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Question
Helium gas is filled in two identical bottles A and B. The mass of the gas in the two bottles is 10 gm and 40 gm respectively. If the speed of sound is the same in both bottles, what conclusions will you draw?
Solution
Let the gas filled bottles be A and B respectively.
Mass of the gas in bottle A = 10 g
Mass of gas in bottle B = 40 g
The velocity of sound in gas is related to the density of gas as
`v ∝ 1/sqrt(ρ)` ...... (i)
and the velocity of sound in a gas is related to the temperature of the gas as
`v ∝ sqrt(T)` ...... (ii)
Combining (i) and (ii), we get
`v ∝ sqrt(T)/sqrt(ρ)`
The bottles are identical, which means the volumes of the gases are equal. Let the volume of the bottle be V. Let M1 and M2 be the masses of gases in bottles A and B, respectively and v1 and v2 be the velocity of the sound in the two bottles, respectively. Also, let T1 and T2 be their respective temperatures. Therefore,
`v_1/v_2 = (sqrt(M_2/V) sqrt(T_1))/(sqrt(M_1/V) sqrt(T_2))`
Now, given that, `"v"_1 = "v"_2`
⇒ `sqrt(M_1)sqrt(T_2) = sqrt(M_2)sqrt(T_1)`
or , `T_2 = (M_2T_1)/M_1`
Given , `M_1 = 10 "g" , M_2 = 40 "g"`
= `T_2 = (40T_1)/10 = 4T_1`
Thus, it can be concluded that the temperature of bottle B is 4 times the temperature of A.
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