Perpetual Motion Machines of the 3rd Kind

This article is highly unusual because it deals with a perpetual motion machine that the laws of Thermodynamics forgot to forbid. Perpetual motion machines, in their most common definition, are those that produce a high-grade kind of energy, such as electricity or mechanical work, for free. Naturally this cannot be or the world would go topsy-turvy: fortunes would be reduced to poverty, governments would collapse, and the earth would start warming up for good. But thankfully, Thermodynamics has laws to prevent such mayhem.

The First Law forbids anything from yielding more energy than is put in, effectively nullifying machines made out of magnets, unbalanced levers, and self-regenerating motors. The Second Law, on top of that, sets a strict limit on the efficiency of the heat to work conversion, so those who thought to solve all the world’s problems by extracting plentiful power from the environment—where it eventually returns only to be recycled back to use—were forced to put their considerable creativity to better use. A perpetual motion machine is fool’s gold, our textbooks say, meant for the doom of those who did not bother to stay awake during Thermo class. Of course, the flaws of such machines are hard to see sometimes. This one, for instance, which is based on light.

Another perpetual machine (called by some of the “third kind” to distinguish it from those of first and second kind, loosely described above), has been proposed. The trick here is to get a mass (let’s say it’s an ice-cream bar) down to absolute zero. Then one could go to the nearest hardware store, buy a perfect Carnot cycle, and stick it on top of the absolute-zero ice-cream bar as shown in the figure below

The Second Law gives, for the efficiency of such a machine, this expression, where the temperature must be expressed in Kelvins of any other absolute scale:

Where TH and TL are the high and low temperatures of the cycle, respectively. If the low temperature in the cycle is absolute zero, then the efficiency of the set is exactly one, meaning that all the heat extracted to the environment will be converted into usable power. An additional benefit is that the heat rejected to the ice-cream bar will be exactly zero, so our precious property will never see its temperature raised above absolute zero. We can keep our machine running forever (remember, it’s a perfect Carnot cycle), creating useful power out of the environment. This, of course, is forbidden by the Second Law, so this machine is actually a special type of perpetual motion machine of the second kind, and giving it its separate kind is not really warranted.

Is there, then, such a thing as a true perpetual motion machine of the third kind?

Enter exergy. This very useful concept “discounts” energy to give its potential to produce useful work. For instance, the exergy of heat of value Q is less than Q, because not all of it can be converted into work, according to the Second Law, but rather:

where T0 is the temperature of the environment, which is usually the heat sink in common thermodynamic systems. Everything that contains energy contains also exergy, including things that contain no energy at all. For instance, an evacuated tank contains no material, and therefore no energy, but it can be used to generate power by causing the environment to push a piston or putting a paddle wheel in front of the onrushing air, should the tank be punctured. The important thing is not whether the power comes from the system or not, but rather that the system is the opportunity by which the system itself or the environment will be able to produce power.

The exergy of substances can be calculated in many ways, and it is usually related to its thermodynamic state and that of the environment, as given by its temperature, T0, pressure, p0, and other properties. The particular case of interest for our perpetual motion machine of the third kind is the exergy of a substance whose constituent molecules are able to evaporate into the environment. If that substance, say, behaves like an ideal gas (and every substance will, once its vapor pressure becomes sufficiently small), its exergy is given by the following expression (assuming its specific heat Cp is constant, for simplicity):

Where y and y0 are its mole fractions within the system and in the environment, respectively, and p and p0 are the pressures. The upshot of this is that the above formula, which is derived in strict compliance with thefirst and second laws of Thermodynamics, would give an infinite exergy whenever y0 is zero, that is, whenever a substance is completely absent from the environment. Creating a substance that is completely absent in the environment is not such a far-fetched concept, though. Pharmaceutical companies are doing it all the time when they synthesize new medicines. Physicists do it routinely, at a subatomic level, when they collide particles traveling at a high speed to create new particles. It takes more or less energy to form a new substance, but it is always a finite amount. The amount of work required is, at a minimum, equal to its “chemical exergy,” which is obtained when the compound is allowed to react producing work (say, in a fuel cell), or maybe absorbing work, down to compounds that are present in the environment, and those are then allowed to diffuse into this environment, contributing more work through terms of the same form as equation (3), but which are now finite because none of the y0 concentrations in the environment is zero. Of course, equation (3) is not supposed to be applied when a substance is completely absent from the environment, but rather one must first calculate how much exergy is required to generate the substance by chemical reaction, starting from substances that are present, and then add the exergy that those substances would have before they expand into the environment, as explained above. But then the paradox remains that a substance that required no infinite exergy to synthesize, would appear to have an infinite capacity to do work if it is simply allowed to expand into the environment.

