Pressure of air in cylinder under water

[frostchaos123]

Homework Statement

Referring to the attachment, a cylinder of 2.5m filled with air is submerged to a dept of 82.3m into the sea such that sea water in cylinder is now x m. Water at bottom is 277.15K.

The Attempt at a Solution

Using ideal gas law PV=nRT of air,

the answer gives (1atm + 82.3*rho*g) * (2.5 - x) * Area of cylinder = nR(277.15)

However i don't understand why the pressure of the gas is 1 atm + 82.3*rho*g. Since the pressure of gas is dependant on height of water, shouldn't it be 1 atm + (82.3-x) * rho*g instead?

Or another reasoning is shouldn't it be like pressure of air + pressure of water trapped in cylinder = pressure at the bottom of the sea?

 

[tiny-tim]

hi frostchaos123!

(have a rho: ρ )

However i don't understand why the pressure of the gas is 1 atm + 82.3*rho*g. Since the pressure of gas is dependant on height of water, shouldn't it be 1 atm + (82.3-x) * rho*g instead?

yes, i think you're right …

P is the pressure on the volume of air, which is the pressure at the air-water surface, which is at height 82.3-x

(x will be very small compared with 82.3, but I'm not sure it's small enough to be negligible)

 

[frostchaos123]

Thanks for the help :)

 

[donutz610]

pressure is based on both volume and temperature of water, that might help u understand the problem a little better

 

[rwisz]

I would like to clarify the variables and assumptions used in this scenario to provide an accurate response. Firstly, it is important to note that the pressure of a gas is not solely dependent on the height of the water, but also on the temperature and volume of the gas. In this case, the ideal gas law is being used to calculate the pressure of the air in the cylinder, which takes into account all of these variables.

The equation provided in the attempt at a solution is correct, as it considers the pressure of the gas at the surface (1 atm) and the additional pressure due to the weight of the water above it (82.3*rho*g). This is known as the hydrostatic pressure and is a result of the weight of the water column above the gas.

The assumption made in this scenario is that the temperature and volume of the gas remain constant as it is submerged. This may not always be the case in real-world situations, but for the purposes of this calculation, it is a reasonable assumption.

Additionally, the pressure at the bottom of the sea is a combination of the hydrostatic pressure from the water column above and the atmospheric pressure at the surface of the sea. This is known as the total pressure and is equal to the sum of these two pressures.

In summary, the equation provided in the attempt at a solution is correct, taking into account the pressure of the gas at the surface, the additional hydrostatic pressure from the water column, and the constant temperature and volume of the gas.