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Made on 3/10/20
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Music and mathematics are closely related. This was discovered by Pythagoras so it’s nothing new, and yet a majority of people don’t know something as simple as why the keys in the piano are what they are. I promise to tell you in a minimum of words things that you’d need many pages, from many sources, to get otherwise. Read More
We all know that perpetual motion of the 1st or 2nd kind is impossible, but this has not deterred inventors from coming up with new ways to attain this dream. The US Patent Office, for instance, has granted several perpetual motion patents in recent history, against its long-standing policy. Perhaps one of the reasons for this is that inventors are getting more sophisticated. Unbalanced wheels and magnets are giving way to holograms, forcing scientists to make connections between fields that would not have been made otherwise. Perpetual motion of the Third Kind, which seeks to produce infinite exergy rather than infinite energy, has made its appearance, and it seems that new laws of Thermodynamics would need to be added in order to forbid it. This paper studies not only the “science” behind perpetual motion, but also the psychological and philosophical underpinnings of a pursuit that would not go away.
A few years ago, the school where I work decided to change its Alma Mater because nobody could remember the tune, and much less the words. There was a contests and, what do you know, I won (against very little competition, truth be said). Read on to see the words and hear the music of “IIT, You Mean to Me” (or, “IIT, you’re mean to me,” as most people sing it). Read More
You think you need a computer to have a high-security code? Think again. I am about to tell you how to encrypt a message with as much security as you want, using just a common calculator. The name of this code is “root”, because the calculator’s square root key is what makes it secure.
Let’s say you have an important message you need to transmit, but your enemies have tapped your phone, fax, and email, and you suspect they have infected your computer with a worm that will relay to them everything inside. Let’s say your message is:
MEET ME AT THE TRAIN STATION
The first thing you have to do is convert each character into a two-digit number from 00 to 99, following a code you have agreed with your friends. This gives you room for all letters in the alphabet, plus numbers, plus a whole lot of special characters. Let’s say the code gives 00 for a space (though often it is better to eliminate spaces and punctuation altogether), and then the letters are numbered in alphabetical order: A = 01, B = 02, C = 03, and so on. Then the message becomes:
13050520 00 1305 00 0120 00 200805 00 2018010914 00 19200120091514
Of course, cracking this would be child’s play for just about anybody, so we must conceal it under something really difficult to guess. Here’s where the calculator comes in handy. But before you begin, you need to have two more things. The first is a secret passnumber, which you have agreed with your friends. You have memorized this passnumber and never ever written it down. If you find it hard to memorize a long number, then use a password made of text, but you must convert it to numbers, as described above, before you can use it.
Let’s say your passnumber is:
Its length is a function on how many digits your calculator can handle. It should be a little shorter than a multiple of the calculator’s precision. Here I assume you have a 12-digit calculator (many desktop calculators can do that; the more digits, the better). Your friends should have an identical calculator, or the code will break down.
The second thing you must have is a serial number for your message, which you must produce right now. The only trick is that you should never exchange two messages with the same serial number (in either direction), or the code will be exposed. The length of the serial number is equal to a multiple of the calculator’s precision, less the passnumber length. Since we use 12-digit calculators and a 30-digit passnumber, that leaves 12×3 – 30 = 6 digits for the serial number, with enough room for a million different messages.
