Question

Hydrogen gas is filled in two identical bottles, A and B, at the same temperature. The mass of hydrogen in two bottles is $12gm$ and $48gm$ respectively. In which bottle will sound travel faster? How many times as fast as the other?

Answer 1

As we know that velocity or rate is inversely proportional to the square root of molecular mass of the gas when the temperature is kept constant and as the bottles are identical, the volume will also remain the same.

Complete Step by step answer:
As we know the fact that velocity or rate of any gas is inversely proportional to the square root of molecular mass of that gas as depicted in the formula:
$V = \dfrac{1}{{\sqrt M }}$, also we know that molecular mass is the ratio of given mass to the moles of the given gas. So we use the mass quantity instead of molecular mass as the gas is the same in two bottles. The identical bottles will have the same volume so volume and temperature both are constant.

Therefore using the formula for mass of bottles A we get:
${V_A} = \dfrac{1}{{\sqrt {{m_A}} }}$
And for bottle B: ${V_B} = \dfrac{1}{{\sqrt {{m_B}} }}$
We are given with mass of Hydrogen in bottle A $= 12gm$ and mass of hydrogen in bottle B $= 48gm$ on comparing both the velocities we get:
$\Rightarrow \dfrac{{{V_A}}}{{{V_B}}} = \dfrac{{\sqrt {{m_B}} }}{{\sqrt {{m_A}} }}$
$\Rightarrow \dfrac{{{V_A}}}{{{V_B}}} = \dfrac{{\sqrt {48} }}{{\sqrt {12} }} \\\ \Rightarrow \dfrac{{{V_A}}}{{{V_B}}} = 2 \\\ \Rightarrow {V_A} = 2{V_B} \\\$
So we can say that the sound will travel faster in bottle A and will travel twice as fast as that of bottle B.

Therefore the correct answer is in bottle A the sound will travel faster by twice the speed of bottle B.

Note: If the temperature is not constant the velocity of the gas molecules or their rate would change according to the change in temperature. If temperature is increased, the rate or velocity of the gas molecules will increase as the kinetic energy of the gas molecules will increase. Therefore the velocity of gas molecules is directly proportional to the temperature.