Question

At which temperature the speed of sound in hydrogen will be same as that of speed of sound in oxygen at $100^{\circ} C$
a) $-148^{\circ} C$
b) $-212.5^{\circ} C$
c) $-317.5^{\circ} C$
d) $-249.7^{\circ} C$

Answer 1

Hint: Sound, similar to all waves, travels at a definite speed and has the characteristics of frequency and wavelength. The speed of sound varies significantly in different media. The sound speed in a medium depends on how fast vibrational energy can be conveyed through the medium. Therefore, the derivation of the sound speed in a medium depends on the medium and the medium state.

Complete step-by-step solution:
The speed of sound is given by
$v = \sqrt{\dfrac{\gamma RT}{M} }$
Where, $\gamma$ is adiabatic index.
R is gas constant.
T is the temperature.
In the question, the speed of sound is the same.
Hydrogen and Oxygen, both are diatomic gas. So, adiabatic index is the same for both.
Now, T and M are not constant.
By formula, we get:
$T \propto M$
$\implies \dfrac{T_{H_{2}}}{ T_{O_{2}}} = \dfrac{ M_{H_{2}}}{ M_{O_{2}}}$
Ratio of mass of Hydrogen to the mass of Oxygen is $2 : 32 = 1 : 16$
$T_{O_{2}} = 100^{\circ} C = 373 K$
$T_{H_{2}} = \dfrac{ M_{H_{2}}}{ M_{O_{2}}} \times T_{O_{2}} $
$\implies T_{H_{2}} = \dfrac{ 1}{16} \times 373 $
$\implies T_{H_{2}}= 23.32 – 273 = -249.68^{\circ} C$
Option (d) is right.

Note: The speed of sound in the air is low because air is compressible. Because liquids and solids are comparatively rigid and very hard to compress, the sound speed in such media is regularly more significant than in gases. The sound speed is independent of the frequency.