Question

Calculate the ratio of the speed of the sound in neon to that in water vapour at any temperature. Given that molecular weight of neon gas = $2.02\times {{10}^{-2}}kg\,mol{{e}^{-1}}$ and molecular weight of water vapour is = $1.8\times {{10}^{-2}}kg\,mol{{e}^{-1}}$
(A). $0.944$
(B). $1.944$
(C). $2.944$
(D). $3.944$

Answer 1

Hint: Sound is mechanical waves which require a medium to travel. Sound travels slowest in a gas and the speed of sound in gas depends on the adiabatic factor, temperature, gas constant and the molecular mass of the gas. Dividing the speed of sound in neon to the speed of sound in water vapour, we can calculate the ratio.
Formulas used:
$v=\sqrt{\dfrac{\gamma RT}{M}}$

Complete answer:
Sound is a mechanical wave that requires a medium to travel forward in space. It cannot travel without a medium. Speed of sound is slowest in gases and fastest in solids. It propagates forward as compressions and rarefactions.
The velocity of sound in a gas is given by the following equation-
$v=\sqrt{\dfrac{\gamma RT}{M}}$
Here, $v$ is the velocity of sound
$\gamma $ is the adiabatic index
$R$ is the gas constant
$T$ is the absolute temperature
$M$ is the molecular mass
Using eq (1),
The velocity of sound in neon gas will be-
${{v}_{1}}=\sqrt{\dfrac{\gamma RT}{{{M}_{N}}}}$ - (1)
The velocity of gas in water vapour will be-
${{v}_{2}}=\sqrt{\dfrac{\gamma RT}{{{M}_{W}}}}$ - (2)
The ratio of velocity of speed of sound in neon gas to water vapour, by dividing eq (1) and eq (2) is-
$\dfrac{{{v}_{1}}}{{{v}_{2}}}=\dfrac{\sqrt{\dfrac{\gamma RT}{{{M}_{N}}}}}{\sqrt{\dfrac{\gamma RT}{{{M}_{W}}}}}$
As $\gamma ,\,R,\,T$ are constants they will cancel out. Therefore,
$\dfrac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\dfrac{{{M}_{W}}}{{{M}_{N}}}}$ - (3)
Given, ${{M}_{N}}=2.02\times {{10}^{-2}}kg\,mol{{e}^{-1}}$, ${{M}_{W}}=1.8\times {{10}^{-2}}kg\,mol{{e}^{-1}}$
In eq (3), we substitute given values to get,
$\begin{align}
 & \dfrac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\dfrac{1.8\times {{10}^{-2}}}{2.02\times {{10}^{-2}}}} \\
 & \Rightarrow \dfrac{{{v}_{1}}}{{{v}_{2}}}=0.94 \\
\end{align}$
The ratio of velocity of sound is $0.94$.

So, the correct answer is “Option A”.

Note: The adiabatic index is the ratio of specific heat at constant pressure to the specific heat at constant volume. As the molecules of a solid are very close to each other, the vibrations travel faster and hence, the speed of sound is fastest. The formula for speed of sound in a gas is called the Newton-Laplace equation.