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 38.0 m. Atmospheric pressure at the surface of the lake is 1.01 multiplied by 105 Pa. Find the fraction of the tank's volume that is filled with air.
I need help getting started. Ive looked in the book and have no idea where to begin.
Hint: What do you think is the pressure of the air in the tank?
just guessing but normal atmospheric pressure
PV=nRT ideal gas law is the only thing i got n R T will stay constant i believe so that leaves me with PV so PV of initial vs. PV of final... maybe
To solve this problem, you can use the concept of hydrostatic pressure. The pressure at any point in a fluid is determined by the weight of the fluid column above it.
First, let's define some variables:
- P_air: pressure of the air inside the tank
- P_water: pressure of the water column above the air in the tank
- P_atm: atmospheric pressure
Since the air is trapped inside the tank, P_air will remain constant throughout the descent. The pressure due to the water column can be calculated using the formula:
P_water = ρgh
Where:
- ρ is the density of water
- g is the acceleration due to gravity
- h is the height of the water column
In this case, h is the distance between the surface of the water in the tank and the surface of the lake, given as 38.0 m. The density of water, ρ, is approximately 1000 kg/m³, and the acceleration due to gravity, g, is approximately 9.8 m/s².
To find the fraction of the tank's volume filled with air, we need to compare the pressures inside the tank. The ratio of the pressure of the air to the atmospheric pressure will be equal to the fraction of the tank's volume filled with air.
Now, let's solve the problem step by step:
1. Calculate the pressure due to the water column:
P_water = ρgh
2. Calculate the pressure of the air inside the tank:
P_air = P_atm - P_water
3. Calculate the fraction of the tank's volume filled with air:
Fraction = P_air / P_atm
By substituting the given values into these equations, you can find the answer.
The ideal gas law:
PV = NRT
is certainly valid, but this equation simply tells you that temperature, volume and pressure are related. In this problem the temperature is constant, so what you need to know is the pressure. If you know the pressure then the equation will tell you by what factor the volume has changed.
So, you need to think about the fact that the oressur at a certain dept is higher than at the surface of the water. Why is that? Can you calculate the pressure at a certain dept from first principles (don't look in your book for an equation, you can't learn physics that way)
Hint: forget this particlar problem with the tank for a moment. Consider the water in the lake between the surface and some dept h. This water has a certain weight. Gravity acts on the water. Why doesn't the water accelerate downward? Clearly it doesn't, so there must be a force acting on it in te opposite direction.
This must be the force exerted by the water below accros the surface, i.e. the pressure times the area. But note that the air above the surface also exerts a downward force on the water.
So, you can find the pressure at some dept by demanding that the total force is zero.