Question

The speed of sound in hydrogen at NTP is $1270m{s^{ - 1}}$. Then the speed in a mixture of hydrogen and oxygen in the ratio $4:1$ by volume will be:
A) $317m{s^{ - 1}}$
B) $635m{s^{ - 1}}$
C) $830m{s^{ - 1}}$
D) $950m{s^{ - 1}}$

Answer 1

First, balance the density of the gas mixture with the densities of hydrogen and oxygen. Then, calculate the value of the density of the gas mixture of hydrogen and oxygen. Thereafter, calculate the velocity of sound in the gas mixture of hydrogen and oxygen in the ratio $4:1$ by volume using the formula ${v_{mix}} = \sqrt {\dfrac{{\gamma P}}{{{\rho _{mix}}}}}$.

Complete step by step solution:
Let us consider, the volume and the density of hydrogen be ${V_H}$ and ${\rho _H}$; and the volume and the density of oxygen be ${V_O}$ and ${\rho _O}$.
$\therefore {V_H}:{V_O} = 4:1$ From the given problem. So, take, ${V_H} = 4V$ and ${V_O} = 1V$.
We know, the total mass of the gas mixture is equal to the combined mass of hydrogen and oxygen.
Now, if ${\rho _{mix}}$ be the density of the gas mixture, then, we can write,
$\Rightarrow {\rho _{mix}} \times 5V = ({\rho _H} \times 4V) + ({\rho _O} \times V)$
Here, ${\rho _H} = \dfrac{{{m_H}}}{{{V_H}}}$ and ${\rho _O} = \dfrac{{{m_O}}}{{{V_O}}} = 16{\rho _H}$
Therefore, from the above equation, we can have,
$\Rightarrow {\rho _{mix}} \times 5V = 4{\rho _H}V + 16{\rho _H}V$
So, density of the gas mixture, ${\rho _{mix}} = 4{\rho _H}$
Now, let ${v_{mix}}$ be the velocity of sound in the gas mixture of hydrogen and oxygen; and ${v_H}$ be the velocity of sound in hydrogen at NTP, i.e., $1270m{s^{ - 1}}$.
Therefore, the velocity of sound in the gas mixture,
$\Rightarrow {v_{mix}} = \sqrt {\dfrac{{\gamma P}}{{{\rho _{mix}}}}}$
We put ${\rho _{mix}} = 4{\rho _H}$ and get-
$\Rightarrow \sqrt {\dfrac{{\gamma P}}{{4{\rho _H}}}}$
Since, ${v_H} = \sqrt {\dfrac{{\gamma P}}{{{\rho _H}}}}$, we have-
$\Rightarrow \dfrac{1}{2}{v_H}$
And we have, ${v_H} = 1270m{s^{ - 1}}$
$\Rightarrow \dfrac{{1270}}{2}m{s^{ - 1}}$
$\Rightarrow 635m{s^{ - 1}}$

The correct answer is (B), $635m{s^{ - 1}}$.

Additional information:
The speed of sound in gas mixtures depends on temperature, and is independent of the pressure. To calculate the speed of sound in a gaseous mixture, we use adiabatic index and mean molecular mass of that mixture.

Note: The density is directly proportional to the molar mass of an element. So, the density of oxygen is $16$ times the density of hydrogen. As the ratio of volume of hydrogen and oxygen is $4:1$ total volume of the gas mixture of hydrogen and oxygen is taken as $5V$.