Depth of a lake

Here are the assumptions I made:

• the temperature $T$ is constant

• air is an ideal gaz

• the bubble is always a sphere

Therefore $PV(=nRT)$ is constant where $P$ is the pressure, $V$ the volume of the bubble.

Let:

• $h$ the depth of the lake

• $r_i$ the initial radius

• $r_f$ the final one

• $P_0$ the atmospheric pressure

• $V_i$ and $V_f$ initial and final volumes of the bubble

• $\gamma$ the surface tension water/air

Usefull formulas:

• at the bottom of the lake, $P=P_0+\rho g h$

• Laplace formula for pressure between an interface: (2D ie the interface is 1D) $P_A-P_B=\frac{\gamma}{R}$ where $R$ is the radius of curvature of the interface (you may sum 2 radius of curvature if you are in 3D, ie the interface is a 2D surface) . In our case, the pressure inside the bubble is $P_{bubble}=P_{water}+2\frac{\gamma}{r}$

Using $PV$ constant we get:

$(P_0 +\rho g h +2\frac{\gamma}{r_i})r_i^3=(P_0+2\frac{\gamma}{r_f})r_f^3$

Solving for $h$ :

$h=\dfrac{(P_0+2\frac{\gamma}{r_f})(r_f/r_i)^3-2\frac{\gamma}{r_i}-P_0}{\rho g}=\boxed{1.28 \ \text{km}}$

Other assumptions could have been made, like $PV^{\gamma}$ is constant (this is not the same $\gamma$ ).