# magnetic separation of air

As some of you will know, oxygen is paramagnetic, attracted slightly by a magnet. Oxygen’s paramagnetism is due to the two unpaired electrons in every O2 molecule. Oxygen has a triple-bond structure as discussed here (much of the chemistry you were taught is wrong). Virtually every other common gas is diamagnetic, repelled by a magnet. These include nitrogen, water, CO2, and argon — all diamagnetic. As a result, you can do a reasonable job of extracting oxygen from air by the use of a magnet. This is awfully cool, and could make for a good science fair project, if anyone is of a mind.

But first some math, or physics, if you like. To a good approximation the magnetization of a material, M = CH/T where M is magnetization, H is magnetic field strength, C is the Curie constant for the material, and T is absolute temperature.

Ignoring for now, the difference between entropy and internal energy, but thinking only in terms of work derived by lowering a magnet towards a volume of gas, we can say that the work extracted, and thus the decrease in energy of the magnetic gas is ∫∫HdM  = MH/2. At constant temperature and pressure, we can say ∆G = -CH2/2T.

The maximum magnetization you’re likely to get with any permanent magnet (not achieved to date) is about 50 Tesla, or 40,000 ampere meters. At 20°C, the per-mol, magnetic susceptibility of oxygen is 1.34×10−6  This suggests that the Curie constant is 1.34 ×10−6 x 293 = 3.93 ×10−4. Applying this value to oxygen in a 50 Tesla magnet at 20°C, we find the energy difference, ∆G is 1072 J/mole = RT ln ß where ß is a concentration ratio factor between the O2 content of the magnetized and un-magnetized gas, C1/C2 =ß

At room temperature, 298K ß = 1.6, and thus we find that the maximum oxygen concentration you’re likely to get is about 1.6 x 21% = 33%. It’s slightly more than this due to nitrogen’s diamagnetism, but this effect is too small the matter. What does matter is that 33% O2 is a good amount for a variety of medical uses.

I show below my simple design for a magnetic O2 concentrator. The dotted line is a permeable membrane of no selectivity – with a little O2 permeability the design will work better. All you need is a blower or pump. A coffee filter could serve as a membrane.

This design is as simple as the standard membrane-based O2 concentrator – those based on semi-permeable membranes, but this design should require less pressure differential — just enough to overcome the magnet. Less pressure means the blower should be smaller, and less noisy, with less energy use.  I figure this could be really convenient for people who need portable oxygen. With current magnets it would take 4-5 stages or low temperatures to reach this concentration, still this design could have commercial use, I’d think.

On the theoretical end, an interesting thing I find concerns the effect on the entropy of the magnetic oxygen. (Please ignore this paragraph if you have not learned statistical thermodynamics.) While you might imagine that magnetization decreases entropy, other-things being equal because the molecules are somewhat aligned with the field, temperature and pressure being fixed, I’ve come to realize that entropy is likely higher. A sea of semi-aligned molecules will have a slightly higher heat capacity than nonaligned molecules because the vibrational Cp is higher, other things being equal. Thus, unless I’m wrong, the temperature of the gas will be slightly lower in the magnetic area than in the non-magnetic field area. Temperature and pressure are not the same within the separator as out, by the way; the blower is something of a compressor, though a much less-energy intense one than used for most air separators. Because of the blower, both the magnetic and the non magnetic air will be slightly warmer than in the surround (blower Work = ∆T/Cp). This heat will be mostly lost when the gas leaves the system, that is when it flows to lower pressure, both gas streams will be, essentially at room temperature. Again, this is not the case with the classic membrane-based oxygen concentrators — there the nitrogen-rich stream is notably warm.

Robert E. Buxbaum, October 11, 2017. I find thermodynamics wonderful, both as science and as an analog for society.

# How Tesla invented, I think, Tesla coils and wireless chargers.

I think I know how Tesla invented his high frequency devices, and thought I’d show you, while also explaining the operation of some devices that develop from in. Even if I’m wrong in historical terms, at least you should come to understand some of his devices, and something of the invention process. Either can be the start of a great science fair project.

The start of Tesla’s invention process, I think, was a visual similarity– I’m guessing he noticed that the physics symbol for a spring was the same as for an electrical, induction coil, as shown at left. A normal person would notice the similarity, and perhaps think about it for a few seconds, get no where, and think of something else. If he or she had a math background — necessary to do most any science — they might look at the relevant equations and notice that they’re different. The equation describing the force of a spring is F = -k x  (I’ll define these letters in the bottom paragraph). The equation describing the voltage in an induction coil is not very similar-looking at first glance, V = L di/dt.  But there is a key similarity that could appeal to some math aficionados: both equations are linear. A linear equation is one where, if you double one side you double the other. Thus, if you double F, you double x, and if you double V, you double dI/dt, and that’s a significant behavior; the equation z= atis not linear, see the difference?

Another linear equation is the key equation for the motion for a mass, Newton’s second law, F = ma = m d2x/dt2. This equation is quite complicated looking, since the latter term is a second-derivative, but it is linear, and a mass is the likely thing for a spring to act upon. Yet another linear equation can be used to relate current to the voltage across a capacitor: V= -1/C ∫idt. At first glance, this equation looks quite different from the others since it involves an integral. But Nicola Tesla did more than a first glance. Perhaps he knew that linear systems tend to show resonance — vibrations at a fixed frequency. Or perhaps that insight came later.

And Tesla saw something else, I imagine, something even less obvious, except in hindsight. If you take the derivative of the two electrical equations, you get dV/dt = L d2i/dt2, and dV/dt = -1/C i . These equations are the same as for the spring and mass, just replace F and x by dV/dt and i. That the derivative of the integral is the thing itself is something I demonstrate here. At this point it becomes clear that a capacitor-coil system will show the same sort of natural resonance effects as shown by a spring and mass system, or by a child’s swing, or by a bouncy bridge. Tesla would have known, like anyone who’s taken college-level physics, that a small input at the right, resonant frequency will excite such systems to great swings. For a mass and spring,

Basic Tesla coil. A switch set off by magnetization of the iron core insures resonant frequency operation.

resonant frequency = (1/2π) √k/m,

Children can make a swing go quite high, just by pumping at the right frequency. Similarly, it should be possible to excite a coil-capacitor system to higher and higher voltages if you can find a way to excite long enough at the right frequency. Tesla would have looked for a way to do this with a coil capacitor system, and after a while of trying and thinking, he seems to have found the circuit shown at right, with a spark gap to impress visitors and keep the voltages from getting to far out of hand. The resonant frequency for this system is 1/(2π√LC), an equation form that is similar to the above. The voltage swings should grow until limited by resistance in the wires, or by the radiation of power into space. The fact that significant power is radiated into space will be used as the basis for wireless phone chargers, but more on that later. For now, you might wish to note that power radiation is proportional to dV/dt.