Let’s say you pick this serial number:
Now append the serial number to the passnumber and break the whole thing into groups of 12 digits, like this:
Now comes the tricky part. You must convert these into three kinds of mathematical entities: multipliers, adders, and a seed. Since you have only three numbers, you’ll have one of each. The multipliers have half of its figures as integers and half as decimals (because of the 12-digit precision, since 12/2 = 6 integer figures; if we had a 14-digit calculator, then you would take 14/2 = 7 integer figures). The adders have two fewer integer figures, in our case, four. The seed is all decimals, and it contains the serial number as its tail. Thus, you have:
Multiplier = 123456.789012
Adder = 3456.78901234
Seed = .567890555555
Now you enter these numbers into the calculator and perform the following calculation:
Result = fract(sqrt(Seed * Multiplier + Adder))
In other words, you take the Seed, multiply it by the Multiplier, and add the Adder. Then you take the square root of that result and subtract the integer part so that only decimals remain. With the numbers of our example, this is:
|.5678905555 x 123456.789012 =||70109.944492|
|+ 3456.78901234 =||73566.7335043|
|do square root =||271.231881430|
|– integer part (that is, 271) =||0.231881430|
The last number obtained becomes the new Seed, and the process can be repeated. If there are more adders and multipliers, the next time a different set of Multiplier and Adder is used. When all the adders and multipliers have been used once, you look at the digits of the final result, which add to the encrypting “keystream”. In the example, they are:
231881430 (all zeros count, except the one on the left of the decimal point)
Then you write those numbers underneath the message (turned into numbers), and add the two together neglecting carryovers (so you can do it from left to right). If there are not enough digits in the keystream, you generate more with the calculator, starting with the last result as Seed but using the same adders and multipliers, until there are enough. Your message has 56 digits but the calculator has given you only 9, so you need to do it six more times. It is convenient to store adders and multipliers in memories (which you will erase when you’re done, right?) to save work. In this way, you get:
|+ encrypting keystream from calculator:||23188143012042259736534369536548841140602240348004490344….|
where the last sequence of numbers is obtained by adding the other two digit by digit, neglecting any carryovers (that is, 8+7 = 5, not 15)
Now you only have to send your friends this last sequence of numbers, plus the serial number (as a list of phone numbers, or stock prices, or whatnot), confident that nobody who doesn’t have the password will be able to decipher it. So, this is what you would send more or less in the open, assuming you have agreed to place the serial number at the very beginning:
Your friends, after retrieving the serial number from the message itself and the passnumber from their own brain, will be able to produce the exact same encrypting keystream from their calculator, and they will subtract it (again, neglecting any carryovers, so that 5 – 7 = 8, and so on) from the transmitted message, in this way:
|– encrypting keystream from calculator:||23188143012042259736534369536548841140602240348004490344….|
|= retrieved message:||13050520001305000120002008050020180109140019200120091514|
And now they can easily convert every two-digit group back into characters:
That’s it! Now anyone can enjoy high-strength encryption without the help of a computer.
Now, you’re probably wondering, how strong is this cipher, really? The short answer to that question is: that depends on the length of the passnumber and on how many digits your calculator can handle. The more digits in either, the better.
A long passnumber ensures that a lot of possibilities must be tried by anyone trying to crack the message by brute force. If the passnumber is thirty digits long, that’s 10^30 different numbers to try. This is comparable to using a 99-bit binary key, which is just about twice as long as the 56-bit key of DES encryption, used by the US Government until recently. A team of determined hackers cracked DES by brute force in just about 23 hours in the year 2000. Assuming that Moore’s Law holds (that computer power increases by a factor of ten every five years, for a constant inflation-adjusted cost), the same team wouldn’t be able to crack your 30-digit encription by brute force in under a day until sometime in 2065 AD. Even giving the NSA a ten-year advantage, that’s still pretty good for most of us.
If this isn’t enough for you, you can always use a longer passnumber, since every digit added increases the cracking difficulty by a factor of ten. Now, this is best done by chunks as big as can fit in the calculator. So, if you use a 12-digit calculator, the next thing to do is to use two multipliers instead of just one, and then the next to use two multipliers and two adders. The digits added to the keystream are those obtained after all the adders and multipliers have been involved. And so on, and so forth.
Additional security can be gained by using a special “message key” instead of passnumber+serial to generate adders, multipliers, and seed (just in case you make a mistake and use the same serial number twice, more on that later). In this case the process goes the same way, except that the first series of digits coming out of the calculator are used to make a new set of adders, multipliers, and seed, and then the keystream is produced starting from those. This added step forces your generating numbers to be completely different from message to message (not just the few last digits of the seed), and thus your master passnumber is never exposed in any way. I recommend using this precaution, which does not add that much extra work anyway (because you’re going to burn the paper containing your calculations, right?).
