Question

Oxygen is $ 16 $ times heavier than hydrogen. Equal volume of hydrogen and oxygen are mixed. The ratio of speed of sound in the mixture to that in hydrogen is
(A) $ \sqrt 8 $
(B) $ \sqrt {\dfrac{1}{8}} $
(C) $ \sqrt {\dfrac{2}{{17}}} $
(D) $ \sqrt {\dfrac{{17}}{{32}}} $

Answer 1

Hint :Let us get some idea about the speed of sound. A sound wave's speed is defined as the distance it travels per unit of time as it propagates through an elastic medium. However, sound travels at different speeds in different substances: sound travels the slowest in gases, the fastest in liquids, and the fastest in solids. While sound travels at $ 343{\text{ }}m{s^{ - 1}} $ in air, it travels at $ 1,481{\text{ }}m{s^{ - 1}} $ in water (almost $ 4.3 $ times faster) and $ 5,120{\text{ }}m{s^{ - 1}} $ in iron (almost $ 15 $ times as fast).
Velocity of sound is given as $ c = \sqrt {\dfrac{{\gamma RT}}{M}} $ .
Where,
 $ \gamma = $ Specific heat ratio.
 $ M = $ Molecular mass of the gas mixture.
 $ R = $ Gas constant.
 $ T = $ Temperature of the mixture.

Complete Step By Step Answer:
So, as given in the question, we have to calculate the ratio, so for that let us first calculate the velocity of sound in hydrogen and then in mixture.
Velocity of sound in hydrogen $ = \sqrt {\dfrac{{\gamma RT}}{{{M_1}}}} $
 $ {M_{mixture}} = \dfrac{{{n_1}{M_1} + {n_2}{M_2}}}{{{n_1} + {n_2}}} $
 $ {n_1} = {n_2} $ as both are equal volume at NTP are used
As given in the question, the oxygen is $ 16 $ times heavier than hydrogen.
So we can say that: $ {M_2} = 16{M_1} $
 $ {M_{mixture}} = \dfrac{{{M_1} + 16{M_2}}}{2} = \dfrac{{17{M_1}}}{2} $ .
Velocity of sound in mixture $ = \sqrt {\dfrac{{\gamma RT}}{{{M_{mixture}}}}} $
Now let’s calculate the ratio of velocity of sound in mixture that of hydrogen $ \sqrt {\dfrac{{{M_1}}}{{{M_{mixture}}}}} = \sqrt {\dfrac{2}{{17}}} $
So, option C is correct.

Note :
Let us know some more points regarding this. Because the density of gases is determined by their molar mass, oxygen gas is heavier than hydrogen gas. As a result, $ {H_2} $ has a higher molar mass of $ 2gmol{e^{ - 1}} $ than oxygen. $ {O_2} $ Has a value of $ 32 $ . As a result, hydrogen is $ 16 $ times lighter than oxygen.