Sunday, May 1, 2011

Sorting out Inconsistencies

This quote has stuck with me since I read it.
"I don’t trust myself to be an effective inquiry-based teacher if I’m not living an inquiry-based life. I don’t trust either of us." Dan Meyer
On that theme, I have recently been sharing a series of posts about my own inquiry into light and water, which began with a simple juxtaposition of two photos. This inquiry has led me to seek out and further juxtapose a range of natural and contrived phenomena. It has also led me to try to sort through various pieces of explanation, which have arisen through contemplation and conversation with peers.

In this post, I want to share another piece of inquiry that I think is important and important to infuse into one's inquiry life: seeking coherence and spotting inconsistency. This puzzle has stuck with me as fun and engaging.

The puzzle:

If we (a) approximate a shallow chunk of water (near earth) as being incompressible
and (b) we assume that piece of water is in thermal equilibrium
and (c) that piece of water is in a constant gravitational field

then (d) macroscopically, we can show that the pressure increases linearly with depth.

but (e) microscopically, the particle density and kinetic energy would appear to be the same be, via our starting assumptions (a) and (b);

so (f) how can the pressure be different, if the particles have the same kinetic energy and particle density?

You have a couple of options here to approach trying to sort this out, and I encourage you to explore this a bit before you settle on a quick answer. Any takers?


  1. I'm perplexed. Pressure is a macroscopic property that comes from the collisions of microscopic molecules with a container. You can increase the pressure by either increasing the frequency of collisions, or the momentum transfer of individual collisions. You could increase the frequency by either increasing the average velocity of the molecules (greater kinetic energy and temperature), increasing the density (more collisions). You could increase the momentum transfer by increasing the kinetic energy of the molecules, which could come from increasing either the velocity or the mass of the molecules. However, none of these seem to vary given the initial constraints, so I don't see how this is possible. Very puzzling.

  2. It's a good sign that you are perplexed. I was in and out of perplexity with this puzzle for a couple years.

    I feel like this kind of thing gets a bit at what I mean by building "globally coherent explanations".

  3. Incompressibility implies there are strong intermolecular forces. The increase in pressure comes from these forces. All you need is a very large spatial gradient for the intermolecular potential. Then you get both (near) incompressibility and a increase in pressure with depth.

  4. @Chris Yeah, the intermolecular forces are strong in some ways, but not in others. The water is easily parted, for example. Or put another way, they are easy to pull apart, but not easy to force closer together.

    One piece of inconsistency that you seem to be noting is with the notion of incompressibility. If it were PERFECTLY incompressible, it would seem impossible to have increased pressure with depth. So in your resolution of the puzzle, it seems like you would throw out assumption (a). If we throw out assumption (a), then it really does get denser, even if just a little bit.

    But what's still odd is that the macroscopic derivation of pressure changing with depth doesn't assume approximate incompressibility (it assumes complete incompressibility), yet microscopically it would seem that compressibility is required.

    There are other places to poke holes at the argument,as well, besides assumption (a)

  5. The exact same thing comes up with rigid bodies and normal forces. How does a rigid body exert a normal force? The answer is that it compresses slightly, and the normal force is the reaction force that results from this compression.

    Actually, now that I think about it, it IS the exact same thing, since the derivation you refer to analyzes the forces on a small volume of the fluid in static equilibrium.

  6. Yeah, this gets at more of puzzle indeed. With a solid, it's easy for us to think of the force as arising from only (or mostly the) "potential" energy of springs being compressed. With a gas, on the other hand, it's easier for us to think of the pressure as arising from momentum exchange during collisions of particles (flying around with kinetic) energy. In other words, with a solid to get more upward force you need to compress the springs to get more potential energy. With the gas, to get more pressure, you either need to pack in more molecules OR increase the temperature thereby increasing the kinetic energy of the particles.

    So liquids fall in between...?

  7. I would say that under the conditions for which the derivation is valid and the result holds (incompressible, static, confined) that the fluid is identical to the solid.

  8. Ok, I'm still a bit perplexed by what this would look like on a molecular level.Imagine a cylinder with a flexible covering (like a drum). Clearly, as you drag the drum to deeper depths, the drum would flex more and more inward. So what would this look like if you were molecularly sized and standing on the surface of the drum? How would you be able to tell that the pressure is greater?

    How would the rate of water molecules hitting the drum and the impulse of those particles change as you go down? If we're saying that the molecules compress slightly more with depth, then the frequency of collisions would be higher, and this would account for the increased pressure right?

    I still don't quite see how this jibes with the idea that the density of the liquid is constant and it is incompressible.

    I get the normal force comparison, but in that case, we don't say the table is incompressible, right? We actually require it to compress elastically just the slightest bit in order to exert the normal force to support the object resting on it. So I think I'm missing a bit when connecting to the liquid example.