It is important to make sure that the serial number is never reused (by either side of the transmission), because then the same encrypting keystream would be used for two different messages. An enemy could then eliminate it by subtracting one message from the other, leaving a superposition of two unencrypted messages that would be quite easy to crack by letter frequency methods. And on top of that the keystream would be revealed, from which it might be conceivably possible to extract the numbers used to generate it. This is a bad feature of all “stream ciphers,” like this one, but they also have good features that other ciphers don’t have. For instance, in most ciphers any mistake garbles the end result completely, but in stream ciphers only the characters encrypted after the mistake end up garbled, which makes it a lot easier to spot the error and correct it. Handy when it’s just you punching keys.
And how about the calculator? If it uses too few digits (many pocket calculators display only eight), then the pseudo-random properties of the keystream suffer, and the message is exposed to statistical analysis. The sequence will also start repeating after a while, which is bad. In my experience, eight digits is no good, ten is barely acceptable, and twelve and above work fine. The more digits, the better. I have a Palm-based calculator that can operate with thirty-eight digits at a time, so it gives me better security than the current AES governement standard (256-bit) without even using an adder.
You have to thank irrational numbers for this cipher. The digits of irrational numbers obtained from a square root, like the square root of two (1.41439799432119332181380109536190378…) never repeat, and appear random to anyone who doesn’t know where they came from. It is this apparent randomness that conceals your message in noise, rendering it impossible to read.
Perpetual Motion Machines are those that would produce free, endless power, thus ending all of humanity’s energy problems (and maybe some political ones at the same time). I am not going to embarrass myself showing pictures of my early perpetual motion machines on this page, since you can find some very similar ones (guess which ones they are) going to this excellent website. What this article is about is a kind of perpetual motion machine that so far I haven”t been able to prove how it doesn’t work. Maybe you will…
What is Perpetual Motion?
Perpetual motion refers to a device that produces free power forever. This does not apply to things like the motion of the earth or the moon, because those would eventually stop if we extracted power from them (in fact, the moon always shows the same face toward earth precisely because, when it was young and soft, the energy of its rotation was “extracted” into tides that eventually died down). The motion of the heavenly bodies is actually inertia, not “perpetual motion.”
Many people have tried to invent (or, more likely, pretend to have invented) devices that produce free energy. Such devices, or “perpetual motion machines,” can be classified into two broad categories:
- First Kind: devices that create power out of nothing.
- Second Kind: devices that extract power out of the environment.
The first and second laws of Thermodynamics derive from the impossibility (so far) to build these devices. The first law tells us that energy must be conserved, and thus it is impossible to make a device that will generate power without consuming some other kind of energy. The second law states that, once energy has gone into the environment, usually converted in to heat, we cannot get it back as useful power: energy is indeed conserved, but it can be degraded.
Throughout history, attempts at perpetual motion of the first kind have usually centered around imbalanced levers, where a clever mechanism shortens one of the sides of a see-saw, for instance, as it rolls around to the opposite position from where it started, or on magnets. I can claim the dubious honor of having invented (or rather, re-invented) machines of both types, before I knew any better (or even after ;-).
Putative perpetual motion of the 2nd kind is a bit more sophisticated and hard to detect. The giveaway usually is the lack of a heat sink in machines that absorb heat from the environment. The same website has a number of pretty interesting machines of this type, so I will only add here those you cannot find there. I happen to know personally the inventor of a few of them, which have obtained (fairly recent) patents from the US Patent Office, despite their long-standing policy. This is either a witness to the lack of training of the examiners or to the insistence of my friend the inventor. Maybe to both. Here are links to those patents:
But I want to tell you about different types of perpetual motion. Here I will tell you about how light can be used to seemingly create perpetual motion of the 2nd kind. A new type of perpetual motion, which I would call “of the 3rd kind,” and which does not seem to be violating any physical law so far in force, is explained in this page.
