Question

Under similar conditions of temperature and pressure, in which of the following gases the speed sound will be the largest
A. ${{\text{H}}_2}$
B. ${{\text{N}}_2}$
C. ${\text{He}}$
D. ${\text{C}}{{\text{O}}_2}$

Answer 1

Try to establish the relation of the speed of sound in gases with the temperature, pressure and the molecular mass of the gases. Apply the Laplace formula to find out which gas is mono-atomic and which one is the diatomic gases.

Formula used:
Laplace’s formula, speed of sound $(v)$ is given by
${\text{v = }}\sqrt {\dfrac{{\gamma P}}{\rho }}$
$\Rightarrow {\text{v = }}\sqrt {\dfrac{{\gamma RT}}{M}}$
Where, $P$ is pressure of gas, $T$ is temperature of gas, $\gamma$ is ratio of molar specific heats of gas at constant pressure and constant volume respectively, $M$ is the molecular mass of the gas and $R$ is the molar gas constant.

Complete step by step answer:
According to Laplace’s formula, speed of sound $(v)$ is given by
${\text{v = }}\sqrt {\dfrac{{\gamma P}}{\rho }}$
$\Rightarrow {\text{v = }}\sqrt {\dfrac{{\gamma RT}}{M}}$
By the above relation, we have,
$v \propto \,\dfrac{1}{M}$ (given that temperature and pressure remain same for all gases)
Now the molecular masses of the given gases are as follows:
${M_{{H_2}}}\, = 2\;$
$\Rightarrow {M_{{N_2}}}\, = \,28\;$
$\Rightarrow {M_{He}}\, = \,4$
$\Rightarrow {M_{C{O_2}}} = 44$

Now ${\text{He}}$ is mono-atomic gas, therefore $\gamma {\text{ = 1}}{\text{.67}}$,
And ${v_{He}} = \,\,\sqrt {\dfrac{{1.67RT}}{4}} \, = \,\sqrt {0.42RT\;\;}$
All other given gases are diatomic gases, therefore $\gamma = 1.4$ for all gases
${v_{{H_2}}}\, = \,\,\dfrac{{1.4RT}}{2}\, = \,\,0.7RT$
$\Rightarrow {v_{{N_2}}}\, = \,\,\dfrac{{1.4RT}}{{28}}$
$\therefore {v_{C{O_2}}}\, = \,\dfrac{{1.4RT}}{{44}}$
Out of diatomic gases, speed of hydrogen is greatest, as molecular mass of hydrogen gas is minimum. Therefore the speed of sound in it will be greatest because speed is inversely proportional to molecular mass. Now, if we compare speed in hydrogen and helium, we can see that speed in hydrogen is greater than helium , therefore in hydrogen gas, speed of sound is greatest .

Therefore, A is the correct option.

Note: $\gamma$ is the ratio of molar specific heats of gas at constant pressure and constant volume respectively and it has different values for mono-atomic and diatomic gases. For mono-atomic gases its value is $1.67$ and for diatomic its value is $1.4$ and temperature and pressure is given to be constant in the question.