A cylindrical diving bell 5 m in diameter and 10 m tall with an open bottom is submerged to a depth of 150 m in the ocean. The temperature of the air at the surface is 20 degrees Celsius, and the air temp 150 m down is 2 degrees Celsius. The density of seawater is 1025 kg/m^3. How high does the sea water rise in the bell when the bell is submerged?

Question

A cylindrical diving bell 5 m in diameter and 10 m tall with an open bottom is submerged to a depth of 150 m in the ocean. The temperature of the air at the surface is 20 degrees Celsius, and the air temp 150 m down is 2 degrees Celsius. The density of seawater is 1025 kg/m^3. How high does the sea water rise in the bell when the bell is submerged?

Answer 1

The number of moles of air in the bell remains the same. Use the reationship

PV/T = constant. Let 1 represent sea level condtions and 2 the condistions at 150 m depth.
P1*V1/T1 = P2*V2/T2

Calculate P2-P1 from the depth.
P2 - P1 = (density)*g*100 m = 1.506*10^6 N/m^2
P1 = 1 atm = 1.015*10^5 N/m^2
P2/P1 = 14.8
T2/T1 = 275/293
Solve for V2/V1 . That will equal L2/L1, the ratio of the lengths of the air column in the bell. L1 = 10 m. The distance the water rises is L1 - L2.

Answer 2

To find out how high the seawater rises in the bell when it is submerged, we need to consider the pressure difference between the surface and the depth of the ocean.

First, let's calculate the pressure at the surface of the ocean. The pressure at any depth in a fluid is given by the formula:

P = P₀ + ρgh

where P is the pressure at the given depth, P₀ is the pressure at the surface, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth.

Since the diving bell is open at the bottom, the pressure inside the bell at the surface is equal to the atmospheric pressure, which we'll assume to be 1 atm (atmosphere) or 101,325 Pa. Thus, P₀ = 101,325 Pa.

Now, let's calculate the pressure at the depth of 150 m:

P = P₀ + ρgh
P = 101,325 Pa + (1025 kg/m^3) * (9.8 m/s^2) * (150 m)

Using this formula, we calculate the pressure at a depth of 150 m:

P = 101,325 + (1025) * (9.8) * (150)
P ≈ 1,510,575 Pa

The pressure inside the bell will also be equal to the pressure at the depth, which is approximately 1,510,575 Pa.

To find the height of the seawater rise in the bell, we need to consider the difference in pressure between the inside and outside of the bell.

ΔP = P_inside - P_outside
ΔP = 1,510,575 Pa - 101,325 Pa
ΔP = 1,409,250 Pa

Now, we can use the hydrostatic equation:

ΔP = ρgh

Solving for h:

h = ΔP / (ρg)
h = (1,409,250 Pa) / ((1025 kg/m^3) * (9.8 m/s^2))

Simplifying this equation, we find:

h ≈ 137.58 m

Therefore, the seawater rises in the bell to a height of approximately 137.58 meters.