An incredible news story today is about a man who survived for two days at the bottom of the sea (~30 m deep) in a capsized boat, in an air bubble that formed in a corner of the boat. He was eventually rescued by divers who came to retrieve dead bodies. Details here. Since gases diffuse through water (and are dissolved in it) the composition of air in the bubble should be close to the atmosphere outside, if the surface of the bubble is large enough; so the excessive carbon dioxide is removed and oxygen is brought in to support life of a human.
Question: How large does the bubble have to be so that a person in it can have indefinite supply of breathable air?
Best Answer
Summary: I find a formula for the diameter of a bubble large enough to support one human and plug in known values to get $d=400\,{\rm m}$.
I'll have a quantitative stab at the answer to the question of how large an air bubble has to be for the carbon dioxide concentration to be in a breathable steady state, whilst a human is continuously producing carbon dioxide inside the bubble.
Fick's law of diffusion is that the flux of a quantity through a surface (amount per unit time per unit area) is proportional to the concentration gradient at that surface,
$$\vec{J} = - D \nabla \phi,$$
where $\phi$ is concentration and $D$ is the diffusivity of the species. We want to find the net flux out of the bubble at the surface, or $\vec{J} = -D_{\text{surface}} \nabla \phi$.
$D_{\text{surface}}$ is going to be some funny combination of the diffusivity of $CO_2$ in air and in water, but since the coefficient in water is so much lower, really diffusion is going to be dominated by this coefficient: it can't diffuse rapidly out of the surface and very slowly immediately outside the surface, because the concentration would then pile up in a thin layer immediately outside until it was high enough to start diffusing back in again. So I'm going to assume $D_{\text{surface}} = D_{\text{water}}$ here.
To estimate $\nabla \phi$, we can first assume $\phi(\text{surface})=\phi(\text{inside})$, fixing $\phi(\text{inside})$ from the maximum nonlethal concentration of CO2 in air and the molar density of air ($=P/RT$); then assuming the bubble is a sphere of radius $a$, because in a steady state the concentration outside is a harmonic function, we can find
$$\phi(r) = \phi(\text{far}) + \frac{(\phi(\text{inside})-\phi(\text{far}))a}{r},$$
where $\phi(\text{far})$ is the concentration far from the bubble, assumed to be constant. Then
$$\nabla \phi(a) = -\frac{(\phi(\text{inside})-\phi(\text{far}))a}{a^2} = -\frac{\phi(\text{inside})-\phi(\text{far})}{a}$$
yielding
$$J = D \frac{\phi(\text{inside})-\phi(\text{far})}{a}.$$
Next we integrate this over the surface of the bubble to get the net amount leaving the bubble, and set this $=$ the amount at which carbon dioxide is exhaled by the human, $\dot{N}$. Since for the above simplifications $J$ is constant over the surface (area $A$), this is just $JA$.
So we have $$\dot{N} = D_{\text{water}} A \frac{\phi(\text{inside})-\phi(\text{far})}{a} = D_{\text{water}} 4 \pi a (\phi(\text{inside})-\phi(\text{far})).$$
Finally assuming $\phi(\text{far})=0$ for convenience, and rearranging for diameter $d=2a$
$$d = \frac{\dot{N}}{2 \pi D_{\text{water}} \phi(\text{inside})}$$
and substituting
I get $d \approx 400\,{\rm m}$.
It's interesting to note that this is independent of pressure: I've neglected pressure dependence of $D$ and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at STP.
Finally, a bubble this large will probably rapidly break up due to buoyancy and Plateau-Rayleigh instabilities.