The full equation is PV=nRT for ideal gases but you have to use matching units.
P=pressure (atmopsheres)
V=volume (liters)
n=moles (related to the number of molecules)
R = ideal gas constant
T=temperature (degrees Kelvin)
we have a comparison of 2 of conditions and the container is sealed:
Initial PV/nRT = final PV/nRT
we can reduce it to these 2 equations since n and R are constant. No gas leaves or enters after sealing. R is always a constant.
PV/T = PV/T Where the values on one side are where you start and the value on the other is in the cold water.
And this reduces further since the volume is identical in your container. We are ignoring the trace shrinkage from getting cold. Its not enough to matter for this.
P/T = P/T
P initial is 1 atm
T initial is 37 C plus 273 = 310K (kelvin - a unit of temp measurement that matches the equation)
P final is what we want to know = X
T final is 10C plus 273
So you have: (1)/310 = X/283
solve for X, 283/310 = 0.91 atm
So the pressure drops from 1.0 atm to 0.91 atm.
When you dive, there is about 1 atm of pressure for each 33 ft or 10 meters of seawater.
And there is one atm pressure at the surface
So at 30 meters, you have 1 atm at the surface plus 3 for the depth = 4.
The 0.09 atm change inside the container is not likely to make a difference.
I have used these cheap housings for yrs and not had a leak ever - except when I did not seal it correctly once. But I change cameras every few yrs and do not get to dive that often so they do not get that much work out.
Hope the equations were what you were looking for.
By the way, I am a nerd for knowing and you could be one for wanting to know.
IMHO,
Dave