Solve the following example.
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? (Ans: Temperature of B is 4 times the temperature of A.)
Solution
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 density of gas as
`v ∝ 1/sqrt(ρ)` ...... (i)
and the velocity of sound in gas is related to temperature of gas as
`v ∝ sqrt(T)` ...... (ii)
Combining (i) and (ii), we get
`v ∝ sqrt(T)/sqrt(ρ)`
The bottle are identical, this 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_2`
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.