  9. John, I like the way you phrase, "How would you be able to tell if you were molecularly sized?" and I agree that thinking about an object and the buoyant force it experiences is helpful for clarifying the puzzle.

    Because, I think the goal isn't to resolve the issue by avoiding the issue. The puzzle resides in figuring out what a liquid is doing differently to exert or less more pressure at the microscopic level and how that connects to issues of temperature and energy.

    I do think it's important to recognize that many of our instincts are formed around these two canonical examples of "an ideal gas" and "static solid". For me, neither of these models fully help to explain the puzzle. In fact, they are the reason for the puzzle.

  10. Hmm, but it seems to me that the pressure can only be a function of the molecules colliding with the drum head (or whatever is submerged). I don't really see how the nature of these collisions chances depending on whether you are dealing with a gas, liquid, or solid. If anything, my guess would be that the stronger inter-molecular forces in liquids and gases would tend to retard the motion of molecules impacting the drum head, and reducing the pressure overall.

    Also, if I think just of the ideal gas case, I think you can say that it is the compressibility (or the density) of the gas that is responsible for the increased pressure at lower altitudes (assuming constant temperature). Molecularly, you would see may more collisions (with the same individual impulse) at lower altitude than you would at higher altitudes.

  11. The pressure in an incompressible liquid isn't generated by the collision of molecules like in a gas, it's generated by the intermolecular forces. I think this puzzle results from applying the wrong mental model to an incompressible liquid. Why do you think that pressure is generated the same way in an incompressible liquid as it is in an ideal gas?

    Here's a riddle for you that might help: What's the mean free path of a molecule in an incompressible liquid?

  12. Down the rabbit hole we go:

    Here's what I think. Whether a gas, liquid, or solid, we should be able to think of the "forces" exerted on the surface of an object nearby as being the net result of molecules getting close enough to the surface to exchange appreciable amounts of momentum. In the ideal gas situation, we model the individual momentum exchanges as being zero until that molecule makes contact with the surface. Thus we think of it as a "collision". In this case, either increasing the momentum of particles or the density or particles, increases the net force on the surface.

    In the solid situation, like with a book on the table, we might think of many molecules oscillating on springs, each bumping into the book's molecules, so that the net momentum exchange keeps the book up. In this case, if we heat up the table (within reason) or use a denser table, the "average" net force on the book surface isn't going to change. So it's quite different from the gas.

    Chris, I do agree that a big part of the puzzle results from appropriating some ideas from the ideal gas that shouldn't be. Other problems result from a misappropriation of the idea that temperature is measure of average kinetic energy. While I think it's to safe to say "the pressure in a liquid isn't generated exactly like in a gas", it's can't be entirely different either because everything is generated by intermolecular forces (not just liquid). The extent to which they are similar and different, and how these mechanisms differently lead to the emergent properties of pressure and temperature, is the crux of the matter.

  13. I agree, but I don't think momentum plays exactly the role you seem to say here. Consider a block attached to a spring. If I push on the block and compress the spring, there will be a force on my hand even when both the block and my hand are at rest. So pressure can be generated by forces even in the absence of motion and without any exchange in momentum.

    I think that this has got to be the dominant mechanism in the original situation: the change in pressure with depth in an incompressible fluid.

  14. I feel like I am channeling Andy Rundquist here, but I think it's important to distinguish "a zero net exchange of momentum" from no momentum exchanges. There is momentum being exchanged with particles, it's just in equilibrium with other exchanges. Even at a macroscopic level, we can say the momentum exchange with the spring (via normal force) is the balanced by momentum exchange with the earth (gravity).

  15. Now we really are down the rathole, but in the block/hand example I gave above, where everything is static, I don't see how there is any momentum exchange at all, net or otherwise. But maybe I don't understand what you mean by "momentum exchange."

  16. Ok, now that I've been channeled, I thought I'd jump in. A lot of this reminds me of the conservation of the king when you throw a ball at a wall. We need to say that the wall takes some momentum or else we can't explain the bounce. However the wall's motion is so small we can ignore it. Isn't that like John's point about the table needing to move just a little? -Andy

  17. Ok, if I'm thinking about this on a molecular level, all of the molecules in either a solid, liquid or gas are in constant motion. The macroscopic pressure is a property that arises from the collisions of molecules with the surface of the drum, or whatever is submerged.

    If I'm understanding Chris correctly, the intermolecular forces must somehow add to this pressure. But I don't quite see how, since it can't either increase the frequency of collision or the momentum transfer by individual collisions.

