Question

The velocity of sound in air at ${{20}^{\circ }}C$ and 1 atm pressure is $344.2m{{s}^{-1}}$. At ${{40}^{\circ }}C$ and at 2 atm pressure the velocity of sound in air is approximately,
A). $350\text{ m/s}$
B). $356\text{ m/s}$
C). $363\text{ m/s}$
D). $370\text{ m/s}$

Answer 1

Hint: We are given the speed of sound at a certain temperature and pressure, and we are required to find the speed of sound for another set of pressure and temperature. So, we can write down the equation for sound in the first condition and in the second condition. Then take the ratio of the two equations from which we can find the desired velocity of sound at a particular pressure and temperature.

Formula Used:
The velocity of sound in air, is given by the formula,
$v=\sqrt{\dfrac{\gamma RT}{M}}$
M is the molar mass of air.
R is the universal gas constant.
T is the temperature for the particular pressure (in Kelvin)

Complete step by step answer:
We know that the velocity of the sound in air depends on the pressure, the ratio of specific heat capacities and the density of the medium. So, we can write,
$v=\sqrt{\dfrac{\gamma P}{\rho }}$
v is the velocity of sound.
P is the pressure.
$\rho $ is the density of the medium.
$\gamma $ is the ratio of specific heats.
So, at constant pressure, the above temperature can also be written as,
$v=\sqrt{\dfrac{\gamma RT}{M}}$
M is the molar mass of air.
R is the universal gas constant.
T is the temperature for the particular pressure (in Kelvin)
So, at ${{20}^{\circ }}C$ and 1 atm pressure the velocity of sound is $344.2m{{s}^{-1}}$. So, we can write the equation of sound in this pressure as,
${{v}_{1}}=\sqrt{\dfrac{\gamma R\left( 293K \right)}{\rho }}$ …. equation (1)
At ${{40}^{\circ }}C$ and at 2 atm pressure the equation for the velocity of sound in the air can be written as,
 ${{v}_{2}}=\sqrt{\dfrac{\gamma R\left( 313K \right)}{\rho }}$ …. equation (2)
In this case, the value of $\gamma $ and $\rho $ is constant. So, taking the ratio between equation (2) and equation (1), we get,
$\dfrac{{{v}_{2}}}{{{v}_{1}}}=\sqrt{\dfrac{313}{293}}$
$\begin{align}
  & \Rightarrow {{v}_{2}}=1.02\times {{v}_{1}} \\
 & \Rightarrow {{v}_{2}}=1.02\times \left( 344.2m{{s}^{-1}} \right) \\
\end{align}$
$\therefore {{v}_{2}}\approx 350m{{s}^{-1}}$
So, the velocity of sound in air at ${{40}^{\circ }}C$ and at 2 atm pressure is approximately $350m{{s}^{-1}}$.
So, the answer to the question is option (A).

Note: The velocity of sound in air is directly proportional to the square root of temperature. So, regions having high atmospheric temperature, the velocity of sound will be greater. Sound is a longitudinal wave, and it needs a material medium for its propagation. Sound waves are propagated in the form of compression and rarefactions in the medium. Compression is a point in the medium where the particle density is maximum, while the rarefaction is a point in the medium where the particle density is low.