Atmospheric pressure experiment using a cup with a fluid to hold a glass plate

Question

When I was in high school, my teacher did an experiment to show the power of atmospheric pressure.

Experiment: Prepare a glass bottle, fill with water, put a glass plate on the bottle, make sure there is no air in the bottle, just water. Hold the glass bottle and plate, and invert them. Slowly, release the hand which hold the plate. Because the atmospheric pressure, the glass plate will not fall down.

Question 1: As we know, if we didn't put water into the glass bottle, just air in the glass bottle the glass plate will fall down. So, If we use other liquid instead of water, this liquid has smaller density than water, like alcohol, the plate will not fall down. But, if we continue to chose the smaller density liquid until the liquid has density like air the plate will fall down. So, if there is a threshold density of liquid make the plate between falling down and not falling down?

Question 2: If we put the bottle and plate into water slowly, when the plate and part of bottle are in the water but part of the bottle still above the water, will the plate fall down?

Question 3: Continuing question 2, when both bottle and plate are in the water I know the plate will fall down. So, how does the pressure of water and atmosphere change?

Is there a good description of this phenomena? Why does the trick continue to work with lower density fluids and why does placing the plate in the water cause it to fall?

Answer 1

There are two unrelated effects at work here. One is the atmospheric pressure, and the other is the surface tension of the water.

Start with you holding the plate in place, and consider what happens when you release the plate. For the plate to fall down one of two things must happen. Either the volume of the water in the glass must increase, to allow the plate to move down, or air must flow into the glass at the contact line between the glass and the plate.

Consider the first of these. If you pull the plate down slightly (and no air leaks in) the volume inside the glass must increase. Water has such a high bulk modulus that we can approximate it as incompressible. You would need an immense force pulling down on the plate to stretch the water to any significant degree. In practice the water would boil before its volume increased significantly, but even the lesser force required to boil the water is far greater than the weight of the glass plate.

So the only way the plate can move down is for air to leak in through the contact between the glass and the plate. However this means forming small bubbles at the contact line, and small bubbles have a very high pressure due to the surface tension at the air/water interface. This means the bubble formation requires a greater pressure than the weight of the plate can generate, so the plate can't move down this way either.

Incidentally, the effect of surface tension explains why the plate won't stick if the glass/plate contact isn't very good, or if there's a chip in the glass. In both cases there is a relatively large gap where a bubble can form, and large bubbles have a smaller pressure than small bubbles (the bubble pressure is inversely proportional to the bubble radius). The weight of the plate can generate enough pressure to form the large bubbles, and the plate falls off.

Now we can explain why the plate falls off when you immerse the glass and plate in water. If you do this there is no air/water interface at the contact between the glass and the plate, so there is no surface tension effect. Water can leak through even the tiniest gap between the rim of the glass and the plate so the plate falls off (though it may take a few seconds as the water won't flow in instantly).

The trick will work with most liquids because most liquids will neither expand nor boil under the weight of the plate. However it wouldn't work with very volatile liquids like ether, because ether boils too easily and vapour bubbles will form in the glass. You'd probably also find it wouldn't work if the air/liquid interface has too low a surface tension, because a low surface tension allows air to lean in between the glass and the plate.

Answer 2

QUESTION 1: I guess the different results with water and air are due to the difference in compressibility of water and air, rather than in their densities. When there is water in the bottle, a small deformation of the glass plate will result in small change of the volume of water in the bottle, which, however, will result in significant drop of pressure of the water, so the difference of pressure between the atmosphere and the water in the bottle will be enough to counteract the weight of the water and the glass plate. If there is air in the bottle, small deformation of the glass plate will result in a small change of air pressure, so the difference of pressure between the atmosphere and the air in the bottle will not be enough to counteract the weight of the glass plate.

Answer 3

A1: The pressure inside is hydrostatic $\rho g h$ plus the pressure of saturated water vapor $p_{sat}$, which is much smaller than the atmospheric pressure $p_a$ at temperature much below the boiling temperature of water, 100 C, so we neglect $p_{sat}$. Then the force balance will hold up to h~10 m which corresponds to $p_a \sim $ 100 kPa. For lower density liquid the height would be larger if $p_{sat}$ is still small. But low density liquids like alcohol tend to evaporate easily, so one should be careful if the vapor pressure can be neglected or not, depends on the temperature.

A2: Submerging the system slowly into water should not reduce the net force pushing the board to the bottle, actually the opposite - it would add external pressure on the board $\rho g H$. But in practice I think the board would quickly detach because the water would easily sip in from outside.

A3: Same as A2.