The volume of the bubble is due to the exapansion or contraction of the air inside it, that it, the volume of air. If we assume that the air inside the bubble is ideal, we can predict its dependency on several parameters using the ideal gas equation. The ideal gas equation is
PV = nRT
where P = pressure (Pa)
V = volume (m^3)
n = number of moles (mol)
R = universal gas constant (8.314 J/ mol K)
T = temperature (K)
At the bottom of the lake, the pressure should be the hydraulic pressure + atmospheric pressure
P (bottom) = p*g*h + 101325
where p is the density of the lake which is assumed to be equal to that of water = 1000 kg/m3
g is the gravitational acceleration (9.8 m/s^2)
h is the depth of the lake (m)
101325 Pa = atmospheric pressure
therefore P(bottom) = 528060 Pa
therefore the number of moles n using the ideal gas equation is
n = 2.28 x 10^-4 mol
at near the surface P is approximately 101325, therefore the volume is V = 5.5 cm^3