Question

The velocity of sound in hydrogen is 1224 m/s. Its velocity in a mixture of hydrogen and oxygen containing 4 parts by volume of hydrogen and 1 part oxygen is
A. 1224 m/s
B. 612 m/s
C. 2448 m/s
D. 306 m/s

Answer 1

Hint: We use the relation between the speed of sound and the density of the medium. To find the speed of the sound in mixture we find the density of mixture using mass-volume proportionality.

Formula used:
\[v = \sqrt {\dfrac{\beta }{\rho }} \]
where v is the speed of sound in a medium ,\[\beta \] is the Bulk modulus of the medium and \[\rho \] is the density of the medium.

Complete step by step solution:
Assuming the hydrogen and oxygen to be at the same temperature and pressure when mixed, the volume of both will be the same. Hence the density of the gases will be proportional to the molecular mass. So, the density of hydrogen and oxygen will be in ratio 1:16.

If the density of the hydrogen is \[\rho \]then the density of oxygen will be \[16\rho \]. The mixture contains 4 parts by volume of hydrogen and 1 part of oxygen. If the volume of oxygen is V then the volume of hydrogen will be 4V. So, the total volume of the mixture is 5V. Then the density of mixture will be,
\[{\rho _{mix}} = \dfrac{{\left( {4V \times \rho } \right) + \left( {V \times 16\rho } \right)}}{{5V}} \\ \]
\[\Rightarrow {\rho _{mix}} = 4\rho \]
So, the density of the mixture of hydrogen and oxygen is 4 times the density of hydrogen.

As the speed of the sound is inversely proportional to the density of the medium,
\[\dfrac{{{v_{mix}}}}{{{v_H}}} = \sqrt {\dfrac{{{\rho _H}}}{{{\rho _{mix}}}}} \\ \]
\[\Rightarrow \dfrac{{{v_{mix}}}}{{{v_H}}} = \sqrt {\dfrac{\rho }{{4\rho }}} \\ \]
\[\Rightarrow \dfrac{{{v_{mix}}}}{{{v_H}}} = \dfrac{1}{2} \\ \]
\[\Rightarrow {v_{mix}} = \dfrac{{{v_H}}}{2}\]
The speed of sound in hydrogen is given as 1224 m/s.
\[{v_{mix}} = \dfrac{{1224\,m/s}}{2} = 612\,m/s\]
Hence, the speed of sound in the mixture of hydrogen and oxygen is 612 m/s.

Therefore, the correct option is B.

Note: Hydrogen and Oxygen gas exist as diatomic molecules and the volume of the gas is the volume of the container in which it is kept. So, the density of the gas depends on the molecular mass of the gas at given physical conditions.