# If a container of water from some depth is brought to the surface, what would be the pressure in the container?

Say we lower a container to depth $$h$$, and completely fill it with water. We then seal the container, and bring it up to the surface.

I know that the pressure at depth $$h$$ is $$P=\rho gh$$, where $$\rho$$ is the density of the water. This pressure is due to the weight of the column of water above the depth. When the container is brought to the surface, would the pressure inside it still be $$P=\rho gh$$, or would it be some smaller value?

The column of water doesn't exist anymore, so in contrast to before, there is no weight that produces pressure in the container. However, I am thinking that the compression of the water would produce some pressure.

On one hand, I am thinking that because of the initial pressure, the water would be compressed by some amount. When it would be brought to the surface, it would still be compressed by the same amount, hence the pressure would still be $$P$$.

On the other hand, I know that liquids aren't as compressible as gases, so I'm not sure if identical compression implies identical pressure for liquids. If it doesn't, then what would be the pressure? Would it be negligible, or would there still be a sizable pressure, although smaller than $$P$$?

• I thought about the opposite case. Assume that we go up to higher altitudes with a container which has an opening. The air with certain pressure will fill into the container. Now assume that we seal the opening perfectly and come down to earth's surface. I see no reason why the air pressure inside the container to go up to sea level air pressure as long as the container is perfectly sealed. So, I think if you trap water with high pressure inside a container when you come up to the surface the water will still have the same high pressure as long as the container is perfectly sealed. Jul 19 at 11:44
• This question is currently impossible to provide an actual answer. You will need to provide some boundary conditions to the question. Is the container assumed to be perfectly rigid (not reality)? Then the internal water pressure at surface is identical to depth. If you're aiming for something accurate, you're talking about an engineering calculation, comparing the bulk modulus, tensile strength, malleability, ductility, and other properties of the container. Any manner of how the container flexes, stretches, or compresses will all influence the water's measured pressure at the surface. Jul 19 at 21:21

That depends on the compressibility of the medium and the rigidity of the container.

Let's first look at a highly compressible medium like air before I answer you question regarding water.

Fill your container with air at sea level and seal it, then travel up into the high mountains. At sea level, you collected air at a pressure of 100 kPa. If you keep everything else constant (especially the temperature), pressure can change only by expanding the volume that contains the air mass. And your container is rigid enough (much more than the air), does not change its volume, so the pressure stays the same. When you open the container, you give the air mass a chance to expand until it reaches the ambient pressure. [I had such an experience with a sunscreen tube while travelling in the Andes.]

Now water is much less compressible than air. Though in fact, water under high pressure occupies a smaller volume than under low pressure, this difference is tiny (1 percent for a depth of 2000m, if I didn't mis-calculate). Again, with a perfectly rigid container, the pressure stays the same until you open it. Then the water expands (by a tiny fraction) until it reaches the volume corresponding to the ambient pressure. If the container wasn't rigid enough (think of a balloon), it allowed for the volume change (and thus the pressure change) already before opening it.

So, while in theory the pressure is kept until you open the container, in practice it will be difficult to get a container rigid enough to see the effect.

I did a quick estimation, how much force the container must withstand (without expanding). If you fill a 20cm-diameter sphere (~ 4 liters) at a depth of 2000m, this will give roughly 250 (metric) tons of force pushing the container walls outwards.

• So even e.g. a steel container would expand enough due to the water inside that the water pressure would drop appreciably? Do you have an example of a type of container that would be incompressible enough to maintain the water pressure? Jul 19 at 14:17
• No passive material can exactly maintain the water pressure found at depth, as no material has the infinite stiffness needed to avoid deformation when the exterior pressure drops. However, you could design an active device that would maintain a constant volume, and this would maintain the pressure exactly. Know what I mean? Jul 19 at 16:57
• My recollection is that after the sent the original Bathyscape down for an uncrewed test dive, they were alarmed by a lethally-intense jet of water than came out when they started to open it up. I think one has to conclude that the was the result of slightly-compressed water, held in that state by the tension of the pressure vessel; I think that the best way of considering some of Constantinescu's demonstrations is similar. Jul 19 at 21:25
• @MarkMorganLloyd The presence of air in the Bathyscaphe changes the calculation, since the pressure of a given quantity of gas is much less sensitive to its volume than the pressure of a given quantity of liquid. Thus a little flexibility in the container doesn't have much effect on gas pressure, but it can have a large effect on liquid pressure. Jul 19 at 22:09
• @JohnDoty Good point. OTOH, when some pressure is reached it would be reasonable to expect that all the air has been forced into solution at which point only the (slight) compressibility of the water and (larger) elasticity of the hull are relevant. Jul 20 at 6:53

When container is not sealed it is a part of entire system, so you can conclude that pressure at the bottom of your container is $$P=\rho{g}{h}$$.

