Question

The temperature at which the speed of sound in air becomes double of its value at ${0^ \circ }C$ is
a) $273K$
b) $546K$
c) $1092K$
d) $0K$

Answer 1

Hint:The speed of sound in a solid depends on the Young’s modulus of the medium and also speed of sound is inversely proportional to the root of molecular mass. Speed of sound depends on the density of the material, and the density depends on the temperature.

Complete step by step answer:
In an ideal gas, the equation speed of the sound in air is given by

$v = \sqrt {\dfrac{{\gamma RT}}{M}} $ ……………...( 1)

In this formula, $\gamma $ is the adiabatic index at room temperature (value depends on material to material) which is constant for air, \[R{\text{ }} = {\text{ }}8.31{\text{ }}J/mol{\text{ }} \cdot K\] is the gas constant, $T$ is the absolute temperature in kelvins, and $M$ is the molecular mass.
Velocity of sound is given by

$v \propto \sqrt T $.........................(2)

$\gamma ,R$, and $M$ are constant,

According to given statement,
Speed of sound in air become double,

say ${v_1}$ =$2 \times {v_s}$ at ${0^ \circ }C$
$\dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}} $ ……………...(3)

Now, $\dfrac{{{v_1}}}{{{v_{{0^ \circ }c}}}} = 2$ (Speed of sound become double)
Temperature conversion, ${0^ \circ }C$ to Kelvin $({0^ \circ }C + 273.15) = 273.15K$

$2 = \sqrt {\dfrac{{{T_1}}}{{{0^ \circ } + 273.15}}} $ ………………...(4)
${(2)^2} = \dfrac{{{T_1}}}{{273.15}}$
$4 = \dfrac{{{T_1}}}{{273.15}}$

Now Solving above equation we get,

$\dfrac{4}{1} = \dfrac{{{T_1}}}{{273.15K}}$................( 5)

Cross multiply equation (5)
$\begin{gathered}
  4 \times 273.15 = {T_1} \\
  {T_1} = 1092.6K \\
\end{gathered} $

 Hence, The speed of sound will double at temperature of $1092.6K$

So, The correct option is C

Note: The speed of sound is variable and depends on the properties of the substance through which the wave is travelling. In solids, the speed of transverse waves depends on the shear deformation under shear stress, and the density of the medium. For air at sea level, the speed of sound is$331m/s$.