A more -modern version of the above excited by AC current. In this version, you achieve resonance by adjusting the coil, capacitor and resistance to match the forcing frequency.

The device above provides an early, simple way to excite a coil -capacitor system. It’s designed for use with a battery or other DC power source. There’s an electromagnetic switch to provide resonance with any capacitor and coil pair. An alternative, more modern device is shown at left. It  achieves resonance too without the switch through the use of input AC power, but you have to match the AC frequency to the resonant frequency of the coil and capacitor. If wall current is used, 60 cps, the coil and capacitor must be chosen so that  1/(2π√LC) = 60 cps. Both versions are called Tesla coils and either can be set up to produce very large sparks (sparks make for a great science fair project — you need to put a spark gap across the capacitor, or better yet use the coil as the low-voltage part of a transformer.

Another use of this circuit is as a transmitter of power into space. The coil becomes the transmission antenna, and you have to set up a similar device as a receiver, see picture at right. The black thing at left of the picture is the capacitor. One has to make sure that the coil-capacitor pair is tuned to the same frequency as the transmitter. One also needs to add a rectifier, the rectifier chosen here is designated 1N4007. This, fairly standard-size rectifier allows you to sip DC power to the battery, without fear that the battery will discharge on every cycle. That’s all the science you need to charge an iPhone without having to plug it in. Designing one of these is a good science fair project, especially if you can improve on the charging distance. Why should you have to put your iPhone right on top of the transmitter battery. Why not allow continuous charging anywhere in your home. Tesla was working on long-distance power transmission till the end of his life. What modifications would that require?

Symbols used above: a = acceleration = d2x/dt2, C= capacitance of the capacitor, dV/dt = the rate of change of voltage with time, F = force, i = current, k = stiffness of the spring, L= inductance of the coil, m = mass of the weight, t= time, V= voltage, x = distance of the mass from its rest point.

Robert Buxbaum, October 2, 2017.

# Future airplane catapults may not be electric

President Trump got into Hot Water with the Navy this week for his suggestion that they should go “back to god-damn steam” for their airplane catapults as a cure for cost over-runs and delays with the Navy’s aircraft carriers. The Navy had chosen to go to a more modern catapult called EMALS (electromagnetic, aircraft launch system) based on a traveling coil and electromagnetic pulses. This EMAL system has cost \$5 Billion in cost over-runs, has added 3 years to the program, and still doesn’t work well. In response to the president’s suggestion (explosion), the Navy did what the rest of Washington has done: blame Trump’s ignorance, e.g. here, in the Navy Times. Still, for what it’s worth, I think Trump’s idea has merit, especially if I can modify it a bit to suggest high pressure air (pneumatics) instead of high pressure steam.

Tests of the navy EMALS, notice that some launches go further than others; the problem is electronics, supposedly.

If you want to launch a 50,000 lb jet fighter at 5 g acceleration, you need to apply 250,000 lbs of force uniformly throughout the launch. For pneumatics, all that takes is 250 psi steam or air, and a 1000 square inch piston, about 3 feet in diameter. This is a very modest pressure and a quite modest size piston. A 50,000 lb object accelerated this way, will reach launch speed (130 mph) in 1.2 seconds. It’s very hard to get such fast or uniform acceleration with an electromagnetic coil since the motion of the coil always produces a back voltage. The electromagnetic pulses can be adjusted to counter this, but it’s not all that easy, as the Navy tests show. You have to know the speed and position of the airplane precisely to get it right, and have to adjust the firing of the pushing coils accordingly. There is no guarantee of smooth acceleration like you get with a piston, and the EMALS control circuit will always be vulnerable to electromagnetic and cyber attack. As things stand, the control system is thought to be the problem.

A piston is invulnerable to EM and cyber attack since, if worse comes to worse, the valves can be operated manually, as was done with steam-catapults throughout WWII. And pistons are very robust — far more robust than solenoid coils — because they are far less complex. As much force as you put on the plane, has to be put on the coil or piston. Thus, for 5 g acceleration, the coil or piston has to experience 250,000 lbs of horizontal force. That’s 3 million Newtons for those who like SI units (here’s a joke about SI units). A solid piston will have no problem withstanding 250,000 lbs for years. Piston steamships from the 50s are still in operation. Coils are far more delicate, and the life-span is likely to be short, at least for current designs.

The reason I suggest compressed air, pneumatics, instead of steam is that air is not as hot and corrosive as steam. Also an air compressor can be located close to the flight deck, connected to the power center by electric wires. Steam requires long runs of steam pipes, a more difficult proposition. As a possible design, one could use a multi-stage, inter-cooled air compressor connected to a ballast tank, perhaps 5 feet in diameter x 100 feet long to guarantee uniform pressure. The ballast tank would provide the uniform pressure while allowing the use of a relatively small compressor, drawing less power than the EMALS. Those who’ve had freshman physics will be able to show that 5 g acceleration will get the plane to 130 mph in only 125 feet of runway. This is far less runway than the EMALS requires. For lighter planes or greater efficiency, one could shut off the input air before 120 feet and allow the remainder of the air to expand for 200 feet of the piston.

