Question

Helium gas is filled in two identical bottles $A$ and $B$. The mass of the gases are $10g$ and $40g$ respectively. If the speed of sound is same in both the bottles, what conclusion will you draw ?

Answer 1

Mass of the gas in bottle $A=10g$
Mass of gas in bottle $B=40g$
The velocity of sound in gas is related to density of gas as
$v\propto \frac{1}{\sqrt{\rho }}$ .........(i)
and the velocity of sound in gas is related to temperature of gas as
$v\propto \sqrt{T}$ ........(ii)
Combining (i) and (ii) , we get
$v\propto \frac{\sqrt{T}}{\sqrt{\rho }}$
The bottles are identical , this means the volume of the gases are equal. Let the volume of the bottle be V.
Let $M_1$ and $M_2$ be the masses of gases in bottles A and B , respectively and v${}_1$a and v${}_2$ be the velocity of the sound in the two bottles, respectively.
Also let $T_1$ and $T_2$ be their respective temperatures. Therefore,
$\frac{v_1}{v_2}=\frac{\sqrt{\frac{M_2}{V}}\sqrt{T_1}}{\sqrt{\frac{M_1}{V}}\sqrt{T_2}}$
Now , given that , ${\text{v}}_1={\text{v}}_2$
$\Rightarrow$ $\sqrt{M_1}\sqrt{T_1}=\sqrt{M_2}\sqrt{T_1}$
or, $T_2=\frac{M_2{T}_1}{M_2}$
Given , $M_1=10g,M_2=40g$
$=T_2=\frac{40T_1}{10}=4T_1$
Thus , it can be concluded that the temperature of bottle $B$ is $4$ times the temperature of $A$ .