The Rubber band engine, Part I

Rubber bands possess interesting thermal behaviour. Tugging hard at one of these causes an easily discernible warming- which, inasmuch as the reader has tried this before, is likely to find reasonable. (If intuition fails you on this count, you could try it now.) What is considerably harder to swallow, though, is that the stretched band cools as it snaps back when you let go. (Try it!)

Things get stranger when you rub your palms with an extended band in between- it tightens. To make this more perceptible, you could hang a weight at the end of a band, and hold a candle flame to it- the Feynman way. Assuming you manage to prevent the band from burning away (and snapping), it would shrivel up, pulling the weight!

And thus begins the next verse in this itinerary of improbable engines, and in the quest to make sense of the second law.

1. A weight is attached to the (stretched) rubber band, and it is exposed to a source of heat at a temperature infinitesimally higher than the band’s.2. As a result, the band shrivels up, raising the weight in the process. We slip the weight off the rubber band, and move it away, sidewards, so we can neglect the work involved.

3. The band then cools off at the same temperature, releasing heat to the source and stretching back to the original (not extended) length.

(Steps 2 and 3 are akin to the freezing and melting steps in the afore-linked ice heat engine.)

4. We attach another weight to it, and it stretches back to state 1, completing the cycle. The outcome is that we put in what we took from the source in the form of heat (at the same temperature), and raised a weight while at it.

Too good to be true, of course.

As it is with all clever text arguments, this contraption appears to function akin to the ice heat engine detailed earlier. The rubber band engine is shamelessly ripped from the Feynman Lectures on Physics- but let us venture further and see what comes out.

The loophole in the reasoning? Unlike before, the argument here is visibly specious- and if you haven’t figured it out yet, read further. (If you have, read further anyway for a calculation of rubber characteristics.)

The heat engine doesn’t work this way, of course. The rubber does not, cannot, contract/extend by the same amount upon the addition/removal of heat- the second law imposes restrictions on the behaviour of all matter- rubber bands (and ice/water) included. It follows that increasing the tension in the rubber band (by adding the weight) must increase the temperature. This is, in fact, the warming effect that you feel when someone stretches a band when in contact with the back of your palm.

To drive the point home, we see that if we run the rubber band engine reversibly*, like we did above, we can actually calculate how the tension in the rubber band varies with the heat added. The actual cycle functions as follows:

1. The extended band (with a weight attached) is heated by a source at an infinitesimally higher temperature. It contracts, raising the weight in the process.2. The weight is removed, slowly, bit by bit, and the (now insulated) rubber band cools in this process**.

3. The band rejects heat at the lower temperature- again over an infinitesimal temperature difference. (To ensure that the temperature remains constant, we must tug at the band)

4. We reattach a(-nother) weight to the band, (slowly, bit by bit, etc!), and watch as it heats up to the original temperature. The cycle is now complete, and the net result is that a weight was lifted, and some heat was absorbed at the high temperature, and some was rejected at the low temperature. A Carnot cycle executed to perfection.

The net work done in the cycle is easy to calculate- since everything at the end of the cycle is the same as that at the beginning- save for the raised weight, the work done is weight times the distance by which it was lifted. Well, not quite. Remember that we had to tug at the band when it was losing heat. So, if F_1 is the weight, and F_2 (< F_1) (why?) is the force we had to apply when tugging at it, and if we neglect the change \delta (\Delta L) in the change in length in the heat gain and loss processes, we obtain:

\Delta Q \frac{\Delta T}{T} = (F_1 - F_2) \Delta L = (\Delta F) (\Delta L)

where (\Delta Q) is the heat added, \Delta T is the difference between the hot and cold temperatures, and T is the hotter temperature. The expression, is of course, that for the efficiency of a Carnot cycle.

This innocuous expression is more powerful than it seems. Imagine, for instance, rubbing your palms together vigorously a few times, with the rubber band in between- you put in roughly a Kilojoule of heat- and you measure the decrease in length and the increase in tension- both are quite easy to do. You now know the temperature rise!

\Delta T = \frac{\Delta Q}{T \Delta L \Delta F}

The exclamation mark is warranted. Realize that we know absolutely nothing about the rubber band- we don’t know its elastic properties, its crystal (or amorphous) structure; we don’t even know the composition- it could be a steel wire, and every piece of reasoning in this article, every calculation, would still hold!

The ability to make predictions on the behaviour of matter with no clue of the stuff that’s inside makes thermodynamics an indispensable tool in the hands of engineers and scientists. (It also makes for some very confusing hypothetical cycles and nightmares for patent offices, but that’s a story for another day.)

In Part II, we take a deeper look at the rubber band engine, and question the assumption of constant temperature heat transfer- and obtain precise workable relations for rubber- all without having to worry about what the cogs driving the rubber look like.

*Reversibility?

**We do this so the tension in the band always equals the weight, with no heat transfer- this keeps things smooth and quasi-equilibrium. If the weight is removed bit by bit, the work done in this process is a complicated function of, among other things, how the weight was removed. As long as we add the weight in the exact same manner in step 4, the effect cancels out if the temperature difference and the displacements are small. (The load-displacement curve for rubber can be linearlized).

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30 Responses to “The Rubber band engine, Part I”

  1. anasua Says:

    “The exclamation mark is warranted.”
    isn’t it? the second law specially always boggled my mind a bit.
    And carnot was brilliant.
    where’s part two?

  2. lizzie Says:

    this is so confusing? if i wanna do this for my science fair project, what variables can i change?

  3. squidgy Says:

    Hello lizzie. This article does not explain how to make a rubber band engine- it explains the principle on which it works.

    If you want to make a working rubber band engine for your project, I suggest you check out working prototypes of such an engine here and here.

    On the other hand, if you want to explain how it works, look up the chapter on Thermodynamics in “The Feynman Lectures on Physics, Volume I” at your local library. If you want to estimate the efficiency of a rubber band engine, you can use the derivations given in this article.

  4. tanambi Says:

    im doing this for a science fair project, but the formula is confusing… i need to find a formula that explains/shows the number of rotations per minute depending on how much heat i applied to the engine. i don’t know how i would explain the formula… this is so hard! :(

  5. tusti Says:

    >:(
    :’(
    :D

    :I

  6. Bob Says:

    Why did you set up a strawman argument by stating that the band is exposed to a source of heat at a temperature only “infinitesimally” higher ?

    No one would expect one to make such an argument and it seems only necessary to make a point that is not central to making a rubber band heat engine which is certainly possible. You are arguing against a perpetual motion machine which was not the topic.

  7. squidgy Says:

    Bob, I quote from the article above:

    “…we can actually calculate how the tension in the rubber band varies with the heat added…”
    The assumption of an infinitesimally higher temperature is not a straw-man, it is central to this calculation, which assumes reversible heat transfer.

    At no point did I say rubber band engines only work with infinitesimal temperature differences, nor is this article about making one. (See my reply to lizzie above). The topic of this article, as set up in the second paragraph, is:

    “And thus begins the next verse in this itinerary of improbable engines, and in the quest to make sense of the second law.”

    So the issue of a perpetual motion machine is quite germane.

  8. Bob Says:

    Thanks. I was Googling how to make one and came across your page. I think I misunderstood what you were attempting to expain. Sorry.

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