Question

The approximate speed of sound in air is equal to (in m/s):
(A) 33.2
(B) 3320
(C) 332
(D) 3.32

Answer 1

In this question, we are going to apply the concept of speed of sound is the speed (distance per unit time) of the sound waves travelling in a medium.We are going to use Newton-Laplace equation for speed of sound in any fluid and after putting the values of constants, we can arrive at the correct result.

Formula used:
Newton-Laplace equation for speed of sound in any fluid is given as:
$c = \sqrt {\dfrac{K}{\rho }}$
Where, K is the bulk modulus of the medium (or the coefficient of stiffness) and,
$\rho$ is the density of the medium
On further simplification, the speed of sound for air for a given temperature is defined as
$c = 20.05\sqrt T m/s$
Where T is in Kelvin

Complete step by step answer:
Initially, it was Newton who in 1667 calculated the speed of sound as 298m/s which was wrong by factor of 15%. He described the speed of sound in gases as
$c = \sqrt {\dfrac{p}{\rho }}$
Where p was pressure and $\rho$ was the density.
The problem was that he did not account for the rapid temperature changes in the sound waves which in due time was not able to escape and resulted in pressure change. This factor made Newton’s answer wrong.
Later, Laplace added an adiabatic index ($\gamma$) in the formula to compensate the above left-out factor, so,
$c = \sqrt {\gamma \dfrac{p}{\rho }}$
Using the ideal gas law, $p = \dfrac{{nRT}}{V}$ and substituting $\rho = \dfrac{{nM}}{V}$ we get,
$c = \sqrt {\dfrac{{RT}}{M}}$
Where,
R is the gas constant and M is the molar mass of the gas.
For a particular gas, R and M are constant, so for air it can be simplified as:
$c = 20.05\sqrt T m/s$
Substituting T as 273K for 0 degree Celsius, we get

$$c = 20.05 \times \sqrt {273} \\\ c = 332m/s \\\$$

Hence, the correct answer is option C.

Note: We can use the first formula as well in terms of coefficient of stiffness and density. But that is a generic formula applicable for any medium whether gas, solids or liquids. Since, in the question it was given air so we have used the simpler formula. It is not practically possible to remember coeff. of stiffness and density of every medium so, whenever asked the speed of sound in any other medium, these quantities will be given.