Doc Al said:
Simple: Va + Vw + Remaining "Empty" Space = Vc.
Empty space? If it's empty, it's vacuum. But a vacuum can't be in equilibrium with water, it will boil. For the case of a liquid that can't boil, that's a slightly different issue. It doesn't really matter as long as the surface of the water doesn't boil.
Regardless; let's change the liquid for the sake of discussion. I think this should help shed some light on things.
Thus far we’ve discussed water which has some vapor pressure at ambient temperature that’s well above a perfect vacuum. It’s around 0.3 to 0.4 psia at 70 F.
What if we used a liquid that could not convert to a gas? Vacuum pump oils for example are highly refined, long chain hydrocarbons with a vapor pressure in the micron range. For the sake of argument, let’s just consider a hypothetical liquid such as hypothetical vacuum pump oil that does not boil/does not vaporize at any low pressure. We'll aslo hypothesize that this liquid is completely incompressible so it rules out any bulk modulus affects. I think these affects are probably getting in the way of understanding what's going on anyway, so let's just ignore them.
For our experiment, we use the balloon and this non-boiling liquid. Let’s start at atmospheric pressure at the upper surface, say 15 psia for the sake of argument, and the balloon is at the top of the container. Let’s also assume the liquid is roughly the same as water, so for every 2 feet of head we get 1 psi. So 30 feet down in this liquid, the pressure is 30 psia (ie: rho*g*h +P
s where P
s is the pressure at the surface = 15 psia at the surface, plus 15 psi head).
As we pull the balloon down 2 feet, we might think the balloon shrinks due to increased head. But if it did, we’d need to come up with that volume somewhere else in our system (ie: We may think there's an additional volume = V
v (vacuum volume)).
-> If a vacuum space formed above the liquid, and assuming Bernoulli’s still holds (which it does) then the surface pressure would be 0 psia, so the pressure 2 feet down (per rho*g*h + P
s where P
s =0 for vacuum) would be only 1 psia. But the balloon had 15 psia in it to begin with, hence it won’t be at 1 psia if it shrinks. Pressure has to be greater than 15 psia and the conclusion is it hasn’t shrunk.
In other words, if the surface pressure drops instantly from 15 psia to 0 psia, because there's a vacuum there, we have a sudden change in pressure ALL THE WAY DOWN to the bottom of the container.
So we are left with the balloon not shrinking and not expanding. We are left with the surface pressure decreasing by 1 psi, not suddenly going into vacuum. There is no sudden step change in pressure at the upper surface.
When the balloon gets down to 30 feet, the pressure at the top of the column finally reaches 0 psia (rho*g*h = 15 psi). At this point, a vacuum just starts to form, and the pressure 30 feet down is only 15 psia instead of the 30 psia originally. Everything is still in equilibrium, and the balloon hasn't changed volume yet.
As the balloon goes deeper, a vacuum space finally begins to form above the liquid as the depth of the balloon goes down more than 30 feet and pressure in the balloon exceeds 15 psia. Now, as the balloon goes deeper and contracts, the volume that it decreases by can be made up by the volume of the vacuum at the top of the container.
What everyone is missing is the Bernoulli equation for a static column of liquid, P(at some point below the surface) = P
s + rho*g*h