When the U.S. submarine Squalus became disabled at a depth of 80 m, a cylindrical chamber was l

When the U.S. submarine Squalus became disabled at a depth...

When the U.S. submarine Squalus became disabled at a depth of $80 \mathrm{~m}$, a cylindrical chamber was lowered from a ship to rescue the crew. The chamber had a radius of $1.00 \mathrm{~m}$ and a height of $4.00 \mathrm{~m}$, was open at the bottom, and held two rescuers. It slid along a guide cable that a diver had attached to a hatch on the submarine. Once the chamber reached the hatch and clamped to the hull, the crew could escape into the chamber. During the descent, air was released from tanks to prevent water from flooding the chamber. Assume that the interior air pressure matched the water pressure at depth $h$ as given by $p_{0}+\rho g h$, where $p_{0}=1.000 \mathrm{~atm}$ is the surface pressure and $\rho=1024 \mathrm{~kg} / \mathrm{m}^{3}$ is the density of seawater. Assume a surface temperature of $20.0^{\circ} \mathrm{C}$ and a submerged water temperature of $-30.0{ }^{\circ} \mathrm{C}$. (a) What is the air volume in the chamber at the surface? (b) If air had not been released from the tanks, what would have been the air volume in the chamber at depth $h=80.0 \mathrm{~m}$ ? (c) How many moles of air were needed to be released to maintain the original air volume in the chamber?

Key Concepts

Hydrostatic Pressure

This concept explains how pressure in a fluid increases with depth due to the weight of the fluid above. In many practical problems involving underwater environments, the pressure at a given depth is determined by the sum of the atmospheric pressure and the product of the fluid density, gravitational acceleration, and depth. This principle is essential for understanding the conditions acting on submerged structures and chambers.

Ideal Gas Law and Combined Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas, and it underpins many calculations about how gases behave under changing conditions. When conditions change, such as in pressure and temperature simultaneously, the Combined Gas Law (P1V1/T1 = P2V2/T2) is used to predict how a gas adjusts its volume. This is crucial when determining how much the gas will compress or expand under varying underwater pressures and temperatures.

Molar Quantity Calculations

Calculating the number of moles of gas required in a system involves relating the amount of gas to its pressure, volume, and temperature using the Ideal Gas Law. This concept is particularly important in applications where maintaining a constant volume or specific pressure is necessary, such as in regulated air supply systems in submerged environments.

Transcript

00:03 Hello, so this question we want to find the volume on the surface so that's going to be given by the area times the height so the area because it's circular is going to be pi on squared we're giving the radius as one and the height is four so our volume is going to be approximately 12 .6 miz cube okay so that's that volume for the first one the second part you were given the temperature as 20 degrees so you convert that to kelvin and that gives you 293 so we are adding that to 273 so you get 293 the initial pressure p1 is the same as p0 which is the atmospheric pressure which is just one atm okay so in this question in this part of the question we are looking at the height of 80 meters uh we also have a second temperature t2 which is negative 30 so if you convert that to kelvin that's going to be 243 kelvin okay so before we can uh find the final volume we want to find the pressure okay the final pressure which is given by p.

01:47 Not i mean that's p2 is p0 plus row g adjust density times gravity times height you will give it the density gravity is a constant the height we will also give it but remember that if you do this part of the calculation it's going to give you pascal okay so the one in bracket is also pressure we're going to give you pascal and you want to convert that to a atm because we cannot add pascal to atm.

02:20 You see atm right here.

02:22 So that is why you have this conversion right here.

02:29 So if you do that, then you can get atm and then add to 1.

02:35 And then that's going to give you 8 .95.

02:38 So that's p2.

02:40 So now that we have p2, we can find v2 using the i2 gas equation, which is given by this.

02:51 So just isolate v2.

02:54 You get this and then we can now put in our values.

02:59 Of course, we have everything.

03:02 And then we're going to get the volume to be 1 .16.

03:10 Let's look at the last part of the question.

03:12 We want the number of moles...