For an example of how a true perpetual motion machine of the third kind would work, look at the figure below:

The “synthesizer” is a black box system where a certain new substance (let’s call it “novium”) is being synthesized, starting from substances present in the environment. This process, as we saw above and know from experience, takes a finite amount of energy, consisting of work and heat. The novium formed in the synthesizer now travels to an expansion chamber maintained at the same temperature as the environment, where it is vaporized and meets a membrane that is permeable to all the substances present in the environment, but not to novium. The membrane, therefore, will be subject to the vapor pressure of novium on one side, and no force on the other. If it is allowed to move, the membrane will produce a work as the novium gas expands at constant temperature, absorbing thermal energy from the environment. The process can move at a vanishingly small rate, approaching equilibrium at all times. The process is also reversible, since it is always possible to push the membrane against the novium vapor pressure until it is concentrated into a small volume. Under these conditions, and having removed friction and other irreversibilities, the expansion chamber will produce a work per unit mass equal to its exergy, given in the equation above. Since y0 of novium (in the environment) is zero, then the work produced, given an infinite stroke for the membrane displacement, will also be infinite. Another way to look at it is that the novium undergoes a constant temperature process. Since it is an ideal gas at low concentrations, the work produced will be given by:

leading to a logarithmic relationship between work and volume. Eventually, for a sufficiently large volume (it does not have to be infinite, in fact), the expansion chamber will have produced enough work to generate the required novium sample, and then some. It should be noted that equations (3) and (4), far from breaking down as the expansion progresses, would be less and less of an idealization, for all substances approach ideal gas behavior as their vapor pressure tends to zero.

The energy, of course, comes from the environment, mainly through heat interactions in the synthesizer and the expansion chamber. But the environment is at a constant temperature, so it should not be possible to produce any work by extracting heat from it, according to the Second Law. And yet, an analysis of the equations above says that this result comes directly from that law, since the logarithmic term that gives the infinite result can also be derived from:

where

is the entropy difference of an ideal gas (with constant specific heats) between a given state and the environmental state. Here the pressure used is the partial pressure of the gas, in case there are other gases mixed with it, which is the usual situation.

What has happened here? How did the Second Law manage to produce a result that seems to contradict the Law itself?

It should be noted that the fact that nobody has made, nor likely ever be able to make, such machine is no argument against the paradox. Likewise, nobody has been able to build something as simple as a Carnot cycle, because there are always irreversibilities such as friction and heat transfer across finite temperature gaps, and yet the science of Thermodynamics is based on it. No, the relevant fact here is that the machine proposed above, in its ideal form, seems a self-contradiction of the Second Law, which this law’s propriety cannot tolerate even in its most ideal form.

The reason why the machine is not a perpetual motion machine violating either the first or the second laws is because it does not really work in cycles. Indeed, after the first novium sample has expanded, producing as much work as we cared to collect, it is necessary to bring it back to its initial state. A valve opens on its far wall, and the membrane is allowed to move back under no pressure differential, venting the novium into the environment. But that means that, next time we try to expand a sample of novium gas, it will no longer be totally absent from the environment, and thus no infinite work will be possible.

Yes, but the same could be said about the Carnot cycle, which absorbs heat from a constant temperature thermal source without lowering its temperature, and rejects heat to a thermal sink without raising its temperature, either. The mental construct used is that those two “thermal reservoirs” are infinite for these purposes, and so a bit more or less heat does not change their temperature. It would not be fair not to give a similar capacity to absorb novium to the environment surrounding the machine, so that the concentration of novium in the environment will not be altered because a few (or a million) strokefuls are dumped in.

In addition, nothing prevents the machine operator from changing the composition of novium for the next stroke (along with that of the membrane). This is something that takes a finite amount of work, so that the next cycle works very much the same as the first, as far as the machine is concerned. There is no fear to run out of different substances to make, so the machine would keep working indefinitely. This is, however, strictly not the same process if the composition of novium has changed. We will revisit this aspect later on.

But perhaps such machine is intrinsically impossible, because no membrane will ever be able to distinguish novium from the other gases in contact with it, and thus it will stop performing as it should. The job we are asking the membrane to perform is indeed quite delicate, and not much different of the job performed by our good old friend, Maxwell’s demon.

