Volume of a bubble rising in a lake

[Nathanael]

Homework Statement

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?

Homework Equations

$PV=kT$
$ΔP=40ρg$

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?

 

[Bystander]

bottom of a lake 40 m deep,

rises to the surface

 

[Nathanael]

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

 

[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."

 

[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 $V_f=V_i(\frac{T_f}{T_i})(\frac{P_{atm}+40ρg}{P_{atm}})$

 

[Bystander]

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