A primitive diving bell consists of a cylindrical tank with one end open and one end closed. The tank is lowered into a freshwater lake, open end downward. Water rises into the tank, compressing the trapped air, whose temperature remains constant during the descent. The tank is brought to a halt when the distance between the surface of the water in the tank and the surface of the lake is 31.9 m. Atmospheric pressure at the surface of the lake is 1.01 x 10^5 Pa. Find the fraction of the tank's volume that is filled with air.
To find the fraction of the tank's volume that is filled with air, we need to consider the relationship between the pressure of the trapped air and the pressure of the water at the given depth.
In this case, the diving bell is submerged in a freshwater lake, so we can use the hydrostatic pressure formula:
P = P₀ + ρgh,
where:
P is the pressure at a given depth,
P₀ is the atmospheric pressure,
ρ is the density of water, and
h is the depth of the water.
Given:
P₀ = 1.01 x 10^5 Pa (atmospheric pressure),
h = 31.9 m (depth of the water).
First, let's calculate the pressure at the given depth using the hydrostatic pressure formula:
P = P₀ + ρgh.
The density of water, ρ, is approximately 1000 kg/m³.
P = 1.01 x 10^5 Pa + (1000 kg/m³) x (9.81 m/s²) x (31.9 m).
Now, we can find the volume of the air in the tank using Boyle's Law, which states that the product of the pressure and volume of a gas is a constant, as long as the temperature remains constant.
P₁V₁ = P₂V₂,
where:
P₁ and V₁ are the initial pressure and volume of the air in the tank, and
P₂ and V₂ are the final pressure and volume of the air in the tank.
The initial pressure, P₁, is equal to the atmospheric pressure:
P₁ = 1.01 x 10^5 Pa.
Since the temperature remains constant, we can write:
P₁V₁ = P₂V₂.
Let's assume that the initial volume of the air in the tank is V₁ and the final volume is V₂. We want to find the fraction of the tank's volume filled with air, which is V₂/V₁.
From the previous calculation, we have the final pressure, P₂:
P₂ = P = 1.01 x 10^5 Pa + (1000 kg/m³) x (9.81 m/s²) x (31.9 m).
Now, let's solve the equation P₁V₁ = P₂V₂ for V₂:
V₂ = (P₁V₁) / P₂.
Finally, we can calculate the fraction of the tank's volume filled with air:
Fraction = V₂ / V₁.
Substitute the values and calculate the fraction to get the answer.