The same pistons could be used for capturing an airplane. It could start at 250 psi, dead-ended to the cylinder top. The captured airplane would push air back into the ballast tank, or the valve could be closed allowing pressure to build. Operated that way, the cylinder could stop the plane in 60 feet. You can’t do that with an EMAL. I should also mention that the efficiency of the piston catapult can be near 100%, but the efficiency of the EMALS will be near zero at the beginning of acceleration. Low efficiency at low speed is a problem found in all electromagnetic actuators: lots of electromagnetic power is needed to get things moving, but the output work,  ∫F dx, is near zero at low velocity. With EM, efficiency is high at only at one speed determined by the size of the moving coil; with pistons it’s high at all speeds. I suggest the Navy keep their EMALS, but only as a secondary system, perhaps used to launch drones until they get sea experience and demonstrate a real advantage over pneumatics.

Robert Buxbaum, May 19, 2017. The USS Princeton was the fanciest ship in the US fleet, with super high-tech cannons. When they mis-fired, it killed most of the cabinet of President Tyler. Slow and steady wins the arms race.

# just water over the dam

Some months ago, I posted an engineering challenge: figure out the water rate over an non-standard V-weir dam during a high flow period (a storm) based on measurements made on the weir during a low flow period. My solution follows. Weir dams of this sort are erected mostly to prevent flooding. They provide secondary benefits, though by improving the water and providing a way to measure the flow.

A series of compound, rectangular weir dams in Maine.

The problem was stated as follows: You’ve got a classic V weir on a dam, but it is not a knife-edge weir, nor is it rectangular or compound as in the picture at right. Instead it is nearly 90°, not very tall, and both the dam and weir have rounded leads. Because the weir is of non-standard shape, thick and rounded, you can not use the flow equation found in standard tables or on the internet. Instead, you decide to use a bucket and stopwatch to determine the flow during a relatively dry period. You measure 0.8 gal/sec when the water height is 3″ in the weir. During the rain-storm some days later, you measure that there are 12″ of water in the weir. Give a good estimate of the storm-flow based on the information you have.

A V-notch weir, side view and end-on.

I also gave a hint, that the flow in a V weir is self-similar. That is, though you may not know what the pattern will be, you can expect it will be stretched the same for all heights.

The upshot of this hint is that, there is one, fairly constant flow coefficient, you just have to find it and the power dependence. First, notice that area of flow will increase with height squared. Now notice that the velocity will increase with the square root of hight, H because of an energy balance. Potential energy per unit volume is mgH, and kinetic energy per unit volume is 1/2 mv2 where m is the mass per unit volume and g is the gravitational constant. Flow in the weir is achieved by converting potential height energy into kinetic, velocity energy. From the power dependence, you can expect that the average v will be proportional to √H at all values of H.

Combining the two effects together, you can expect a power dependence of 2.5 (square root is a power of 0.5). Putting this another way, the storm height in the weir is four times the dry height, so the area of flow is 16 times what it was when you measured with the bucket. Also, since the average height is four times greater than before, you can expect that the average velocity will be twice what it was. Thus, we estimate that there was 32 times the flow during the storm than there was during the dry period; 32 x 0.8 = 25.6 gallons/sec., or 92,000 gal/hr, or 3.28 cfs.

The general equation I derive for flow over this, V-shaped weir is

Flow (gallons/sec) = Cv gal/hr x(feet)5/2.

where Cv = 3.28 cfs. This result is not much different to a standard one  in the tables — that for knife-edge, 90° weirs with large shoulders on either side and at least twice the weir height below the weir (P, in the diagram above). For this knife-edge weir, the Bureau of Reclamation Manual suggests Cv = 2.49 and a power value of 2.48. It is unlikely that you ever get this sort of knife-edge weir in a practical application. Be sure to measure Cv at low flow for any weir you build or find.

Robert Buxbaum, vote for me for water commissioner. Here are some thoughts on other problems with our drains.

# Boy-Girl physics humor

Girl breaking up with her boyfriend: I just need two things, more space, and time.

Atoms build physicists in an attempt to understand themselves. That’s also why physicists build physics societies and clubs.

Boyfriend: So, what’s the other thing?

Robert Buxbaum. And that, dear friend, is why science majors so rarely have normal boyfriends / girlfriends.

A female engineer friend of mine commented on the plight of dating in the department: “The odds are good, but the goods are odd.”

By the way, the solution to Einstein’s twin paradox resides in understanding that time is space. Both twins see the space ship moving at the same pace, but space shrinks for the moving twin in the space ship, not for the standing one. Thus, the moving twin finishes his (or her) journey in less time than the standing one observes.

# Our expanding, black hole universe

In a previous post I showed a classical derivation of the mass-to-size relationship for black -holes and gave evidence to suggest that our universe (all the galaxies together) constitute a single, large black hole. Everything is inside the black hole and nothing outside but empty space — We can tell this because you can see outside from inside a black hole — it’s only others, outside who can not see in (Finkelstein, Phys Rev. 1958). Not that there appear to be others outside the universe, but if they were, they would not be able to see us.

In several ways having a private, black hole universe is a gratifying thought. It provides privacy and a nice answer to an easily proved conundrum: that the universe is not infinitely big. The black hole universe that ends as the math requires, but not with a brick wall, as i the Hitchhiker’s guide (one of badly-laid brick). There are one or two problems with this nice tidy solution. One is that the universe appears to be expanding, and black holes are not supposed to expand. Further, the universe appears to be bigger than it should be, suggesting that it expanded faster than the speed of light at some point. its radius now appears to be 40-46 billion light years despite the universe appearing to have started as a point some 14 billion years ago. That these are deeply disturbing questions does not stop NASA and Nova from publishing the picture below for use by teachers. This picture makes little sense, but it’s found in Wikipedia and most, newer books.

Standard picture of the big bang theory: A period of faster than light expansion (inflation) then light-speed, accelerating expansion. NASA, and Wikipedia.

We think the creation event occurred some 14 billion years ago because we observe that the majority of galaxies are expanding from us at a rate proportional to their distance from us. From this proportionality between the rate of motion and the distance from us, we conclude that we were all in one spot some 14 billion years ago. Unfortunately, some of the most distant galaxies are really dim — dimmer than they would be if they were only 14 billion light years away. The model “explains this” by a period of inflation, where the universe expanded faster than the speed of light. The current expansion then slowed, but is accelerating again; not slowing as would be expected if it were held back by gravity of the galaxies. Why hasn’t the speed of the galaxies slowed, and how does the faster-than-light part work? No one knows. Like Dr. Who’s Tardis, our universe is bigger on the inside than seems possible.