Perpetual motion from light
Electromagnetic radiation is a strange beast as far as the Second Law is concerned, because it behaves either as heat or as work, depending on the circumstances. It is heat for a hamburger warming up in a microwave oven, but it would be work, capable of moving electrons in an ordered, spark-generating way, for a fork inadvertently left inside. Light is electromagnetic radiation, and because of this it sometimes behaves as heat, as in a black body glowing due to its temperature, and sometimes gets diffracted and aligned like the millimeter waves that link your cell phone to the world.
An interesting result of diffraction is holography. A hologram is a two-dimensional image, usually on film or a similar, fine-grained substrate, that actually contains a three-dimensional image. It is generated by shining a laser on the 3D object to be recorded while at the same time shining an identical laser on the film. The film will be exposed with a series of very fine lines, which look kind of like what you get when you put a semitransparent fabric on top of another, such that the image will be reconstructed by shining yet another identical laser (or a non-laser light of the same color) on it. Holograms have a lot of interesting properties in addition to looking really cool, and one of them is that you can record multiple holograms on the same film and they won’t interfere with one another so long as the laser shining directly on the film during recording (called the reference beam) changes in color or position from shot to shot.
This property is used in US patents 5,877,874 and 6,274,860 by Rosenberg (assigned to Terrasun, Inc.), to create a film that directs sunlight into a narrow beam no matter what direction it comes from. Thus, a solar panel having this film in front of it would not need to turn to track the sun, which is quite handy. This is the intended application of the film, which allegedly has been tested in a number of prototypes. The patent descriptions can be found here and here. Below is a picture from the first patent, which illustrates the concept:
All right, then. Imagine we have a piece of this film, made so that all the light (and infrared radiation) falling on one side of it will be transmitted through to the other side, where it will leave in the direction perpendicular to the surface no matter what direction it came from on the other side. This film would be the heart of the device in the picture below:
In addition to the film, the device includes a large flat black body 1, a small black body 2, and a parabolic mirror, which has black body 2 centered on its focal point. The side surface between the large black body and the film is also polished to mirror finish. All surfaces are thermally insulated on the outside.
For those whose heat transfer courses are a bit rusty, a black body is a type of surface (actually, an ideal surface, but we can get pretty close in reality) that absorbs all the radiation that falls onto it. That’s why it’s called “black,” because it absorbs all light that falls on it and thus our eyes register its location as black, but it is also “black” for thermal radiation as well, which is mostly in the infrared range and our eyes cannot see it. When hot, a black body will emit radiation at a rate given by this formula:
where A is the surface area, s is the Stefan-Boltzmann constant (5.67×10-8 Watt/m2K4), and T is the black body absolute temperature, in Kelvins (temperature in centigrade degrees + 273.15).
Now we do the following: heat up black body 1 to a (high) temperature T1, and wait until thermal equilibrium with black body 2 is established. Thermal equilibrium means that no heat flows from 1 to 2, or from 2 to 1. The 2nd law of Thermodynamics (through a corollary mistakenly called by many “the zeroth law”) commands that this shall only occur when the temperatures of 1 and 2 are the same. But let’s see if that is the case here.
Observe that, given the geometry and the presence of the holographic film, all radiation issuing from 1 will hit the film, where it will be made parallel and transmitted to the other side. Now, a parabolic mirror has the interesting property that all beams parallel to its axis are reflected toward its focus. This means that all the radiation will end up on black body 2, where it will be absorbed. No radiation will bounce back to the surface of 1 (this is important). The heat rate going from 1 to 2, therefore, is given by:
But black body 2 also emits radiation. Some of it will bounce on the mirror and then back to the surface of 2, but most of it will miss it after the reflection and will travel toward the film after that. A lot of the radiation will travel toward the film directly. This means that eventually most of it will end up on the film, where it will either be steered toward the perpendicular direction or will be somehow diffused as it travels to the other side. In either case all of that radiation will end up on surface 1, as given by:
where the factor f, representing the fraction of the heat radiated by black body 2 that ends up falling on black body 1, is less than, but close to 1. When thermal equilibrium is reached, both heat flows balance one another, so that:
and then, it follows that:
Since A2 is smaller than A1 and f is less than 1, the temperature of black body 2 is greater than the temperature of black body 1 when equilibrium is reached. This makes it a perpetual motion machine of the 2nd kind, because now we can run a heat cycle using black body 2 as a heat source and black body 1 as a heat sink, and thus generate some power.
