Question

The speed of sound in hydrogen is $280m/s$. Calculate the speed of sound in the mixture of oxygen and hydrogen in which they are mixed in the ratio of $1:4$, It is given that $\gamma $is the same for both the gases and the density of oxygen equals $16$times the density of hydrogen.
(A) $640m/s$
(B) $280m/s$
(C) $360m/s$
(D) None of these

Answer 1

Hint We have to consider the volume of oxygen and the volume of hydrogen as variables. The relation between the density of oxygen and hydrogen is also given in the question. Now we have to calculate the density of the mixture of gases from which we can calculate the speed of sound in a mixture of oxygen and hydrogen.
Formula used:
$m = V \times d$
Where \[V\] is the volume of the gas and $d$stands for the density of the gas.
$v = \sqrt {\dfrac{{\gamma p}}{d}} $
$\gamma p$instead of adiabatic elasticity, $v$is the velocity of sound, and $d$stands for the density of the medium.

Complete Step by step solution
The ratio of oxygen and hydrogen is given as $1:4$
Let the volume of oxygen be $V$.
Then the volume of hydrogen will be $4V$.
The total volume of the mixture can be written as,
$4V + V = 5V$
Let the density of hydrogen be $d$.
It is given that the density of oxygen equals $16$times the density of hydrogen.
Therefore, the density of oxygen can be written as, $16d$.
Let the total density of the mixture of gas be, $D$.
The total mass of the mixture can be written as,
The total mass of the mixture is the sum mass of hydrogen and the mass of oxygen.
$M = {m_{hydrogen}} + {m_{oxygen}}$
We know that the mass can be written as,
$m = V \times d$
The total mass can be written as,
${V_{mixture}} \times {d_{mixture}} = {V_{hydrogen}} \times {d_{hydrogen}} + {V_{oxygen}} \times {d_{oxygen}}$
The total volume is $5V$and the density of the mixture is $D$
$5V \times D = 4V \times d + V \times 16d$
Separating the common terms,
$V\left( {5D} \right) = V(4d + 16d)$
$5D = 20d$
From this, we get
$D = 4d$
The speed of sound through a gas can be written as,
$v = \sqrt {\dfrac{{\gamma p}}{d}} $
From this, we can write
$v \propto \sqrt {\dfrac{1}{d}} $
The ratio of speed through the mixture to the ratio of the speed of sound through hydrogen can be written as
$\dfrac{{{v_{mix}}}}{{{v_{hydrogen}}}} = \dfrac{{\sqrt {\dfrac{1}{D}} }}{{\sqrt {\dfrac{1}{d}} }} = \sqrt {\dfrac{d}{D}} $
Substituting $D = 4d$,
$\dfrac{{{v_{mix}}}}{{{v_{hydrogen}}}} = \sqrt {\dfrac{d}{{4d}}} = \sqrt {\dfrac{1}{4}} = \dfrac{1}{2}$
The speed of sound through hydrogen is given as,
${v_{hydrogen}} = 280m/s$
Substituting this value in the above equation,
$\dfrac{{{v_{mix}}}}{{280}} = \dfrac{1}{2}$
From this, we can write the speed of sound through the mixture as,
${v_{mix}} = \dfrac{{280}}{2} = 140m/s$

Therefore, The answer is: Option (D): None of these

Note
Laplace assumed that the condensation and rarefaction are taking place adiabatically in gas when the sound wave passes through it. Hence he substituted adiabatic elasticity $\gamma p$ instead of isothermal elasticity$p$ and modified Newton’s formula. This modified formula $v = \sqrt {\dfrac{{\gamma p}}{d}} $ is known as the Newton-Laplace formula.