Question

The velocity of sound in air at room temperature is $340m/s$ and density of air is $1.2kg{m}^{-1}$ .The value of atmospheric pressure in terms of the height of mercury column in meter will be
A. $0.75$
B. $7.5$
C. $75$
D. $750$

Answer 1

Hint: You can start by defining the speed of sound and what factors it depends on. Then use the equation $V_s=\sqrt{{\displaystyle \frac{\gamma p}{\rho }}}$ to find out the pressure. Then finally use the equation $p=h{\rho }_{Hg}g$ to find out what height does the mercury column go up to.

Complete step-by-step answer:
The distance travelled by a sound wave in per unit time is known as the speed of sound. For an ideal gas the speed of sound only depends upon the composition of the medium and the temperature.
In the given question we are provided with the speed of sound in air (air slightly deviate from the ideal gas behavior)
Density of air $({\rho }_{air})$ $=1.2kg{m}^{-3}$
Now we know
 $V_s=\sqrt{{\displaystyle \frac{\gamma p}{\rho }}}$
 $340=\sqrt{{\displaystyle \frac{1.4\times p}{1.2}}}$
 $p={\displaystyle \frac{{(340)}^2\times 1.2}{1.4}}$
 $p=99086N{m}^{-2}$
Now we know, the equation for density is
 $p=h{\rho }_{Hg}g$
Here
 $\rho =$ Pressure
 $h=$ Height of mercury column
 ${\rho }_{Hg}=$ Density of mercury $=23594kg{m}^{-3}$
 $g=$ Acceleration due to gravity
Substituting the given values in the equation for density, we get
$$h={\displaystyle \frac{99086}{13594}}$$
 $h={\displaystyle \frac{99086}{13594\times 9.8}}$
$$h=0.743m\approx 0.75m$$
The height of the mercury column will rise to a height of$$0.75m$$.
Hence, option A is the correct choice.

Note - We discussed above that sound travels in a medium, it can be gas, liquid or solid. Most people believe that sound travels fastest in gases (air), but it is not so. Sound travels faster in solids than it does in liquids. The speed of sound is slowest in the gaseous medium. For sound to travel it has to transfer energy from one molecule to the next and since the molecules in solids are closely packed in solids than in liquids and gases the speed of sound in solids is also the fastest.