Einstein’s oscillating universe: it expands and contracts at some (large) frequency. Oscillations would explain why the universe is near-uniform, but not why it’s so big or moving outward so fast.

Einstein’s preferred view was of an infinite space universe where the mass within expands and contracts. He joked that two things were infinite, the universe and stupidity… see my explanation... In theory, gravity could drive the regular contractions to an extent that would turn entropy backward. Einstein’s oscillating model would explain how the universe is reasonably stable and near-uniform in temperature, but it’s not clear how his universe could be bigger than 14 billion light years across, or how it could continue to expand as fast as it does. A new view, published this month suggests that there are two universes, one going forward in time the other backward. The backward in time part of the universe could be antimatter, or regular matter going anti entropy (that’s how I understand it — If it’s antimatter, we’d run into the it all the time). Random other ideas float through the physics literature: that we’re connected to other space through a black hole/worm hole, perhaps to many other universes by many worm holes in fractal chaos, see for example, Physics Reports, 1992.

The forward-in-time expansion part of the two universes model. This drawing, like the first, is from NASA.

For all I know, there are these many black hole  tunnels to parallel universes. Perhaps the universal constant and all these black-hole tunnels are windows on quantum mechanics. At some point the logic of the universe seems as perverse as in the Hitchhiker guide.

Something I didn’t mention yet is the Higgs boson, the so-called God particle. As in the joke, it’s supposed to be responsible for mass. The idea is that all particles have mass only by interaction with these near-invisible Higgs particles. Strong interactions with the Higgs are what make these particles heavier, while weaker – interacting particles are perceived to have less gravity and inertia. But this seems to me to be the theory that Einstein’s relativity and the 1919 eclipse put to rest. There is no easy way for a particle model like this to explain relativistic warping of space-time. Without mass being able to warp space-time you’d see various degrees of light bending around the sun, and preferential gravity in the direction of our planet’s motion: things we do not see. We’re back in 1900, looking for some plausible explanation for the uniform speed of light and Lawrence contraction of space.

Dr. r µ ßuxbaum. December 20, 2014. The  meaning of the universe could be 42 for all I know, or just pickles down the worm hole. No religion seems to accept the 14 billion year old universe, and for all I know the God of creation has a wicked sense of humor. Carry a towel and don’t think too much.

# A simple, classical view of and into black holes

Black holes are regions of the universe where gravity is so strong that light can not emerge. And, since the motion of light is related to the fundamental structure of space and time, they must also be regions where space curves on itself, and where time appears to stop — at least as seen by us, from outside the black hole. But what does space-time look like inside the black hole.

NASA’s semi-useless depiction of a black hole — one they created for educators. Though it’s sort of true, I’m not sure what you’re supposed to understand from this. I hope to present a better version.

From our outside perspective, an object tossed into a black hole will appear to move slower as it approaches the hole, and at the hole horizon it will appear to have stopped. From the inside of the hole, the object appears to just fall right in. Some claim that tidal force will rip it apart, but I think that’s a mistake. Here’s a simple, classical way to calculate the size of a black hole, and to understand why things look like they do and do what they do.

Lets begin with light, and accept, for now, that light travels in particle form. We call these particles photons; they have both an energy and a mass, and mostly move in straight lines. The energy of a photon is related to its frequency by way of Plank’s constant. E= hν, where h is Plank’s constant and ν is frequency. Their mass is related to their energy by way of the formula m=E/c2. This formulate is surprisingly easy to derive, and is often shown as E= mc2. In classical form, the gravitational force between a star, mass M, and this photon or any other object of mass m described as follows:

F = GMm/r2

where F is force, G is the gravitational constant, and r is the distance of the photon from the center of the star. The potential energy of a photon of the mass increases as it rises from the star surface, but the internal energy (proportional to frequency) decreases — the photon gets redder. The amount of internal energy lost to gravity as it rises from the surface is the integral of the force, and is thus related to the mass of the object and of the star.

∆E =  ∫Fdr = ∫GMm/r2 dr = -GMm/r

Lets consider a photon of original energy E° and original mass m°. Lets figure out the radius of the star r° such that all of the original energy, E° is lost in rising away from the star. That is let calculate the r for which ∆E = -E° as the photon rises to freedom. Lets assume, for now, that the photon mass remains constant at m°.

E° = GMm°/r° = GME°/c2r°.

We now eliminate E° from the equation and solve for this special radius, r°:

r° =  GM/c2.

This would be the radius of a black hole if space didn’t curve and if the mass of the photon didn’t decrease as it rose. While neither of these assumptions is true, the errors nearly cancel, and the true value for r° is double the size calculated this way.

r° = 2GM/c2

r° = 2.95 km (M/Msun).

Karl Schwarzschild 1873-1916.

The first person to do this calculation was Karl Schwarzschild and r° is called the Schwarzschild radius. This is the minimal radius for a star of mass M to produce closed space-time; a black hole. Msun is the mass of our sun, sol, 2 × 1030 kg.  To make a black hole one would have to compress the mass of our sun into a ball of 2.95 km radius, about the size of an SUV. Space-time would close around it, and light starting from the surface would not be able to escape.

As it happens, our sun is far bigger than a SUV and is not a black hole: we can see light from the sun’s surface with minimal space-time deformation. Still, if the mass were a lot bigger, the radius would be a lot bigger and the needed density less. Consider a black hole the same mass as our galaxy, about 1 x1012 solar masses (mostly dark matter), or 2 x 1042  kg. The Schwarzschild radius of a star with the mass of our galaxy would be 3 x 1012 km, or 0.3 light years, about 1/20 the distance to Alpha Centauri. This is far bigger than the six of our solar system, but far smaller than the actual size of the galaxy, 5 x 1017 km, or 50,000 light years. Still, the difference between 0.3 light years and 50,000 light years isn’t that great on the cosmic scale, and it’s worthwhile to consider a black hole comprising something 10 to 100 Billion times more massive than our galaxy — the universe as a whole.

