Thermodynamics of expanding nitrogen bubble as it rises in water Problem
An air bubble is released from a nitrogen tank 20 meters under water. The bubble has a radius of 1 mm. What will be the radius of the bubble when it reaches the surface?
My Questions

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*Is it reasonable to assume the process is adiabatic and use the formula $p_iV_i^{\gamma}=p_fV_f^{\gamma}$ for an ideal gas? Note, p is pressure, V is volume and $\gamma$ is a constant.

*Generally, when can one assume a process is adiabatic when facing a problem such as this?

 A: So you're right, 'adiabatic' is a one possible assumption for this kind of problem. The other is 'isothermal'.  Which one is valid depends on the particulars of the problem (particularly bubble size). Some cases will be in between and neither will work very well.
First, let's get an estimate of how long this takes.  To get that we should find the speed at which the bubble moves where drag force ($F_D$) and buoyancy ($F_g$) are equal and opposite.
$$ F_d - F_g = 1/2 \rho_{w} A C_d v^2 - \rho_{w} g V = 0 $$
using the density of water, cross-sectional area of the bubble, drag coefficient (~1), velocity, gravitational acceleration and the volume of the bubble.  For this I get that v is about .16 m/s.  So it takes about 120 seconds for the bubble to reach the top. (N.B. this drag estimate is pretty crude and I've ignored the expansion of the bubble but this is all just to get a sense of things).
Then we want to take a look at how quickly heat conduction takes place within the bubble itself (i'm assuming that the density of the water is so high that it remains at constant temperature).  For heat conduction, you can estimate the characteristic time of the problem (meaning how long it takes to stabilize after a change) from the length scale of the problem (radius of the bubble in this case) and the thermal diffusivity ($\alpha$) of the material (~ $2.2 \cdot 10^{-5} m^2/s$ for nitrogen).
$$ t_c = \frac{r^2}{\alpha} $$
I get 0.045 seconds for this.
So, it looks like conduction is the gas is pretty fast on this scale, while the rising of the bubble. So I'd say that you can assume that the bubble remains at the temperature of the water.
This wouldn't be the case for a 20 mm bubble. That would take something more like 30 seconds to reach the top but would have a characteristic time of 18 seconds. that would get pretty gnarly to deal with.
You'd get some thin adiabatic with a bubble 100 mm in diameter.  12 s to rise 455 s characteristic time.