But when you seal the container inside water. It is not a part of entire system now. So the pressure inside the container is not affected by external water. It is compensated by the walls of container, assuming it is a rigid container. Now the pressure inside the container depends on water filled.

• But actual containers are not perfectly rigid, so, in reality, the pressure in the sealed container would be less at the surface than at depth. Jul 19 at 13:04
• @JohnDoty Without more information about the container, it can be almost anywhere between the surface pressure and pressure at depth. Sure, some pressure is lost, finding accurately how much is dependent on the properties of the container. Jul 19 at 21:55
• @David Exactly. Reality is more complicated than whiteboard idealization. Jul 19 at 22:04
• The pressure inside the container depends on temperature. Which is why you put the Freon tank either on top of the condenser's discharge or in a bucket of hot water so that it fills faster. DO NOT tip it upside down if you're not charging with liquid. Jul 20 at 2:01

Pressure is just the force that a substance is applying outwards to everything in contact with it; the substance "trying to expand" if you will. If there's no balancing force pushing back the other way, then the substance will expand.

The pressure of water in the open ocean is equal to the weight of the column of water above it, but the presence of a column of water is not the cause of all pressure. The reason pressure is equal to the weight of the water above it is simply that the pressure is what is supporting the water, balancing out the force of gravity. The weight of the water above is what's stopping the pressure from making the deep water expand, and the pressure is what's stopping the water above from moving down.

If the pressure were higher then there would be a net imbalance and the water would experience an upward force. Since there's no more water coming in at the bottom but the top of the water is moving up, the same amount of water has to redistribute over a larger volume; the density goes down, and thus the pressure goes down.

If the pressure were lower than the weight then the water above would be partially unsupported; it would move down. This compresses the water at the bottom; the density goes up, and thus the pressure goes up.

So if the pressure is higher or lower than the weight of the water above it, the situation will self-adjust towards an equilibrium where they are equal. But storing water above some other water doesn't magically cause pressure; it's just two forces that are in balance, and we find them in balance because if they were out of balance they self-adjust until they are balanced again.

Now when you're talking about sealing water in a container, these two forces (the pressure of the contained water and the weight of the water that used to be above it) are no longer in direct contact with each other, and have no mechanism to self-adjust. They are now unrelated.

Instead the pressure of the contained water is pushing outwards on the container. If the container were perfectly rigid and infinitely strong, it would push back on the water with exactly the same force, keeping it contained in exactly the same volume and exactly the same pressure. This situation would continue wherever you took the container - to the surface of the ocean, the vacuum of space, or the centre of the earth. It is the structural forces of the container that maintain the pressure of the water inside.

Real containers, of course, are not perfectly rigid or infinitely strong. The water inside is pushing outwards and trying to expand. While you're still at the depth where you sealed the container, you are surrounded by water at the same pressure, so there is also an equivalent force of pressure on the outside of the container pushing in. That helps to hold the pressure of the water in. But as you rise to the surface, there is less pressure outside than inside the container. This imbalance will deform the walls of the container outward, which increases the interior volume, which lowers the pressure.

Exactly what happens depends on the material properties of the container. It might be able deform very little and exert enough force to balance the pressure inside even with much lower pressure outside (such as sea-level instead of deep ocean), allowing you to have a container of high pressure water. Or it might be able to deform enough without breaking to allow the contained water to expand enough to drop the pressure to sea-level, in which case you'll have a container that is still sealed, but the pressure inside has dropped as you ascended and is basically the same as surface-level. Anything in between is a possible outcome, depending on the properties of the container. Water is fairly incompressible, which means it actually doesn't need to expand/compress very much to change pressure by a lot. (And of course if the container hits the limits of how much it can deform and is not strong enough to contain the pressure difference, it will rupture and release the pressure that way)