Now let’s discuss some things that might possibly go wrong, and why they still wouldn’t keep the machine from breaking the law.
1. It could be said that there is no such thing as a perfect black body, and neither is there a perfect mirror, or a perfect insulator, or a perfectly transparent film. True, but the 2nd law is supposed to apply even if they existed, for it is based on infinitely slow, ideal processes, which are the limit of real processes. It’s just not fair to a candidate perpetual motion machine to force it to deal with non-ideal materials. You can stop anything, including machines that don’t violate any laws, by heaping friction, losses, and leaks onto them.
2. There is a similar putative perpetual motion machine based on elliptical mirrors with black bodies in their foci. You can find it here, and here, (courtesy of Prof. L.H. Palmer, from Simon Fraser University). This is what it looks like:
The mirror geometry supposedly conspires to put more energy on black body 2 than on the other because some of the light coming out of black body 2 will be reflected back onto itself but not so for black body 1, but this is only in appearance. In reality, since the bodies are not mathematical points some of the energy coming out of one will fail to strike the other (this happens for both bodies), so that eventually the device will be filled with a uniform radiation intensity, and therefore both bodies will end up at the same temperature. The problem here is that the same argument does not work for the device we’re dealing with, since all the radiation coming out of black body 1 is guaranteed to fall on 2, even if it is not a mathematical point (and even better if it’s not), and any radiation that, coming out of black body 2, does not strike 1 but falls back onto 2, only makes the temperature difference even greater (through the effect of f < 1).
3. It could also be that the holographic film cannot steer the incoming radiation into exactly parallel beams, but rather into beams deviating from the perpendicular by up to an angle q, so that, after reflection on the parabolic mirror, they will miss black body 2, leading to a uniform radiation field as in the device mentioned in the paragraph above. If q is sufficiently small, however, the rays reflected on the parabolic mirror will concentrate within a sphere of radius e around the focal point, where e is a continuous function of q with e(0) = 0. If follows that one only has to choose a spherical black body 2 with a radius larger than e, for all the radiation from black body 1 to fall onto it, which brings us back to the original situation. For a sufficiently small q, T2 will still be greater than T1.
4. Another argument is that, since the film patents only show films working by reflection, then the machine above will not work because it works by transmission of the light through the film. This is only an apparent problem. First, every reflection hologram has a conjugate transmission hologram, the difference being whether the reference beam used in recording the hologram is placed on the same side of the film as the object to be holographed (for transmission holograms), or on the other side (for reflection holograms). It follows that, even if the film patents do not specifically speak about transmission holograms (and they should), it should be possible to make them by putting the reference beam on the same side as the object beam during recording. In addition, it is possible indeed to make a similar machine as the one above using a hologram that reflects light into a single direction, no matter what direction it originally comes from, as in the next figure:
As in the transmission case, all the light issuing from the large black body 1 will fall onto the small black body 2, and the difference in surface area will ensure that T2 > T1. But this case is even more extreme than the transmission case because, if the film reflects into a single direction all the light that strikes it, then it does it both for the radiation going from 1 to 2 and for the radiation initially going from 2 to 1, which will end up falling back onto black body 2. This means that thermal equilibrium will never be reached, since no energy flows from 2 to 1, and temperature T2 can reach arbitrarily high values.
