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The Toothpaste Puzzle

by Brian Dunning, Aug 16 2012

Most people get the right answer to the toothpaste puzzle, but it never ceases to surprise me how many get it wrong, and even staunchly argue their position. I’ve even seen it tested by taking a tube of toothpaste to the bottom of a swimming pool.

The question: What would happen if you took a tube of toothpaste to the bottom of the sea, and then opened it? It’s a great party question. Try it sometime.

Many people answer that the toothpaste would all squirt out. Some say ocean water would rush in. The right answer, as most of you surmise, is that nothing would happen. The wrong answer I hear most often is that the toothpaste would squirt out. Why? Because there’s so much more surface area on the tube than there is at the little opening — say 1 cm2 compared to 500 — so it would squirt out with 500× power, or something along those lines. Not true.

Some people ask if toothpaste is compressible, or if there are air bubbles inside the tube. Some suspect that if the toothpaste is squeezed and compressed in there, it would squirt out more vigorously, or they say that water would rush in to fill the vacated space. Both are wrong.

Toothpaste only moves from one environment (inside the tube) to another (outside the tube) when it’s in a high-pressure environment and there is a low-pressure environment nearby. At the bottom of the sea, there is one single pressure measurement (say 5 atmospheres, for example, depending on the depth) just as there is a single pressure measurement here on land (1 atmosphere). If the tube is in a 5 atmosphere environment, there is no reason the toothpaste would squirt out in order to reach another 5 atmosphere environment. This is where the argument of relative surface area between the nozzle and the tube comes into play. But it’s nonsensical. Choose any given 1 cm2 patch on a balloon; it does not pop at that point and all the air rushing out because there are so many other patches. The pressure from those other patches does not build up cumulatively. If it did, the same argument could be made for every single point on the balloon’s surface.

When we open and squeeze a tube of toothpaste, our squeeze is creating high pressure inside the tube. The toothpaste then vacates to the lower pressure environment outside the tube. Simple as that. At the bottom of the sea, there is no convenient low pressure region. So nothing would happen.

Discuss. I’m sure you will.

48 Responses to “The Toothpaste Puzzle”

  1. Otto J. Mäkelä says:

    And next, the airplane-on-a-conveyor-belt problem.

  2. gski says:

    It is said we live at the bottom of an ocean of air. Whether air or water the toothpaste acts the same way.

  3. LovleAnjel says:

    If anything, wouldn’t the tube be crushed by the pressure before it is even opened?

    • Max says:

      If toothpaste is compressible, it’ll get compressed and stay that way after the tube is opened.

    • itzac says:

      Max is right. Whatever compressible material (small air bubbles, for example) is in the tube will be compressed, but there’s likely so little, at least in a brand new tube, that you wouldn’t even notice the difference.

  4. Noadi says:

    Having seen what happens to a styrofoam cup at that depth I think “nothing” isn’t quite what would happen. The tube would be crushed due to the pressure difference. The toothpaste would stay in the tube but the tube would never be the same.

  5. Trimegistus says:

    If toothpaste moves out of the tube because of the difference in pressure created by squeezing, does the baseline pressure make a difference in how fast it moves? Is it simple subtraction or a ratio?

    In other words, if I can squeeze a tube to generate (we’ll say) 1 atmosphere of pressure, at sea level the result is 1 atmosphere at my brush bristles, 2 atmospheres in the tube. The paste oozes out and I proceed to whiten and brighten my smile.

    But at the bottom of the Marianas Trench the pressure is 1000 atmospheres, so when I squeeze my tube the pressure inside is 1001 atm. Will my toothpaste dispense normally because there’s still a pressure difference of 1 atm? Or will it come out very slowly because instead of a 2:1 pressure ratio it’s a 1001:1000 ratio?

    Any fluidics specialists know the answer?

    • Jim Shaver says:

      Tri, I’m not claiming to be a fluid dynamics specialist, but my money is on pressure differential as the key, not pressure ratio. Think of performing the same experiment in a vacuum (or near-vacuum). If pressure ratio were to determine the speed of toothpaste evacuation, the slightest touch on the outside of the tube would fire the toothpaste out at high speed, since even a small number divided by nearly zero is huge. But it’s fairly easy to imagine that the shooting-toothpaste scenario would not happen in the vacuum chamber, and neither would it be difficult to squeeze the tube at the bottom of the ocean.

  6. Adam Slagell says:

    It would be pretty hard to open though. And depending on the material itself, there may be deformation.

