# Pressure and molecular average kinetic energy in ideal, incompressible and static liquid in a thermally insulated container

We know that in liquids pressure increases with depth. We also know that pressure is nothing but the rate of change of momentum per unit time per unit area, during the collisions of different molecules with each other and also with the walls of the container. Now if the pressure is increasing with increasing the depth then the number of collisions per unit time must have increased at greater depths (assuming incompressible fluid).

1. It implies that the average molecular kinetic energy of molecules at depths will be greater than the average molecular kinetic energy at the shallow water. Is it true? I do not find any literature confirming this idea, so I need a confirmation on it.

2. If average molecular kinetic energy at depth is greater then what is responsible for it? How does it actually happen?

3. Does it also mean that the temperature of water at depths will be greater than the temperature at the shallow water because the average molecular kinetic energy at the depth is greater? Does it mean that water will itself develop some temperature gradient? Even if its earlier temperature was uniformly distributed?

4. Can the variation of pressure happen without the variation of density and without the variation of temperature? If NO, then how does it happen in ideal incompressible fluids, which we study and are taught everywhere?

Note 1: Here since we are talking about ideal liquid, so it will be incompressible, so the variation of density with depth can be ignored (unless the reader believes the variation of density is indispensable. In that case, I request the reader to also kindly comment on how then pressure variation can happen with depths in ideal incompressible liquids).

Note 2: I have seen similar questions on Stack Exchange to which most of the responses say that at the depth the weight of liquid above it will increase, the force will increase and hence the pressure will increase. I am not against that idea, but that is the macroscopic understanding of this phenomenon. I am asking for a microscopic understanding of this phenomenon.

## 1 Answer

1. It implies that the average molecular kinetic energy of molecules at depths will be greater than the average molecular kinetic energy at the shallow water. Is it true? I do not find any literature confirming this idea, so I need a confirmation on it.

The increase in water pressure with depth is due to the weight of all the water above pushing down on the water below. It is not the result of an increase in the kinetic energy of the water molecules. What you may be thinking of is the increase in pressure that occurs when a fixed volume of gas is heated, i.e., when the temperature of the gas goes up. In that case, in the example of an ideal gas, the increase in pressure is proportional to the increase in average kinetic energy of the molecules

1. If average molecular kinetic energy at depth is greater then what is responsible for it? How does it actually happen?

See answer to Q1.

1. Does it also mean that the temperature of water at depths will be greater than the temperature at the shallow water because the average molecular kinetic energy at the depth is greater? Does it mean that water will itself develop some temperature gradient? Even if its earlier temperature was uniformly distributed?

Again, see answer to Q1.

1. Can the variation of pressure happen without the variation of density and without the variation of temperature? If NO, then how does it happen in ideal incompressible fluids, which we study and are taught

Yes the variation in pressure happens without a variation in density (and without variation in temperature). Although water density increases with depth it does so by an extremely small amount. I read on this exchange SE that at the bottom of the deepest ocean density only increases by about 5%. So it can be ignored with regard to pressure variation. The increase in pressure with depth is strictly due to the weight of the water above.

Note 2....I am not against that idea, but that is the macroscopic understanding of this phenomenon. I am asking for a microscopic understanding of this phenomenon.

Pressure is a macroscopic not microscopic phenomenon for liquids as opposed to gases. But when the pressure increases it does decrease the intermolecular distances between the water molecules below due to the force (weight) of the water above. However this decrease is extremely small and does not translate to any significant increase in density of the fluid.

Hope this helps.

• Why the down vote? – Bob D Dec 10 '19 at 17:53
• No. I did not downvote. I rather upvoted. – Devansh Mittal Dec 10 '19 at 18:26
• @DevanshMittal My message wasn't directed at you. It was to whomever did down vote, as down votes are anonymous. I don't mind a down vote if at least I know the reason. – Bob D Dec 10 '19 at 18:35