\right)+R\right)}{3\left(5.0R\right)}} \\ =\sqrt{\frac{6.0}{15}} \\ =0.63 \end{gathered}$$

Hence, the ratio of speed of sound in the gas to the rms speed of the molecules, for the given temperature T is, 0.63 .