The folks at Cornell estimate the sum of dark and luminous matter in the universe to be about 15 billion times the mass of our galaxy, or 3 x 1052 kg. This does not include the mass of the dark energy, but no one’s quite sure what dark energy is. Considering only this physical mass, the Schwarzschild radius for the universe would be about 4.5 billion light years, or about 1/3 the size of our universe based on its age. The universe appears to be 14 billion years old, so if it’s expanding at the speed of light, the radius should be 14 billion light years. The universe may be 2-3 times bigger than this on the inside (rather like Dr. Who’s Tardis it’s bigger on the inside) but in astronomical terms a factor of 3 or 10 is nothing: the size of the universe is remarkably similar to its Schwarzschild radius, and this is without considering the mass its dark energy must have.

Standard picture of the big bang theory. Dark energy causes the latter-stage expansion.

The evidence for dark energy is that the universe is expanding faster and faster instead of slowing. See figure. There is no visible reason for the acceleration, but it’s there. The significant amount of energy must have significant mass, E = mc2. If the mass of this energy is 3 to 10 times the physical mass, as seems possible, we are living inside a large black hole, something many physicists, including Einstein considered extremely likely. Einstein originally didn’t consider the possibility that the hole could be expanding. But now we know how to calculate the size of a black hole, and we know what a large black hole looks like from the inside. It looks just like home.

Wait for further posts on curved space-time. For some reason, no religion seems to embrace science’s 14 billion year old, black-hole universe (expanding or not). As for tidal forces, they are horrific only for the small black holes that most people write about. If the black hole is big enough, the tidal forces are small.

Dr. µß Buxbaum Nov 17, 2014. You can drink to the Schwarzchild radius with my new R° cocktail.

# Dr. Who’s Quantum reality viewed as diffusion

It’s very hard to get the meaning of life from science because reality is very strange, Further, science is mathematical, and the math relations for reality can be re-arranged. One arrangement of the terms will suggest a version of causality, while another will suggest a different causality. As Dr. Who points out, in non-linear, non-objective terms, there’s no causality, but rather a wibbly-wobbely ball of timey-wimey stuff.

Reality is a ball of  timey wimpy stuff, Dr. Who.

To this end, I’ll provide my favorite way of looking at the timey-wimey way of the world by rearranging the equations of quantum mechanics into a sort of diffusion. It’s not the diffusion of something you’re quite familiar with, but rather a timey-wimey wave-stuff referred to as Ψ. It’s part real and part imaginary, and the only relationship between ψ and life is that the chance of finding something somewhere is proportional Ψ*|Ψ. The diffusion of this half-imaginary stuff is the underpinning of reality — if viewed in a certain way.

First let’s consider the steady diffusion of a normal (un-quantum) material. If there is a lot of it, like when there’s perfume off of a prima donna, you can say that N = -D dc/dx where N is the flux of perfume (molecules per minute per area), dc/dx is a concentration gradient (there’s more perfume near her than near you), and D is a diffusivity, a number related to the mobility of those perfume molecules.

We can further generalize the diffusion of an ordinary material for a case where concentration varies with time because of reaction or a difference between the in-rate and the out rate, with reaction added as a secondary accumulator, we can write: dc/dt = reaction + dN/dx = reaction + D d2c/dx2. For a first order reaction, for example radioactive decay, reaction = -ßc, and

dc/dt = -ßc + D d2c/dx2               (1)

where ß is the radioactive decay constant of the material whose concentration is c.

Viewed in a certain way, the most relevant equation for reality, the time-dependent Schrödinger wave equation (semi-derived here), fits into the same diffusion-reaction form:

dΨ/dt = – 2iπV/h Ψ + hi/4πm d2Ψ/dx               (2)

Instead of reality involving the motion of a real material (perfume, radioactive radon, etc.) with a real concentration, c, in this relation, the material can not be sensed directly, and the concentration, Ψ, is semi -imaginary. Here, h is plank’s constant, i is the imaginary number, √-1, m is the mass of the real material, and V is potential energy. When dealing with reactions or charged materials, it’s relevant that V will vary with position (e.g. electrons’ energy is lower when they are near protons). The diffusivity term here is imaginary, hi/4πm, but that’s OK, Ψ is part imaginary, and we’d expect that potential energy is something of a destroyer of Ψ: the likelihood of finding something at a spot goes down where the energy is high.

The form of this diffusion is linear, a mathematical term that refers to equations where solution that works for Ψ will also work for 2Ψ. Generally speaking linear solutions have exp() terms in them, and that’s especially likely here as the only place where you see a time term is on the left. For most cases we can say that

Ψ = ψ exp(-2iπE/h)t               (3)

where ψ is not a function of anything but x (space) and E is the energy of the thing whose behavior is described by Ψ. If you take the derivative of equation 3 this with respect to time, t, you get

dΨ/dt = ψ (-2iπE/h) exp(-2iπE/h)t = (-2iπE/h)Ψ.               (4)

If you insert this into equation 2, you’ll notice that the form of the first term is now identical to the second, with energy appearing identically in both terms. Divide now by exp(-2iπE/h)t, and you get the following equation:

(E-V) ψ =  -h2/8π2m d2ψ/dx2                      (5)

where ψ can be thought of as the physical concentration in space of the timey-wimey stuff. ψ is still wibbly-wobbley, but no longer timey-wimey. Now ψ- squared is the likelihood of finding the stuff somewhere at any time, and E, the energy of the thing. For most things in normal conditions, E is quantized and equals approximately kT. That is E of the thing equals, typically, a quantized energy state that’s nearly Boltzmann’s constant times temperature.

You now want to check that the approximation in equation 3-5 was legitimate. You do this by checking if the length-scale implicit in exp(-2iπE/h)t is small relative to the length-scales of the action. If it is (and it usually is), You are free to solve for ψ at any E and V using normal mathematics, by analytic or digital means, for example this way. ψ will be wibbely-wobbely but won’t be timey-wimey. That is, the space behavior of the thing will be peculiar with the item in forbidden locations, but there won’t be time reversal. For time reversal, you need small space features (like here) or entanglement.

