Question

Under similar conditions of temperature and pressure, in which of the following gases the velocity of sound will be largest?

• A. $C{O}_2$
• B. $N_2$
• C. $H_2$
• D. $He$

Answer 1

The correct option is C $H_2$

According to Laplace's formula, speed of sound $\left(v\right)$ is given by

$v=\sqrt{\frac{\gamma P}{\rho }}$

$v=\sqrt{\frac{\gamma RT}{M}}$

By the above relation, we have, $v\propto \frac{1}{\sqrt{M}}$ (given that temperature and pressure remain same for all gases). Now the molecular masses of the given gases are as follows:

$M_{H_2}=2$

$M_{N_2}=28$

$M_{H_e}=4$

$M_{C{O}_2}=44$

Now $He$ is mono-atomic gas, therefore $\gamma =1.67$, And

$v_{He}=\sqrt{\frac{1.67RT}{4}}=\sqrt{0.42RT}$

All other given gases are diatomic gases, therefore $\gamma =1.4$ for all gases

$v_{H_2}=\sqrt{\frac{1.4RT}{2}}=\sqrt{0.7RT}$

$v_{N_2}=\sqrt{\frac{1.4RT}{28}}$

$v_{C{O}_2}=\sqrt{\frac{1.4RT}{44}}$

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.

Final answer : (a)