When an (ideal gas) air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 60%. If the temperature at the bottom and top of the lake are 3.0 and 11 °C, respectively, how deep is the lake? Question 10 options: 4.1 m 2.4 m 5.7 m Insufficient information to answer this question
Question
When an (ideal gas) air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 60%. If the temperature at the bottom and top of the lake are 3.0 and 11 °C, respectively, how deep is the lake?
Question 10 options:
4.1 m
2.4 m
5.7 m
Insufficient information to answer this question
Solution
To solve this problem, we need to use the combined gas law which states that the ratio of the product of pressure and volume to temperature remains constant.
The combined gas law is: P1V1/T1 = P2V2/T2
Given that the volume of the air bubble increases by 60%, we can say that V2 = 1.6*V1.
The temperatures are given in Celsius, but we need to convert them to Kelvin (since the gas law uses absolute temperatures). The conversion is K = °C + 273.15, so T1 = 3 + 273.15 = 276.15 K and T2 = 11 + 273.15 = 284.15 K.
The pressure at the bottom of the lake is the atmospheric pressure plus the pressure due to the water column above the bubble. The pressure at the top of the lake is just the atmospheric pressure. We can say that P1 = P0 + ρgh and P2 = P0, where P0 is the atmospheric pressure, ρ is the density of water (approximately 1000 kg/m³), g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the depth of the lake.
Substituting all these values into the combined gas law gives us:
(P0 + ρgh)V1/276.15 = P01.6*V1/284.15
We can cancel out V1 and P0 from both sides, and rearrange the equation to solve for h:
h = ((1.6/284.15) - (1/276.15)) * 276.15 * 284.15 / (ρg * (1/284.15 - 1/276.15))
Substituting the values for ρ and g gives us:
h = ((1.6/284.15) - (1/276.15)) * 276.15 * 284.15 / (1000 * 9.81 * (1/284.15 - 1/276.15))
Solving this equation gives us a depth of approximately 2.4 meters. So, the correct answer is 2.4 m.
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