Equation 5 can be considered a simple steady state diffusion equation. The stuff whose concentration is ψ is created wherever E is greater than V, and is destroyed wherever V is greater than E. The stuff then continuously diffuses from the former area to the latter establishing a time-independent concentration profile. E is quantized (can only be some specific values) since matter can never be created of destroyed, and it is only at specific values of E that this happens in Equation 5. For a particle in a flat box, E and ψ are found, typically, by realizing that the format of ψ must be a sin function (and ignoring an infinity). For more complex potential energy surfaces, it’s best to use a matrix solution for ψ along with non-continuous calculous. This avoids the infinity, and is a lot more flexible besides.

When you detect a material in some spot, you can imagine that the space- function ψ collapses, but even that isn’t clear as you can never know the position and velocity of a thing simultaneously, so doesn’t collapse all that much. And as for what the stuff is that diffuses and has concentration ψ, no-one knows, but it behaves like a stuff. And as to why it diffuses, perhaps it’s jiggled by unseen photons. I don’t know if this is what happens, but it’s a way I often choose to imagine reality — a moving, unseen material with real and imaginary (spiritual ?) parts, whose concentration, ψ, is related to experience, but not directly experienced.

This is not the only way the equations can be rearranged. Another way of thinking of things is as the sum of path integrals — an approach that appears to me as a many-world version, with fixed-points in time (another Dr Who feature). In this view, every object takes every path possible between these points, and reality as the sum of all the versions, including some that have time reversals. Richard Feynman explains this path integral approach here. If it doesn’t make more sense than my version, that’s OK. There is no version of the quantum equations that will make total, rational sense. All the true ones are mathematically equivalent — totally equal, but differ in the “meaning”. That is, if you were to impose meaning on the math terms, the meaning would be totally different. That’s not to say that all explanations are equally valid — most versions are totally wrong, but there are many, equally valid math version to fit many, equally valid religious or philosophic world views. The various religions, I think, are uncomfortable with having so many completely different views being totally equal because (as I understand it) each wants exclusive ownership of truth. Since this is never so for math, I claim religion is the opposite of science. Religion is trying to find The Meaning of life, and science is trying to match experiential truth — and ideally useful truth; knowing the meaning of life isn’t that useful in a knife fight.

Dr. Robert E. Buxbaum, July 9, 2014. If nothing else, you now perhaps understand Dr. Who more than you did previously. If you liked this, see here for a view of political happiness in terms of the thermodynamics of free-energy minimization.

# Toxic electrochemistry and biology at home

A few weeks back, I decided to do something about the low quality of experiments in modern chemistry and science sets; I posted to this blog some interesting science experiments, and some more-interesting experiments that could be done at home using the toxic (poisonous dangerous) chemicals available under the sink or on the hardware store. Here are some more. As previously, the chemicals are toxic and dangerous but available. As previously, these experiments should be done only with parental (adult) supervision. Some of these next experiments involve some math, as key aspect of science; others involve some new equipment as well as the stuff you used previously. To do them all, you will want a stop watch, a volt-amp meter, and a small transformer, available at RadioShack; you’ll also want some test tubes or similar, clear cigar tubes, wire and baking soda; for the coating experiment you’ll want copper drain clear, or copper containing fertilizer and some washers available at the hardware store; for metal casting experiment you’ll need a tin can, pliers, a gas stove and some pennies, plus a mold, some sand, good shoes, and a floor cover; and for the biology experiment you will need several 9 V batteries, and you will have to get a frog and kill it. You can skip any of these experiments, if you like and do the others. If you have not done the previous experiments, look them over or do them now.

1) The first experiments aim to add some numerical observations to our previous studies of electrolysis. Here is where you will see why we think that molecules like water are made of fixed compositions of atoms. Lets redo the water electrolysis experiment now with an Ammeter in line between the battery and one of the electrodes. With the ammeter connected, put both electrodes deep into a solution of water with a little lye, and then (while watching the ammeter) lift one electrode half out, place it back, and lift the other. You will find, I think, that one of the other electrode is the limiting electrode, and that the amperage goes to 1/2 its previous value when this electrode is half lifted. Lifting the other electrode changes neither the amperage or the amount of bubbles, but lifting this limiting electrode changes both the amount of bubbles and the amperage. If you watch closely, though, you’ll see it changes the amount of bubbles at both electrodes in proportion, and that the amount of bubbles is in promotion to the amperage. If you collect the two gasses simultaneously, you’ll see that the volume of gas collected is always in a ratio of 2 to 1. For other electrolysis (H2 and Cl2) it will be 1 to1; it’s always a ratio of small numbers. See diagram below on how to make and collect oxygen and hydrogen simultaneously by electrolyzing water with lye or baking soda as electrolyte. With lye or baking soda, you’ll find that there is always twice as much hydrogen produced as oxygen — exactly.

You can also do electrolysis with table salt or muriatic acid as an electrolyte, but for this you’ll need carbon or platinum electrodes. If you do it right, you’ll get hydrogen and chlorine, a green gas that smells bad. If you don’t do this right, using a wire instead of a carbon or platinum electrode, you’ll still get hydrogen, but no chlorine. Instead of chlorine, you’ll corrode the wire on that end, making e.g. copper chloride. With a carbon electrode and any chloride compound as the electrolyte, you’ll produce chlorine; without a chloride electrolyte, you will not produce chlorine at any voltage, or with any electrode. And if you make chlorine and check the volumes, you’ll find you always make one volume of chlorine for every volume of hydrogen. We imagine from this that the compounds are made of fixed atoms that transfer electrons in fixed whole numbers per molecule. You always make two volumes of hydrogen for every volume of oxygen because (we think) making oxygen requires twice as many electrons as making hydrogen.

At home electrolysis experiment

We get the same volume of chlorine as hydrogen because making chlorine and hydrogen requires the same amount of electrons to be transferred. These are the sort of experiments that caused people to believe in atoms and molecules as the fundamental unchanging components of matter. Different solutes, voltages, and electrodes will affect how fast you make hydrogen and oxygen, as will the amount of dissolved solute, but the gas produced are always the same, and the ratio of volumes is always proportional to the amperage in a fixed ratio of small whole numbers.

