# What is pressure potential energy?

Hello Forum,

potential energy is a form of energy that something has in virtue of occupying a certain position in space (gravitational potential energy, electric potential energy, etc...)

What is, in the case of fluids, pressure potential energy?

A fluid is said to have a certain pressure, which is P=F/A
work is W=Fd so W= P A d= P V where V is volume...

So pressure, in a sense, is Work, energy per unit volume....but why does this energy need to be potential?

Pressure (static or dynamic) seems to indicate the pushing between the molecules composing the fluid..

thanks
fisico30

Gold Member
You can effectively view static pressure as representing potential energy and dynamic pressure as representing kinetic energy. For example, in an inviscid, incompressible flow, Bernoulli's equation does exactly this.

Philip Wood
Gold Member
I'd argue that there's no need to think of an energy specifically associated with fluid pressure. We have gravitational PE and kinetic energy of bulk movement, and the fluid's internal energy, which is a sum of molecular kinetic and intermolecular potential energies. That's surely all we need. The points I venture to make below concern mainly this internal energy for the case of a gas, and what happens when a gas expands pushing a piston.

(1) Potential energy of a system is the work it can do because of the relative positions of its parts.

For example we have the gravitational potential energy of a system consisting of the Earth and an apple. [It's quite common to speak of this as the apple's potential energy, though, really, it's just as much the Earth's. It's a mutual thing. So best to think of it as the system's energy.] The energy arises because of the mutual forces between bodies in the system.

(2) For a gas at 'ordinary' density (up to, say, 30 times that of the atmosphere at sea level) the forces between the molecules are, to a good approximation, negligible (except during collisions), so we can pretty much forget about any intermolecular potential energy. The gas's internal energy is almost all kinetic, the sum of the kinetic energies of random motion of the molecules.

(3) The pressure the gas exerts on its container walls, stationary or moving, is not any sort of energy. Its unit of pressure are N m-2; those of energy are N m.

(4) When the container expands against opposing forces, for example if the gas is in a cylinder fitted with a piston which moves outwards, pushing something, then the gas does work (integral pdV). It therefore loses energy – because it's lost some of its ability to do work! This energy is almost exclusively kinetic. As the piston moves out, the gas molecules lose some of their kinetic energy, and the gas cools (assuming the cylinder is thermally insulated).

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rcgldr
Homework Helper
What is, in the case of fluids, pressure potential energy?
Is there a possiblity that you're thinking of the term in Bernoullis equation that is equal to gravitational potential energy per unit volume, which is ρ g h?

Gold Member
Philip Wood said:
(3) The pressure the gas exerts on its container walls, stationary or moving, is not any sort of energy. Its unit of pressure are N m-2; those of energy are N m

This is wrong. The units of pressure are equivalent to energy per volume. (J/m3 --> N m/m3 --> N/m2 --> Pa)

Think about it for a second. Pressure comes from collisions of gas particles, and these particles collide more often in a gas with higher internal energy. The higher the energy per unit volume, the more collisions you have and the higher the gas pressure. In other words, energy and pressure are very closely related, contrary to your statement.

Philip Wood
Gold Member
Boneh3ad. With respect, the statement of mine which you've quoted is not wrong. The units of pressure and energy are different. And the concept of pressure is quite different from that of energy. Their definitions are quite different.

I'm entirely in agreement that there is a relationship between pressure and internal energy, U, in an a gas, namely that $U = \frac{c_v}{R} pV$. But that doesn't make pressure a sort of energy, any more than temperature is a sort of potential difference just because the temperature of a lamp filament depends on the p.d. across it.

Sorry about the extreme example. I don't suppose there's that much difference in our positions. There is clearly a connection between energy and pressure in a fluid, but it's surely not right to say they mean virtually the same thing.

vanhees71
Gold Member
As usual, this question becomes better understood in (special) relativistic physics. Pressure is a kind of (negative) stress and thus it's clear that it must be part of the energy-momentum tensor. In the most simple situation, for an ideal fluid, the energy-momentum tensor reads
$$T^{\mu \nu}=(\epsilon+p) \frac{u^{\mu} u^{\nu}}{c^2}-g^{\mu \nu} p.$$
Here $\epsilon$ and $p$ are energy density and pressure as measured in the local rest frame of the fluid at the space-time point in question, and $u^{\mu}$ is the four-velocity with
$$(u^{\mu})=(c,\vec{v})/\sqrt{1-\vec{v}^2/c^2}.$$

Potential Energy comes in all shapes and sizes, and when two uneven potentials commingle, the equilibrium is a kinetic event. Man's usual method of creating energy is a manufactured Potential 1 (high pressure) exploding into an atmosphere Potential 2 (low pressure). The greater the disparity between potential 1 and 2, the greater the kinetic energy. The Big Bang was a P1 & P2, (mass/vacuum) with the equilibrium occurring when the energy of the vacuum became greater than the mass itself, (slowed down at least) and the whole Universe exploded in a whole pressure equilibrium event. The Earth is P1 and the Moon a P2. Earthquakes, Volcanoes, Hurricanes, Tornado's, Lightning, Ocean Waves, basically everything is a kinetic fulfillment of two uneven Potentials. Even the stupid wind mills that are popping up are just capturing the energy of P1/P2 weather. So in answer to the question, all energy is found in the structure of the molecule and its compressed status in relation to the Mean Free Path in which it exists. When two uneven Mean Free Paths come across each other they perform a kinetic dance of equilibrium. Like a Man and a Woman (P1/P2) potential energy can exist separately, but they can only create a new kinetic energy when they become one.

The potential energy is 'electrostatic' and it is stored in the container 'walls' in the form of stress. Without walls there can be no pressure- accept in the case of massive objects like planets with large enough gravity. The container is stressed and when the gas expands for example, the stress in the walls is reduced to compensate for that.

davenn
Gold Member
The potential energy is 'electrostatic' and it is stored in the container 'walls' in the form of stress. Without walls there can be no pressure- accept in the case of massive objects like planets with large enough gravity. The container is stressed and when the gas expands for example, the stress in the walls is reduced to compensate for that.

huh ??

Andrew Mason
Homework Helper
I'm entirely in agreement that there is a relationship between pressure and internal energy, U, in an a gas, namely that $U = \frac{c_v}{R} pV$. But that doesn't make pressure a sort of energy, any more than temperature is a sort of potential difference just because the temperature of a lamp filament depends on the p.d. across it.

Sorry about the extreme example. I don't suppose there's that much difference in our positions. There is clearly a connection between energy and pressure in a fluid, but it's surely not right to say they mean virtually the same thing.
Pressure can certainly be viewed as a form of energy density. As has been pointed out, it has units of energy density. Internal energy of an ideal gas per unit volume is directly proportional to pressure: U/V = nCvT/V = P(Cv/R)

AM

Gold Member
It might have been clear to the OP... 3.5 years ago when he/she started this thread. Why are we still posing here in this necrothread?

Jano L.
Gold Member
Pressure can certainly be viewed as a form of energy density. As has been pointed out, it has units of energy density.
AM

This holds only for ideal gas; it is a very special case. There is no general relation between pressure and internal energy density.

Sometimes people call pV "pressure energy", but that is misleading, because fluid in incompressible flow (which is very accurate model for common cases) does not change its internal energy.
There is no additional energy of fluid associated with pressure when dealing with incompressible flow. The pressure term enters purely as work, not as fluid energy. The only energy of the fluid is kinetic energy; only if different heights are compared, potential energy enters as well.

Dale
Mentor
2020 Award
It might have been clear to the OP... 3.5 years ago when he/she started this thread. Why are we still posing here in this necrothread?