# Air Bubble Rising in a Lake

#### erik-the-red

Question
A diver named Jacques observes a bubble of air rising from the bottom of a lake (where the absolute pressure is $$3.50 {\rm atm}$$) to the surface (where the pressure is $$1.00 {\rm atm}$$). The temperature at the bottom is $$4.0{\rm ^{\circ} C}$$, and the temperature at the surface is $$23.0{\rm ^{\circ} C}$$.
1. What is the ratio of the volume of the bubble as it reaches the surface ( $$V_s$$) to its volume at the bottom ($$V_b$$)?
Well, I'm thinking $$P_b = P_s + \rho \cdot g \cdot h$$. But, I don't know the height, so this may not be the right place to start.
Another thing I was thinking was $$\rho_{water} = \frac{m}{V} = 1$$ kg / L.
But, it seems like neither of these will get me started.
What is a good starting point for this question?

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#### Doc Al

Mentor
Hint: Treat the air in the bubble as an ideal gas.

#### Hurkyl

Staff Emeritus
Gold Member
erik-the-red said:
Well, I'm thinking $$P_b = P_s + \rho \cdot g \cdot h$$. But, I don't know the height, so this may not be the right place to start.
The first thing that strikes me when I see this is: "Don't you know everything but the height?"

What is a good starting point for this question?
I would probably begin by trying to think of a principle (and corresponding formula, if appropriate) that would involve the volume of the bubble.

#### erik-the-red

Hurkyl said:
The first thing that strikes me when I see this is: "Don't you know everything but the height?"
What possessed me to not realize that?

Thanks

#### erik-the-red

I played on Doc Al's hint on treating the air in the bubble as an ideal gas. If so, this means I can use the combined gas law.

$$\frac{P_s \cdot V_s}{T_s} = \frac{P_b \cdot V_b}{T_b}$$.

From that, I get $$\frac{V_s}{V_b} = \frac{P_b \cdot T_s}{P_s \cdot T_b}$$.

Plugging in known values results in $$\frac{V_s}{V_b} = \frac{3.50 * (23.0 + 273.15)}{1.00 * (4.00 + 273.15)}$$.

This is 3.74. Is this correct?

#### erik-the-red

The second part of the question is:

Would it be safe for Jacques to hold his breath while ascending from the bottom of the lake to the surface?

The volume at the surface is almost four times the volume at the bottom. I think the answer is no. If he holds it in, the volume of air in his chest will expand, right?

#### NateTG

Homework Helper
erik-the-red said:
The second part of the question is:

Would it be safe for Jacques to hold his breath while ascending from the bottom of the lake to the surface?

The volume at the surface is almost four times the volume at the bottom. I think the answer is no. If he holds it in, the volume of air in his chest will expand, right?
That actually depends on what kind of diving Jaques is doing. If he's using compressed air of any kind (like SCUBA gear), then the answer is definitely no. If, on the other hand he's free diving then he could (mostly) get away with holding his breath. (A difference of 2.5 atmospheres corresponds to about 90 feet, which is quite feasible.)

#### jyeh1989107

The second part of the question is:

Would it be safe for Jacques to hold his breath while ascending from the bottom of the lake to the surface?

The volume at the surface is almost four times the volume at the bottom. I think the answer is no. If he holds it in, the volume of air in his chest will expand, right?

If Jacques were holding his breath, then air would be unable to enter or leave his lungs. As he ascends to the surface, the air in his lungs would expand, like the air in the bubble, and his lungs would have to stretch outward to hold this increased volume, which would be extremely unsafe.

In fact, even if he does not hold his breath, if he ascends too quickly after a particularly long or deep dive, the nitrogen dissolved in his bloodstream could form into small bubbles, which can be equally dangerous to any diver. This condition is known as decompression sickness, or more commonly as the bends.

#### DaveC426913

Gold Member

If Jacques were holding his breath, then air would be unable to enter or leave his lungs. As he ascends to the surface, the air in his lungs would expand, like the air in the bubble, and his lungs would have to stretch outward to hold this increased volume, which would be extremely unsafe.
A bit of an understatement there.

The lungs can only stand an overpressure of about 25%. More will cause the alveoli (air sacs) will rupture and his blood will leak out into his pleura (the space between the lungs and the chest wall). And that's just the start...

"Air Bubble Rising in a Lake"

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