Maxwell’s demon, pictured below, is supposed to stand guard by a little trapdoor, and allow only fast molecules to go from left to right, and slow molecules to go from right to left, with the result that a temperature difference is soon created, against the Second Law. But Maxwell’s demon cannot perform its job unless he analyzes the speed of the approaching molecules, and he creates more entropy doing so than he destroys by classifying the molecules into fast and slow. The question is, does a membrane suffer from a similar limitation? How does a membrane know novium from any other substance?

The answer is not simple, for semi-permeable membranes operate in many different ways. The membrane that is around every one of our bodies’ cells, for instance, has receptors of many kinds on its surface, and certain molecules can latch onto it by means of hydrogen bonds, provided their geometry matches that of the receptors. Scientists have been able to do a similar thing with DNA fragments on a silicon chip, using restriction enzymes that would only bind to specific sequences. If novium is DNA-based, then a substrate coated with a restriction enzyme for its particular sequence would be able to stop it as it tries to get past, while it would not stop any other molecule. The novium bound to the enzyme would eventually reach an equilibrium (controlled by the Second Law) with the free novium, so that as many molecules are released back into the expansion chamber as are captured on its surface. The result would be a barrier for novium, and the perpetual motion machine of the third kind would be able to run.

But it gets worse: making a membrane that would stop novium but would not stop anything else could be as simple as making the novium molecule larger than any other molecule present in the environment. A common wall with holes big enough for those, but not for novium, would do the trick. This is the laziest kind of Maxwell demon. A Maxwell demon who does not need to use any energy to classify the incoming molecules and therefore generates no entropy to operate. This case is different from the spring-operated Maxwell demon on the right side of the above figure, which allows through only those molecules fast enough to open the door against the spring. This may not be too far-fetched: it has become known recently that nanomaterials behave anomalously where the Second Law is concerned, probably due to their microstructure. But even the microscopic sorting door falls under the curse of the second Law. That tiny spring, indeed, would end up taking some of the energy of the incoming molecules, with the result that the door would end up shaking so badly that soon it would not be able to classify the molecules at all. But a membrane with simple holes would not take any more energy than a wall without holes. The wall material can be perfectly rigid, and still it would fulfill its mission to restrain novium safely to one side of it. Molecules too large to pass through would bounce off elastically, while those that pass need not lose any energy as they do so.

Still, our instinct tells us that there must be a reason why this machine cannot work, or the world might go upside down. Consider this: the gas in the expansion chamber does not need to be completely absent from the environment for the machine to produce power; it only needs to be rare enough so that producing it takes less power than it gives when it expands, according to eq. (4). An inventor might speculate on what gases are easy to make but are rare outside, but let us just look at carbon dioxide, for instance. Pure CO2 can be generated by a number of processes well known to freshman students, (such as dripping vinegar on marble), none of which take much energy at all. Yet, the mole fraction of CO2 in earth’s atmosphere is only 0.0003, yielding, from eq. (3), 456.6 kJ per kg of pure CO2 in the expansion chamber, if the temperature is the standard 25ºC. Once liberated, the CO2 is captured back into rocks by biotic processes, so ultimately the power produced by the machine has come from the sun. Is this, therefore, a perpetual motion machine, or isn’t it?

But perhaps this example—though possibly the basis for a power-producing method—only serves to muddy the issue, which is whether the laws of Thermodynamics can allow the infinite exergy case. Perhaps the problem is that the law of Thermodynamics that prevents this perpetual motion machine from working has not yet been promulgated, and therefore the machine would keep happily working, oblivious to any fault or misdemeanor.

The Third Law exists already, so the new law (if God decides to pass it) would have to be called the “Fourth Law” at best. It might read like this:

“It is impossible for the exergy of any system to be infinite.”

Or, more specifically:

“It is impossible for the concentration of any substance to be zero.”

In its second form, the Fourth Law smells suspiciously like Schrodinger’s cat, which is both dead and alive at the same time. In our case, the undead “novium” has acquired the ability to tunnel, ghost catlike, through any wall, so that it is on both sides of it at the same time. There was always a certain amount of tunneling, controlled by Heisenberg’s principle, but now we are imposing a minimum value: there must be at least enough tunneling so that this cat’s exergy drops below the power it takes to make it.

But maybe God will be happy with this loophole: the “novium” in our machine won’t the same in every stroke, and therefore the machine isn’t strictly running in cycles. In that case, he might let us keep running it to produce infinite power as long as we don’t fill this universe with our creations (such as the different kinds of novium). And then, we’ll finally be able to say (in a rather subdued voice, just in case):

Eppure si muove!