    • Jim Shaver says:

      Actually, Adam, I don’t think the toothpaste would be hard to open at ocean depths (assuming you had a diving suit that allowed for reasonable finger articulation). If the tube is closed with a screw-on cap (as pictured), the screw threads would provide leverage with about a 20-to-1 advantage in converting circumferential force to axial force. Also, the tube itself is flexible, so most of the vacancy in the initially expanding volume of the cap would be filled with toothpaste anyway, as the tube collapsed ever so slightly. Then, with the seal broken, the water would rush into the cap, equalizing the pressure.

      A flip-top cap would also open easily, since the water pressure would defeat its weak seal, equalizing the pressure between the inside and outside of the cap. I’d bet my internet reputation on it.

  7. neil jackson says:

    If the toothpaste is slightly compressible & the tube is elastic (as opposed to plastic) then surely the tube would be squidged a bit. When the top comes off it could try to reform its original shape & suck a bit of water in.

    Squidged is a technical term.

    Am I right, (pedantic possibly too)

  8. Chris Howard says:

    I totally got that wrong. (this is why my degrees in a soft science) The explination makes sense, though. Awesome post. Thanks for the knowledge.

  9. Hal says:

    But will there be a toothbrush? Otherwise it will be for not.

  10. Chris Paris says:

    No. NASA has already demonstrated that pressure differentials DO affect toothpaste. This is due to a difference in pressure between that of where the toothpaste was packed and sealed, and then where it is opened. This is the same reason your contact lens cleaner squirts when you open it on an airplane: even though technically the entire airplane has (as this article assumes) a “single pressure environment”, that pressure is different than the that of where you last sealed the container.

    Apollo 16 astronauts reported that their toothpaste squirted like made because it had been packed at 14 PSI, and their capsule had an operating pressure of 5 PSI. (Which means the toothpaste was actually SUCKED out.)

    Because of the HIGH pressure below sea level, the toothpaste would be suddenly compressed backwards, meaning you may still not have anything “squirt out” (opposite of the NASA experience), but the pressure differential is still doing its thing. For one thing, the tube would be really, really hard to open, because of the lower pressure inside.

    • Hal says:

      But now with baggy or flexible containers……

      With the contact lens fluid, the container is rigid when it closed and the air inside will expand at higher altitude and lower outside air pressure and hence you have it squirt out. But if their were no air in it, its expansion would be significantly less. Look at air in a plastic water bottle as you land or take off but if there is no air, it is different.

      • Ryan says:

        Contact lens fluid also needs to be sterile. So the bottle needs to be air/water tight. Toothpaste not so much. Most are sealed with a plastic screw cap, and many might not even be particularly well closed at the back end. Take your contact fluid up in a plane and the interior is gonna be at whatever pressure it was sealed at, so you have a pressure differential. A plastic toothpaste tube with a screw on cap should be able to equalize on the way up or down.

        But in the past (like the 60’s) toothpaste came in metal tubes. Like oil paints and certain ointments now, if was sealed with with a metal skin over the nozzle you’ve got the same problem as the contact lens fluid.

    • MadScientist says:

      No, the toothpaste was pushed out by the higher pressure gas. “Sucking” is a mechanical action which produces a pressure differential (though a baby uses peristaltic action in addition to the pressure). The Apollo capsule may look like a big tit but I doubt anyone or anything had been sucking at it.

  11. oldebabe says:

    Yeh, I knew, but it’s always fun reading about something mundane and non-philosophical, and something one can get ones teeth into (oh, sorry – couldn’t help myself)…

  12. Janet Camp says:

    This is such a relief! I am going to the bottom of the sea next week and I was really worried about my toothpaste exploding.

  13. ridgididge says:

    Mmmm…I’m trying to imagine the device needed to unscrew or flip the cap open.
    I thought at the abyss the toothpaste may freeze, but, on checking, I believe that the temperature is still a degree or two above freezing.

  14. Nyar says:

    I don’t know if we have enough information to solve this puzzle. Is the toothpaste made by Koch Industries? If not, then yeah nothing much will happen. But if yes, then the high level of evilonium in it will cause Cthulhu to rise from his slumber and… well I dunno, but something bad I think.

    • Janet Camp says:

      evilonium–I love it!

      I’m already avoiding a long list of Koch Industries products–hope they aren’t doing my fav toothpaste!

  15. d brown says:

    When opened the cast tube should try and go back to its shape. The paste will do nothing unless it had bubbles in it. But what will the Deep Ones do?