    But, could it be that the these forces, which I'm envisioning as tiny springs, are adding momentum to the molecules before they hit the surface, so that total momentum transfer is greater? This would seem to be a bit plausible, since the molecules are probably oscillating around an equilibrium and when you insert an object into the liquid, you'll disturb this equilibrium ever so slightly and compress the spacing between molecules which will cause that the intermolecular forces to push outward in all directions adding outward momentum to all of the molecules, and conveniently conserving the king.

    Does this make any sense?

  18. @Chris. We can think about force as rates of momentum flow. In static situations, it comes off a bit awkward. So instead of balanced forces, you have balanced rates of momentum swapping.

    @John, Another difference is, with the gas, it's OK to think of each molecules all by itself when it collides. But with liquid and solid, when it collides it isn't isolated, because it's coupled to surrounding molecules that also tug on it. I think this changes the dynamics of the collision.

    I'm recognizing the need for a whiteboard or a piece paper, and the limited ability to talk this through just with words.

  19. @Brian, I think it makes sense to think of net force (on an individual object or system) as rate of momentum flow (that is after all just Newton's law), but I don't think it makes sense to think of individual forces as corresponding to momentum flow. In my example of the block trapped between my hand and the wall, that would lead to the idea that momentum is flowing from the wall into the block and from my hand into the block, right? Ugh :-).

    @John, I think your conception of pressure is a little backward. Pressure is just force per area. The question is where does that force come from? It can come from collisions as in a gas, or it can come from (what I'll call for the moment) static forces, like weight. In a gas, the force on the walls due to collisions with individual atoms is the dominant contribution to the pressure, but that's not going to be true in other situations, like in solids or Brian's original example here.

  20. @Chris—I totally get that pressure is force/area. And I understand how to do these problems macroscopically as well as the derivation that if we assume constant density and gravitational field, pressure increases with depth, but what I'm trying to get a handle on is how this would look different at a microscopic/atomic level. For instance, if you were molecular sized and standing on the surface of a some object, how would you be able to tell the difference between being immersed in a gas or a liquid? I suppose since the density difference is of the order of 10^3, you'd be seeing a tremendously greater number of collisions per unit of time—is that the primary difference? How would you be able to tell the difference between being supported by a buoyant force due to a liquid, or a normal force due to a solid, again, assuming you are molecular sized and looking only at molecules in motion striking a very small part of the surface.

    I also agree that it's cumbersome to think of static object like a book exchanging momentum with objects in the surroundings at the same rate in opposite directions, but I do think this picture is valid, and it might be thinking about my previous question.

    @Brian I'm seeing that the liquid and solid are more difficult because those molecules are affected by their interactions with all the other molecules in the liquid solid. Couldn't you simply model that net interaction between the any particular atom and the surroundings as just some net force of the surrounding molecules? Or even better, as a net momentum transfer between the surrounding molecules in the liquid/solid and the one about to impact the surface? Of course, this would be fantastically complicated in practice, but I guess I'm trying to get to the idea of why this would lead to some microscopically observable phenomenon that would indicate that the pressure of the liquid is increasing with depth.

    I'm so intrigued by this that I might try to put together some illustrations on my own blog—but that might take me a bit of time to do.

  21. @John, I think for these purposes you should model a liquid as molecules connected by springs that are extremely stiff on compression (infinitely stiff for an incompressible fluid) but very fragile on expansion.

    As for a "microscopically observable phenomenon that would indicate that the pressure of the liquid is increasing with depth," for an incompressible fluid I think that there is none (I can elaborate if this doesn't make sense). For an almost-incompressible fluid you would see a slight increase in density as the signal for increasing pressure.

  22. @John. I think some illustrations are certainly in order, and maybe even some Vpython programming. If I find the time, I might try to program something.

    All and all, making correspondences between micro and macro are harder than we often acknowledge. I have a few similar puzzles, which are also intriguing and fun-- maybe I'll post about them later.

    @John. I think this is a bit of the paradox: mechanistically, compression is required in order for pressure to propagate through the liquid.

  23. @Brian—can this paradox be resolved? When we make the incompressible assumption in order to show that pressure varies linearly with depth, are we in essence just assuming a very high "spring constant" between the molecules?

  24. @John So, what I was thinking today was about the difference between constraint-based reasoning and mechanistic reasoning.

    I think it is commitment to the idea of energy conservation and/or static equilibrium that constrains our answer to necessarily be "the pressure must vary with depth with there is gravity"... but that constraint doesn't explain how or why the liquid comes to exert more pressure. It just say that it must. Once we pursue explaining why, we realize that the pressure can only vary if there is something different going on with the molecules, and we are inclined to believe that this implies at least some compression.

    A lot of this boils down to what ideas you are committed to...and how strong your commitments to those ideas are. But I'm still not "sure"