Question

The ratio of velocity of sound in air at 4atm pressure and that at 1atm pressure would be:
A. 2:1
B. 1:2
C. 1:1
D. 1:4

Answer 1

As a first step find out whether the speed of sound in air depends on the pressure. If yes, find the dependency relation between them and hence find the ratio. Also, assume ideal gas condition and recall Newton-Laplace relation.
Formula used:
Newton-Laplace relation,
$c=\sqrt{\gamma \times \dfrac{P}{\rho }}$

Complete answer:
In the question, we are given two pressures and then we are asked to find the ratio of velocity of sound at those pressures.
We know that sound requires a medium for propagation. So, we can define the speed of sound as the distance covered by the sound wave per unit time in an elastic medium.
The speed of sound changes with medium, that is, the speed of sound is different for different mediums. It also shows variations when certain properties of this medium are changed. Since, in the question the medium is specified as air, we shouldn’t worry about the variation caused due to change in medium.
Let us see what characteristics of the medium affect the speed of sound in air. The most important among all is the temperature. As the sound is a vibration that spreads as waves through particles, temperature being the measure of the energy of those particles, strongly influences how fast they are moving. Higher the temperature of air is, the higher the speed of sound will be. Other factors include humidity that shows a very minute effect on the speed of sound and then comes pressure.
Pressure of air has no effect on the speed of sound in air in an ideal gas approximation. Since, the contributions of pressure and density of air are equal, they cancel each other out in case of an ideal gas.
From Newton-Laplace equation the speed of sound in an ideal gas is given by,
$c=\sqrt{\gamma \times \dfrac{P}{\rho }}$
But we have,
$PV=nRT$
And,
$\rho =\dfrac{nM}{V}$
$\Rightarrow c=\sqrt{\gamma \times \dfrac{\dfrac{nRT}{V}}{\dfrac{nM}{V}}}$
$\Rightarrow c=\sqrt{\dfrac{\gamma RT}{M}}$
Clearly, the speed of sound in an ideal gas is independent of pressure.
Therefore, the speed of sound remains constant even if we change the pressure of the medium. So, the ratio of velocity of sound in air at 4atm pressure and that at 1atm pressure will be 1:1.

So, the correct answer is “Option C”.

Note:
We have solved this question under the assumption that the given medium is an ideal gas. However, in non-ideal conditions, the speed of sound shows slight variation with change in pressure. Also, the variation that is observed with increasing altitude is not due to pressure but that due to decrease in temperature.