  16. Ed Seedhouse says:

    Toothpaste is mostly water, with various chemicals dissolved in it and others held in stable suspension, most of that, if I recall right, being food grade diatomaceous earth. Water is an incompressible liquid, so the tube, sealed, will not be compressed because the water in it will counteract the pressure at any depth. Diamatomaceous earth, being fossils of diatoms, is porous so the water will likely permeate the pores. what ever slight amount of gas is in the tube will compress, but in a normal toothpaste tube this will be slight.

    So to a good approximation the tube will be unchanged and when you take the cap off nothing will happen until you squeeze the tube, when toothpaste will issue forth. How exposed toothpaste acts underwater I don’t know enough to say.

    It’s pressure differential that matters, not absolute pressure. Fill the tube with air and seal it and it will compress as it goes down, assuming you prevent it from floating to the surface by some method.

    • MadScientist says:

      I haven’t seen toothpaste with diatomaceous earth for decades now. Most common big brand toothpastes use finely ground calcium fluoride which can act as a soft abrasive (if the intent is indeed to provide a mild abrasive). The old toothpaste was great – the diatoms were ground up to only a few micrometers in size so you could use toothpaste to polish off scratches in plastics once upon a time.

  17. carl dibella says:

    I see a future Mythbusters episode coming to solve this.

  18. Slugsie says:

    Interesting. I was more concerned with how the toothpaste would affect the smiles of any passing sharks. But if you want to be boring and cover the physics side then fair enough.

  19. Jerome Barry says:

    A non-stupid question with a non-stupid answer is: How much would the volume of a full toothpaste tube occupy if it were opened under 5 atmospheres of pressure at the ocean’s bottom?

    And since I don’t have the answer and I’m not stupid, I’ll smoke that in my pipe for a while.

    • Carl says:

      Jerome, why such a shallow sea? Five atmospheres of pressure is only about 150m. Pressure at the bottom of the Marianas Trench is about 1000 atmospheres. At that depth, water compresses IIRC about 5 percent.

      I have no idea what the compressibility of toothpaste vs. water is.

  20. LeickR says:

    FYI, 5 atmospheres is only about 50m, or roughly 150 feet.

  21. Scott Hamilton says:

    A similar misconception is the idea that you’d blow up like a balloon if exposed to the vacuum of space (or Mars!). The pressure difference between being in an atmosphere and being in a vacuum is one atmosphere at most, and I think even our most delicate membranes can handle that.

    • Graham says:

      I totally recall that Arnie started to blow up like a balloon when he was last on Mars. His membranes weren’t handling it very well at all!

    • MadScientist says:

      One atmosphere is about 1 metric ton per square meter (if I recall correctly – if I weren’t lazy I’d do the calculations). Hickeys are produced with far smaller pressure differentials. While I wouldn’t expect someone to blow up like a balloon, I’d expect severe and extensive tissue damage due to dissolution and expansion of gases rupturing the cell membranes.

  22. Louie Hall says:

    The toothpaste would sqeezed to the top of the tube but the pressure on the lid won’t let the toothpaaste out because of the rim on the “click” lid. When and if you get it open, the toothpaste would squrt out slightly because the producers of the item in question leave a small space of air in the tip of the tube and that would allow a small bit of toothpaste. to escape

  23. Nomad says:

    The correct answer seemed so intuitively obvious that all the others responses were an entertaining surprise to me. Thanks.

  24. MadScientist says:

    If there is no air bubble in the tube then the tube will simply be somewhat crushed (depending on the compressibility of the fluid in it). If there’s an air bubble, its buoyancy may be sufficient to push the air (and some toothpaste) out of the opened tube – if you have the opening of the tube at the top end of course. Then again with the cold temperatures down there, the toothpaste will probably be too viscous for the air bubble to move at any appreciable rate.

  25. Not Completely Useless says:

    Seems odd that there’s so much confident verbiage here, in a place where skeptics are known to gather …

    Why doesn’t someone try it out? A swimming pool may be too shallow, but surely we don’t have to go to the Marianas trench to find out for sure.

  26. Richard Blaine says:

    A related story: Some year ago, Kodak produced a waterproof single use camera intended for underwater picture taking. (It had “film” in it; you kids will have to ask your parents what that was.) The camera body was sealed at the factory with the film inside. The camera had a conventional shutter mechanism on the top. The problem was that when divers took the camera below a certain depth, the difference between the pressures inside and outside the camera would depress the shutter button uncontrollably.

    Soon afterwards, the design of the camera was improved by replacing the shutter button with a small lever that stuck out of the back. In this way, there was water on both sides of the lever, and the shutter would not be affected by water pressure.

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