As always, don’t let significant quantities of use hydrogen and oxygen or pure hydrogen and chlorine mix in a closed space. Hydrogen and oxygen is quite explosive brown’s gas; hydrogen and chlorine are reactive as well. When working with chlorine it is best to work outside or near an open window: chlorine is a poison gas.

You may also want to try this with non-electrolytes, pure water or water with sugar or alcohol dissolved. You will find there is hardly any amperage or gas with these, but the small amount of gas produced will retain the same ratio. For college level folks, here is some physics/math relating to the minimum voltage and relating to the quantities you should expect at any amperage.

2) Now let’s try electro-plating metals. Using the right solutes, metals can be made to coat your electrodes the same way that bubbles of gas coated your electrodes in the experiments above. The key is to find the right chemical, and as a start let me suggest the copper sulphate sold in hardware stores to stop root growth. As an alternative copper sulphate is often sold as part of a fertilizer solution like “Miracle grow.” Look for copper on the label, or for a blue color fertilizer. Make a solution of copper using enough copper so that the solution is recognizably green, Use two steel washers as electrodes (that is connect the wires from your battery to the washers) and put them in the solution. You will find that one side turns red, as it is coated with copper. Depending on what else your copper solution contained, bubbles may appear at the other washer, or the other washer will corrode.

You are now ready to take this to a higher level — silver coating. take a piece of silver plated material that you want to coat, and clean it nicely with soap and water. Connect it to the electrode where you previously coated copper. Now clean out the solution carefully. Buy some silver nitrate from a drug store, and dissolve a few grams (1/8 tsp for a start) in pure water; place the silverware and the same electrodes as before, connected to the battery. For a nicer coat use a 1 1/2 volt lantern battery; the 6 V battery will work too, but the silver won’t look as nice. With silver nitrate, you’ll notice that one electrode produces gas (oxygen) and the other turns silvery. Now disconnect the silvery electrode. You can use this method to silver coat a ring, fork, or cup — anything you want to have silver coated. This process is called electroplating. As with hydrogen production, there is a proportional relationship between the time, the amperage and the amount of metal you deposit — until all the silver nitrate in solution is used up.

As a yet-more complex version, you can also electroplate without using a battery. This was my Simple electroplating (presented previously). Consider this only after you understand most everything else I’ve done. When I saw this the first time in high school I was confused.

3) Casting metal objects using melted pennies, heat from a gas stove, and sand or plaster as a cast. This is pretty easy, but sort of dangerous — you need parents help, if only as a watcher. This is a version of an experiment I did as a kid.  I did metal casting using lead that some plumbers had left over. I melted it in a tin can on our gas stove and cast “quarters” in a plaster mold. Plumbers no longer use lead, but modern pennies are mostly zinc, and will melt about as well as my lead did. They are also much safer.

A plaster mold can be made by putting a 50¢ piece at the bottom of a paper cup, pouring plaster over the coin, and waiting for it to dry. Tear off the cup, turn the plaster over and pull out the coin; you’ve got a one-sided mold, good enough to make a one-sided coin. If you enjoy this, you can learn more about casting on Wikipedia; it’s an endeavor that only costs 4 or 5 cents per try. As a safety note: wear solid leather shoes and cover the floor near the stove with a board. If you drop the metal on the floor you’ll have a permanent burn mark on the floor and your mother will not be happy. If you drop hot metal on your you’ll have a permanent injury, and you won’t be happy. Older pennies are made of copper and will not melt. Here’s a video of someone pouring a lot of metal into an ant-hill (kills lots of ants, makes a mold of the hill).

It’s nice to have assistants, friends and adult help in the laboratory when you do science. Even without the castle, it’s what Dr. Frankenstein did.

4) Bringing a dead frog back to life (sort of). Make a high voltage battery of 45 to 90 V battery by attaching 5-10, 9V batteries in a daisy chain they will snap together. If you touch both exposed contacts you’ll give yourself a wicked shock. If you touch the electrodes to a newly killed frog, the frog legs will kick. This is sort of groovy. It was the inspiration for Dr. Frankenstein (at right), who then decides he could bring a person back from the dead with “more power.” Frankenstein’s monster is brought back to life this way, but ends up killing the good doctor. Shocks are sometimes helpful reanimating people stricken by heat attacks, and many buildings have shockers for this purpose. But don’t try to bring back the long-dead. By all accounts, the results are less-than pleasing. Try dissecting the rest of the frog and guess what each part is (a world book encyclopedia helps). As I recall, the heart keeps going for a while after it’s out of the frog — spooky.

If you want to do more advanced science, it’s a good idea to learn math. This is important for statistics, for engineering, for quantum mechanics, and can even help for music. Get a few good high school or college books and read them cover to cover. An approach to science is to try to make something cool, that sort-of works, and then try to improve it. You then decide what a better version would work like,  modify your original semi-randomly and see if you’re going in the right direction. Don’t redesign with only one approach –it may not work. Read whatever you can, but don’t believe all you read. Often books are misleading, or wrong, and blogs are worse (I ought to know). When you find mistakes, note them in the margin, and try to explain them. You may find you were right, or that the book was right, but it’s a learning experience. If you like you can write the author and inform him/her of the errors. I find mailed letters are more respectful than e-mails — it shows you put in more effort.

Robert Buxbaum, February 20, 2014. Here’s the difference between metals and non-metals, and a periodic table cup that I made, and sell. And here’s a difference between science and religion – reproducibility.

# Toxic chemistry you can do at home

I got my start on science working with a 7 chemical, chemistry set that my sister got me when I was 7 years old (thanks Beverly). The chemicals would never be sold by a US company today — too much liability. What if your child poisons himself/herself or someone else, or is allergic, or someone chokes on the caps (anything the size of a nut has to be labeled as a hazard). Many of the experiments were called magic, and they were, in the sense that, if you did them 200 years earlier, you’d be burnt as a witch. There were dramatic color changes (phenolphthalein plus base, Prussian Blue) a time-delay experiment involving cobalt, and even an experiment that (as I recall) burst into fire on its own (glycerine plus granulated potassium permanganate).