A simpler version of the above device would have no parabolic concentrator, like this, using films that reflect light at a 45 degree angle:
Because of the geometry, light from the right side never reaches the left side, so the black body on the right will keep receiving energy by radiation while losing none. This setup acts as a Maxwell demon for photons, rather than gas molecules.
5. One could say that, in practice, there is no such thing as a holographic film that steers all incident light into a single direction. The actual prototypes made by Terrasun (the inventors of the patents mentioned above), which I have seen in action, collect light from just a couple directions and send it in a broad beam in a different direction. It is doubtful whether a perfect or near-perfect light-steering film can be made in practice, since this would involve recording many holograms into a single film, and the superimposed holograms, although theoretically able to coexist, would end up interfering with one another so they could not work. So the next question is: how well do the light-steering holograms have to work in practice so they can still be the basis of a perpetual motion machine? The last machine above violates the 2nd law by impeding backward transmission of light, without any concentration, while the first does it by concentration, without needing one-way transmission at all. If both effects are combined, as in the machine in the middle, it may be possible to still achieve a violation with imperfect holographic films.
For instance, a film that does not steer light at all, but simply prevents it from being reflected into a certain region can still form the basis of a one-way light valve, as in the picture below:
Here the film would not reflect light into a certain angle off the surface. By shaping the film into a logarithmic spiral, it is possible to ensure that none of the light that enters the device from the left (or from the right), would come out of the left port, even if the light would be otherwise dispersed. If the “blind angle” for no reflection does not come all the way to the surface, the device would be more complicated, but I don’t believe it would be impossible to come up with an example.
Likewise, the concentration effect does not need to be perfect in order to work. Even the concentration due to a change of index of refraction can do the trick if the change is strong enough, as in this device:
Here there is no holographic film at all, but partial concentration is achieved by changing the index of refraction of the transmission medium, which we assume to be non-participating in the radiation. After crossing the interface, the radiation coming from black body 1 will be partially aligned along the axis of the parabolic mirror within the total reflection angle of the interface, given by:
where n1 and n2 are the indices of refraction of the transparent media in contact with black bodies 1 and 2, respectively. This angle is measured from the perpendicular direction so that, the greater the ratio of the refraction indices, the more aligned toward the perpendicular (which also happens to be the axis of the parabolic mirror) will the light be. As we saw before, the light does not need to be fully aligned with the axis of the parabolic mirror in order to strike a surface smaller than the emission surface, leading to a temperature imbalance.
On top of all that, holograms, interfaces, and other components in the device do not need to work equally well with all the wavelengths in the spectrum, as indeed they won’t. To see this it is enough to imagine that the blackbody surfaces are coated with a filter film or paint that only allows a certain wavelength to pass through. This narrow-pass film is certainly ideal, but there are existing films that approximate this behavior quite well. In that case, all the radiation involved in the exchange will be of a single wavelength, but everything we have discussed above would still hold and there would still be a heat transfer imbalance and a temperature difference would be created.
I would probably get kicked out of my job as a professor and defender of the 2nd Law of Thermodynamics if I said that these machines could work, so I won’t say that. But I will say this: they’re stumping me so far and sometimes manage to take away my sleep (well, part of it, really :-), so please let me know if you come up with a more convincing argument why they cannot work (other than saying that they seem to violate the 2nd Law of Thermodynamics).
If, on the other hand, you decide to build one and succeed (they seem pretty simple, but I’m not that handy, and forget about writing a proposal to get funding for that), I’d like to hear from you. My email is elsewhere on this website. You will be assured strict confidentiality, because I know what you’re likely to get if your machine actually works and Uncle Sam finds out about it (hint: definitely not a patent, how about a nice vacation at an exclusive resort in southern Cuba?). Should they find me, the only line that’ll cross my lips will be:
Eppure si muove!
What do you know? It seems there is a new kind of Perpetual Motion Machine, which would give free energy for ever, and Nature hasn’t managed to pass a law against it (just yet, at any rate). Read on if you’re strong-hearted (warning: contents under heavy math)…
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:
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!