# Microscopic interpretation of pressure in liquids

Pressure can be explained at microscopic level for a gas with kinetic theory of gases. From that the pressure $p$ is linked to the velocity of molecules (and it is caused by the high amount of collisions in the gas).

$$p=\frac{m N_a}{V} \frac{\bar{v}^2}{3}$$

Where $m$ is the mass of a molecule, $N_a$ Avogadro's number, $V$ volume, $\bar{v}^2$ the quadratic average velocity of molecules.

Nevertheless I did not find a similar microscopic interpretation in the case of liquids. In that case molecules are not as free as in a gas, so it looks like pressure is not linked to the higher or lower velocity of molecules. So what is responsible for liquid pressure, at a microscopic level?

Is there a quite simple microscopic description for pressure in liquids, as there is in the kinetic theory of gases?

• The basic kinetic theory of gases, as I guess you already know, often starts from the ideal gas idea, in which the gas particles are considered not to interact with each other, and unless you take the case of high pressure, the gas density is assumed to be low. With liquids, you can't assume either of these conditions. Molecules form, which provides extra degrees of freedom physics.stackexchange.com/questions/39706/… – user108787 Aug 18 '16 at 11:44
• Possible duplicate of physics.stackexchange.com/questions/39706/… – user108787 Aug 18 '16 at 11:45

Unlike a gas a liquid has a finite volume at zero pressure i.e. a liquid floating in vacuum would not expand beyond a certain volume. This volume is determined by the interatomic/intermolecular forces in the liquid.

If you look at the potential energy between two liquid molecules as a function of intermolecular distance $r$ it will be something like:

And the zero pressure volume will be the one where the intermolecular distances are at the minimum of the potential energy. This will be your zero pressure volume. If you compress the liquid you push the molecules up the higher potential energy curve towards smaller $r$, and that takes work, i.e. a force, which is why the compressed liquid has a pressure.

There is some effect of molecular motion, and indeed that's why liquids (usually) expand when you heat them. The potential well is not symmetric, so as you add thermal energy the mean intermolecular distance moves to larger $r$. However the main mechanism for sustaining a pressure is the intermolecular potential.

• Great answer! If I may ask one thing, studying Bernoulli equation I sometimes heard of pressure as "energy per unit volume", so pressure seems to be strictly linked to energy owned by the fluid. In the case of gases the relation is with kinetic energy, as $(1)$ suggests. In the case of fluid is this energy at wich pressure is linked mainly the potential energy that you described in the answer (as you said the molecular motion is neglegible)? – Sørën Aug 19 '16 at 11:16
• @Sørën: it depends. Suppose you put water at $1$ºC and STP in a sealed box then heat it to $99$ºC. If the box volume stays constant the pressure will go up because the water tries to expand. The expansion, and therefore pressure, is due to vibrational motion of the water molecules moving the mean distance to greater $r$. So molecular motion is involved, but is intimately related to the potential. I don't think it helps to imagine ideal gas like collisions with the box because the mean free path of a water molecule is effectively zero. – John Rennie Aug 19 '16 at 12:04

Pressure in liquids is not a microscopic phenomena as it is in gasses. In liquids there are two sources of pressure: 1) that caused by surface tension (if any), and 2) that caused by gravity. The pressure from surface tension varies with the sample size due to the different surface to volume ratios. You have probably seen videos of astronauts paying with liquid globular at 0 G on the ISS. The globular are spherical because that minimizes the surface to volume ratio. This pressure plays a role in the phenomena of freezing rain. The raindrops are so small that the pressure difference inside the drop leads to the drop remaining in the liquid state at lower than freezing air temperstures. SupercooIing also plays a role in freezing rain.

In the presence of gravity liquids are confined by containers and the pressure is a linear function of the depth. Near the bottom of the container the pressure on the container walls is the same as on the bottom. It is just the weight of the liquid divided by the bottom area. As you move up from the bottom the pressure drops off since the weight of the liquid above that point declines.