“Better the evil you know.” If you get good at this, the military may have use of your services. Yes, the American military does science.

Science kits nowadays don’t do anything magically cool like that, and they don’t really teach chemistry, either, I think. Doing magical things requires chemicals that are reasonably reactive, and that means corrosive and/or toxic. Current kits use only food products like corn-starch or baking soda, and the best you can do with these is to make goo and/ or bubbles. No one would be burnt at the stake for this, even 300 years ago. I suppose one could design a program that used these materials to teach something about flow, or nucleation, but that would require math, and the kit producers fear that any math will turn off kids and stop their parents from spending money. There is also the issue of motivation. Much of historical chemistry was driven by greed and war; these are issues that still motivate kids, but that modern set-makers would like to ignore. Instead, current kits are supposed to be exciting in a cooperative way (whatever that means), because the kit-maker says so. They are not. I went through every experiment in my first kit in the first day, and got things right within the first week — showing off to whoever would watch. Modern kits don’t motivate this sort of use; I doubt most get half-used in a lifetime.

There are some foreign-made chemistry sets still that are pretty good. Here is a link to a decent mid-range one from England. But it’s sort of pricy, and already somewhat dumbed down. Instead, here are some cheaper, more dangerous, American options: 5 experiments you can do (kids and parents together, please) using toxic household chemicals found in our US hardware stores. These are NOT the safest experiments, just cheap ones that are interesting. I’ll also try to give some math and explanations — so you’ll understand what’s happening on a deeper level — and I’ll give some financial motivation — some commercial value.

1) Crystal Drano + aluminum. Crystal Drano is available in the hardware store. It’s mostly lye, sodium hydroxide, one of the strongest bases known to man. It’s a toxic (highly poisonous) chemical used to dissolve hair and fat in a drain. It will also dissolve some metals and it will dissolve you if you get it on yourself (if you do get it on yourself, wash it off fast with lots of water). Drano also contains ammonium nitrate (an explosive) and bits of aluminum. For the most part, the aluminum is there so that the Drano will get hot in the clogged drain (heat helps it dissolve the clog faster). I’ll explain the ammonium nitrate later. For this experiment, you’re going to want to work outside, on a dinner plate on the street. You’ll use additional aluminum (aluminum foil), and you’ll get more heat and fun gases. Fold up a 1 foot square of aluminum foil to 6″ x 4″ say, and put it on the plate (outside). Put an indent in the middle of the foil making a sort of small cup — one that can stand. Into this indent, put a tablespoon or two of water plus a teaspoon of Drano. Wait about 5 minutes, and you will see that the Drano starts smoking and the aluminum foils starts to dissolve. The plate will start to get hot and you will begin to notice a bad smell (ammonia). The aluminum foil will turn black and will continue to dissolve till there is a hole in the middle of the indent.

The main reaction is 2 Al + 3 H2O –> Al2O3 + H2; that is, aluminum plus water gives you aluminum oxide (alumina), and hydrogen. The sodium hydroxide (lye) in the Drano is a catalyst in this reaction, something that is not consumed in this reaction but makes it happen faster than otherwise. The hydrogen you produce here is explosive and valuable (I explain below). But there is another reaction going on too, the one that makes the bad smell. When ammonium nitrate is heated in the presence of sodium hydroxide, it reacts to make ammonia and sodium nitrate. The reaction formula is: NH4-NO3 + NaOH –> NH3 + NaNO3 + H2O. The ammonia produced gives off a smell, something that is important for safety — the smell is a warning — and (I think) helps keep the aluminum gunk from clogging the drain by reacting with the aluminum oxide to form aluminum amine hydroxide Al2O3(NH3)2. It’s a fun experiment to watch, but you can do more if you like. The hydrogen and ammonia are flammable and is useful for other experiments (below). If you collect these gases, you can can make explosions or fill a balloon that will float. Currently the US military, and several manufacturers in Asia are considering using the hydrogen created this way to power motorcycles by way of a fuel cell. There is also the Hindenburg, a zeppelin that went around the world in the 1930s. It was kept aloft by hydrogen. The ammonia you make has value too, though toxic; if bubbled into water, it makes ammonium hydroxide NH3 + H2O –> NH4OH. This is a common cleaning liquid. Just to remind you: you’re supposed to do these experiments outside to dissipate the toxic gases and to avoid an explosion in your house. A parent will come in handy if you get this stuff on your hand or in your eye.

Next experiment: check that iron does not dissolve in Drano, but it does in acid (that’s experiment 5; done with Muriatic acid from the hardware store). Try also copper, and solder (mostly tin, these days). Metals that dissolve well in Drano are near the right of the periodic table, like aluminum. Aluminum is nearly a non-metal, and thus can be expected to have an oxide that reacts with hydroxide. Iron and steel have oxides that are bases themselves, and thus don’t react with lye. This is important as otherwise Drano would destroy your iron drain, not only the hair in it. It’s somewhat hard on copper though, so beware if you’ve a copper drain.

Thought problem: based on the formulas above figure out the right mix of aluminum, NaOH, water and Ammonium nitrate. Answer: note that, for every two atoms of aluminum you dissolve, you’ll need three molecules of water (for the three O atoms), plus at least two molecules of ammonium nitrate (to provide the two NH2 (amine) groups above. You’ll also want at least 2 molecules of NaOH to have enough Na to react with the nitrate groups of the ammonium nitrate. As a first guess, assume that all atoms are the same size. A better way to do this involves molecular weights (formula weights), read a chemistry book, or look on the internet.

Four more experiments can be seen here. This post was getting to be over-long.As with this experiment, wear gloves and eye protection; don’t drink the chemicals, and if you get any chemicals on you, wash them off quick.

Here are a few more experiments in electrochemistry and biology, perhaps I’ll add more. In the meantime, if you or your child are interested in science, I’d suggest you read science books by Mr Wizard, or Isaac Asimov, and that you learn math. Another thought, take out a high school chemistry text-book at the library — preferably an old one with experiments..

Robert Buxbaum, December 29, 2013. If you are interested in weather flow, by the way, here is a bit on why tornadoes and hurricanes lift stuff up, and on how/ why they form.