How can fast moving particles gain energy from slow moving ones?

Imagine a large diameter piston filled with water connected to a small funnel. When you press on the piston slowly but with considerable force the water will move very quickly from the funnel in form of a jet. But how is it possible on a molecular level?

Water molecules are constantly moving about in the piston with various speeds and directions bumping into each other and exchanging momentum like billiard balls, however water molecules from the funnel are moving uniformly at great speed.

I want to know how it is possible for slow molecules to be adding momentum to the ones that are already moving faster than the average. In billiard ball analogy slow moving ball moving in the same direction would never catch up with the faster one to further increase its momentum and if it was moving in the opposite direction then it could only receive momentum from the faster one and therefore only slow it down.

Now I imagine that this question probably sounds silly but I can't find any answer after searching for it, so I decided to ask here.

• What would happen if I applied so much force to the piston (let's assume a very durable one) that water would be going out already at the speed of sound in water (1.4 km/s) and then doubled that force? – Ardath Sep 13 '15 at 17:27
• Speed of sound is only concerned with the velocity relative to the bulk movement of the fluid. In other words, since the water is already going at 1.4 km/s, the extra pressure will simply make it move faster - and after the wave propagates through, the liquid will again have a single bulk speed, faster than the speed of sound (all this assuming nothing slows the flow down, of course). The key here is that no matter the force you apply, as long as it's applied instantly, you're going to have supersonic flow. If you spread it over time instead, you simply boost the bulk flow. – Luaan Sep 14 '15 at 9:27
• Just a comment. Imagine that the funnel is blocked and you press the piston. If the piston is filled with LIQUID (this is really important because liquids are mainly incompressible fluids), the pressure inside will rise, but the billiard balls will not move faster (the temperature of the fluid is not rising). So, in this particular case, you can't think the water molecules only as bouncing balls. It just does not works this way. As others had pointed, molecular forces play a very important role. – algolejos Sep 14 '15 at 22:25
• Though this is on a slightly different topic, I'm very surprised nobody mentioned inverse-compton scattering upon reading the title. eud.gsfc.nasa.gov/Volker.Beckmann/school/download/… – Arturo don Juan Sep 15 '15 at 2:54
• @ArturodonJuan: My first thought when reading the title was about Fermi acceleration, but reading the body neither mine nor yours is what OP wants (sadly, because Fermi acceleration was a good portion of my dissertation). – Kyle Kanos Sep 30 '15 at 11:58

Adjacent molecules in a liquid all repel each other because of the electron clouds that surround the nuclei that they contain. In that sense these molecules never even 'touch' each other (at least not in the intuitive sense of the word).

When you apply pressure to the liquid you're squeezing them into a (very slightly) smaller volume, thereby increasing the repulsive forces between them. Now allow an outlet (your funnel or the hole in the milk carton of the previous answer) and these increased repulsive forces now propel molecules through the outlet in a macroscopic flow. The higher the pressure, the more the volume is decreased (and thus inter-molecular distances are reduced), the more the repulsive forces are increased and the higher the macroscopic flow rate through the outlet.

• -1 Although this is a nice answer, it does not ANSWER the core question of the absolute limit act which you can accelerate a fluid using a Convergent Nozzle. – Aron Sep 14 '15 at 13:04
• @Aron: there is nothing in the question about "the absolute limit act", whatever that means. The question is about how flow originates at the molecular level. – Gert Sep 14 '15 at 13:55

The question isn't silly. The speed of each molecule in the liquid is much higher than the speed of either the piston or the water shooting out from the nozzle. At room temperature, for water molecules the average is on the order of 500m/s. And yet, the speed of sound in water is three times higher than that, which implies that pressure can propagate in water at that speed. If you tried to make a jet of water molecules faster than the speed of sound, then you would run into difficulties with the nozzle method, though. You would get a shock front, strong heating, possibly ionization... all loss mechanisms that would restrict the velocity that you are really after. The better way would be to make a piston that shoots out a cylinder/piston that shoots out water. That's a two stage rocket... to squirt water.

Picking up on Aaron's comment: he is absolutely correct that one can find nozzle shapes which convert a hot, high pressure gas into a clean supersonic flow. That is the major requirement to building efficient rocket engines. The engine nozzle in that case is a thermodynamic machine which converts the internal thermal energy of the gas into a directed flow that is many times faster than the speed of sound in the gas.

• +1 This is the right answer. There is a hard limit on how fast you can make that jet of water, which is closely related to the speed of sound, which in turn is closely related to the speed at which the molecules are randomly moving. – Level River St Sep 13 '15 at 19:07
• Actually pressure can propagate faster than the speed of the molecules - because molecules have finite size and are close together, if you put three of them in a row and move the one on the left, the one on the right will feel the motion very quickly - with a time determined by the slope of the potential curve and the mass of the molecule, not the mean speed of the molecules. – Floris Sep 13 '15 at 21:42
• @Floris: I didn't say that the propagation was limited to the speed of individual molecules. It's not. The speed of sound in water is about three times higher. Once you leave this thermodynamic regime, though, things go crazy and shock waves are not fun stuff... – CuriousOne Sep 13 '15 at 21:48
• Sorry I misread... My up vote stands... – Floris Sep 13 '15 at 21:53
• @Floris: Feel free to edit the answer, you know I trust you. As you know it started out as a comment and it could use some polishing, for sure. – CuriousOne Sep 13 '15 at 22:06

Let's simplify things down to the barest minimum: one dimension, one particle, and a wall.

