Question

Under similar conditions of temperature and pressure, In which of the following gases the speed of sound will be largest.

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

Answer 1

The correct option is A $H_2$

Speed of sound $v$ in a gas is given by Laplace's formula as ,

$v=\sqrt{\gamma P/\rho }$ ,

or $v=\sqrt{\gamma RT/M}$ ,

where , $P=$ pressure of the gas ,

$T=$ temperature of the gas ,

$\gamma =$ ratio of specific heats at constant pressure and temperature ,

$M=$ molecular mass of the gas ,

by above relation , we have ,

$v\propto 1/M$ , (given that , pressure and temperature are same for all gases)

now molecular masses of the given gases are ,

$M_H=2$ ,

$M_N=14$ ,

$M_{He}=4$ ,

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

now He is a monoatomic gase , therefore $\gamma =1.67$ ,

and $v_{He}=\sqrt{1.67RT/4}=\sqrt{0.42RT}$ ,

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

$v_H=\sqrt{1.4RT/2}=\sqrt{0.7RT}$

$v_N=\sqrt{1.4RT/14}$

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

out of diatomic gases , speed in hydrogen is greatest ,as molecular mass of hydrogen is minimum therefore 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 speed of sound is greatest .