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.) Solve the example.
Answers
Gas mass in A bottle = 10 g
Gas mass in B bottle = 40 g
The sound velocity in gas medium is related to gas density as v ∝ 1/√p …… (1)
The velocity of sound in gas is related to temperature of gas is ∝ √T ….. (2)
Combining (1) and (2), then we get v ∝ √T/√p
Both bottles are identical, means the gas volumes present in bottle is equal.
The bottles volume is assumed as V.
Let M_1 and M_2 be the gas masses A and B bottles, respectively and v_1 and v_2 be the sound velocity in two bottles respectively.
T_1 and T_2 be their respective temperatures. Therefore,
v_1/v_2 = (√(M_2/v) × √(T_1))/(√(M_1/v) × √(T_2))
Now given that v_1 = v_2
→ √(M_1) × √(T_2) = √(M_2) × √(T_1)
Or
T_2 = (M_2 × T_1)/M_1
Given, M_1 = 10 g; M_2 = 40 g
→ T_2 = (40 × T_1)/10 = 4T_1
Thus, it can be concluded that the B bottle temperature is 4 times the temperature of A bottle.