# Volume of a bubble rising in a lake

1. Dec 22, 2014

### Nathanael

1. The problem statement, all variables and given/known data
An air bubble of volume 20 cm^3 is at the bottom of a lake 40 m deep, where the temperature is 4.0°C. The bubble rises to the surface, which is at a temperature of 20°C. Take the temperature of the bubble’s air to be the same as that of the surrounding water. Just as the bubble reaches the surface, what is its volume?

2. Relevant equations
$PV=kT$
$ΔP=40ρg$

3. The attempt at a solution
I don't understand how the answer doesn't depend on the atmospheric pressure. If the atmospheric pressure were greater, then wouldn't the change in volume be smaller?

2. Dec 22, 2014

### Bystander

3. Dec 22, 2014

### Nathanael

Right... But that only gives you the difference in pressure. Isn't the atmospheric pressure still relevant?

4. Dec 22, 2014

### Bystander

Yes. What are the initial and final pressures? Don't feel that adding the "stack" is an unjustified ad hoc step to take. Pressure is the result of the sum of all masses above a certain point x "g."

5. Dec 22, 2014

### Nathanael

Initial pressure would be $P_{atm}+40ρg$ (where ρ is the density of water) and the final pressure would be $P_{atm}$

This gives me a final volume of $V_f=V_i(\frac{T_f}{T_i})(\frac{P_{atm}+40ρg}{P_{atm}})$

6. Dec 22, 2014

### Bystander

That's the way to play it. Hopefully whoever wrote the problem remembered it the same way.