         O      |


The particle moves to the right, hits the wall, and rebounds, perfectly elastically. If the wall is fixed in place, the particle will leave the collision with exactly the same kinetic energy as it came in with.

But what if the wall is moving to the left at the time of the collision? Do you see that the particle must gain kinetic energy in the collision, even if the wall is moving much slower than the particle?

(A real-world scenario where this actually happens: striking a tennis ball with a racquet.)

Your misconception is that the water particles are moving very slowly or are stationary because they are not escaping. In fact, they are moving very quickly and are constantly bouncing off each other and the walls of the container. The pressure is basically how many collisions occur over a given time period. As you squeeze the piston, you are increasing the pressure, meaning that there is more energy inside.

The water molecules in the stream are also moving at different speeds relative to each other and are bouncing off each other. They are only moving at high speed relative to the average speed of the water in the container. If the water molecules were all moving at the same speed you would have ice at absolute zero. Even in ice just a few degrees below freezing the molecules are moving quickly and bouncing off each other, just not fast enough to break the weak bonds which bind them into a solid structure.

So, the individual molecules of water in the stream are mostly moving much faster than the stream itself and are bouncing off each other (sometimes in the opposite direction of the stream); it is only the average speed you can see with your naked eye, which is the speed of the stream.

Try picking a tiny hole in a milk cardboard container. Squeeze the container. You only press is slightly, but milk is poured fast out of the tiny hole.

This system is simply redistributing the total force that you provide onto a much smaller area. Same force on a smaller area equals larger pressure:

$$p=\frac{F}{A}$$

The pressure on the particles at the opening is much greater and so those particles are accelerating much more and gains a much higher speed.

• Yes, I agree with what you said, but you haven't answered my question. What I want to know is how it all happens on the molecular scale. How slow moving molecules give momentum to faster ones. – Ardath Sep 13 '15 at 17:22

The problem of your conclusion lies in a flawed premise, which I think is a very common one. None of the other answers have addressed this.

Kinetic Theory of Matter does not explain by itself all macroscopic behavior.

This means that although we can understand much of the properties of macroscopic matter from the point of view of Kinetic Theory of Matter KTM (mainly gas like systems, made of weakly interacting bodies) we cannot make up all properties of stronger interacting systems, like liquids.

For example the fact that a liquid does not spread over a larger volume, like a gas, can be qualitatively understood by saying that the molecules are more bound to one another. Still in a KTM framework potentials between molecules are not different in the gaseous and liquid states of the macroscopic system, so we cannot explain how from molecules being closer, the macroscopic system behaves so drastically different.

There is another misconception related to the above one, or at least appearing as commonly, which is the relation between movement of microscopic components and movement of the macroscopic system. They are not at all related, and there are many examples, but a simple one can convey the idea: the speed of sound, that is mechanical vibrations of the macroscopic system where relative macroscopic parts move w.r.t to each other, smaller in gasses than in solids, yet average speeds of molecules in the former are drastically large. This is another phenomenon the KTM cannot explain by the way.

So no slower molecules give momentum to faster ones in your example, they all have the same speed distribution both those in the slower part of the liquid and in the faster part in the tunnel.

The description of this behavior is contained in Pascal's Law of redistribution of pressure, and contained in the more complete Navier-Stoke's Equation.

As for the explanation of how this connects to microscopic behavior, which is what you search for as it seems to me, well you are in the right path to find it, keep asking questions.

• You have a very simplistic understanding of kinetic theory - one in which molecules are apparently point masses which exert no force on other molecules except at the point where they impact (which they ought not to be able to do, being just points). To broadly state that that KTM is the KTM, and is woefully flawed, is doing a disservice to your readers. – Floris Sep 13 '15 at 20:44
• I do think KTM is simple, I also think there lies its power. Let me give my view on what KTM is, which I have not given and which is not exactly what you stated. KTM does see particles as point masses which move and interact, certainly with a potential which can be more complex than just contact. In which way is this view of KTM incorrect? – rmhleo Sep 13 '15 at 21:17
• By having a repulsive component of their potential, molecules in KTM do not behave like a point mass any more - and that gives rise to things like "speed of sound greater than speed of molecules" which you claim it cannot do. – Floris Sep 13 '15 at 21:21
• I also do not say KTM is flawed, but maybe I was not clear in the answer. What is flawed is to think that when a liquid moves faster its molecules move faster as well. This is not completely true: the average speed w.r.t. the center of mass is the same and the velocity distribution does not change, hence internally the liquid is the same. I mean as long as the system is in thermodynamic equilibrium. – rmhleo Sep 13 '15 at 21:26
• Quoting from your answer: "Still in a KTM framework potentials between molecules are not different in the gaseous and liquid states of the macroscopic system, so we cannot explain how ...", and "This is another phenomenon the KTM cannot explain by the way." - makes it sound like you say it is far less capable than it is. That was my point. – Floris Sep 13 '15 at 21:29

protected by Qmechanic♦Sep 13 '15 at 21:07

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