Is the reason why water in a cup rises if a solid is placed within it the ideal gas law?

Question

Let's say you have a cylinder with water that takes up volume V and has temperature T. When a solid mass is placed within the cylinder, the water momentarily takes up a smaller volume within the cylinder and in $$P=\frac{nRT}{V}$$, when V decreases, pressure increases. When pressure increases, the water pushes on its surroundings with more force, including the air above it, so the equilibrium had before the mass was placed is disturbed and the water rises until atmospheric pressure downwards is equal to the water's pressure upwards.

This sounds reasonable to me, but with my limited experience, I'm aware I could be wrong and the ideal gas law may not even apply to fluids like I've done so. Would anyone care to correct me if I'm wrong?

Answer 1

The ideal gas law isn't a good model to describe a fluid like water, because the ideal gas law predicts that the number density of a fluid, $$\frac nV = \frac P{RT},$$ depends linearly on the pressure and the temperature. Were that the case, ice water at 273 K would have only about 73% the volume of boiling water at 373 K. You are invited to verify this is not the case using a glass measuring cup and five minutes with a microwave. For that matter, reducing the pressure above a cup of water doesn't increase its volume appreciably, either, certainly not the way that reducing the pressure outside a balloon does.

The liquid in the cup rises because the water is essentially an incompressible fluid with constant density: when the fluid is displaced by the object, the fluid can no longer fit into the reduced volume and has to go somewhere else. Rising up is common because open-topped containers are a convenient way to hold fluids, but it's possible to image other geometries as well.

Here's an example to illustrate the difference. Suppose you fill a glass with air and seal the top with plastic wrap. Make it a good airtight seal. Now set the glass upside-down on a countertop on top of some foreign body, like a stack of a few coins. The plastic wrap will deform to wrap around the coins, but you shouldn't see any air leaking out of your seal: you can compress the ideal gas into a modestly smaller volume by introducing a modest change of temperature or pressure.

If you instead fill the glass with water and seal the top with plastic wrap (perhaps you do this whole operation under water, so that there is zero compressible air trapped inside) then you will find that you cannot decrease the volume allotted to the water without spilling any.

You're right that fluid pressure is still the motive force here; the difference is that the ideal gas law doesn't describe liquids.

Answer 2

As was mentioned in some comments, water is a mostly incompressible liquid - that is, if you want to make the volume of water smaller, you need to apply a LOT of pressure. How much pressure is given by the bulk modulus, which for water at room temperature is about 2.2 GPa. Comparing with the bulk modulus for air at 1 atm, it is about 100 kPa - 2200 times smaller.

So if you try to reduce the volume of water, it will try very hard to find another place to go, rather than occupy a smaller volume. One way to think about it is this: if you have a small car that can seat four people, and you need to transport a large suitcase in it, you might end up with somebody sitting on the roof. That is the situation of water: there is not a lot of space between the molecules. By contrast, an ideal gas can be thought of as a large bus with four people in it: if you need to transport a suitcase, somebody may have to sit a little bit close to someone else - but nobody has to sit on the roof.

We can do the same analysis a bit more formally by considering the microscopic picture. If you have an elastic ball bouncing back and forth between two walls, then the average force on each wall is going to depend on the speed of the ball, and the time between collisions - which is a function of the distance between the walls. If the walls are closer, the ball bounces more frequently, and you have a higher average pressure (more collisions with the wall per unit time). That is the "ideal gas" situation.

For a liquid, the molecules are so close together that they experience a significant attraction - the Van der Waals force. This means that the molecules cannot be considered as individual balls that spend most of their time flying around without hitting anything. And because they are so close together, if you reduce the distance even by a tiny amount, the repulsive force increases significantly - much more than for a gas. As is seen by the difference in bulk modulus.

So the ideal gas law cannot be applied to liquids. It would have had to be called something else, I suppose.