Category Archives: Science: Physics, Astronomy, etc.

Thermodynamics of hydrogen generation

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Perhaps the simplest way to make hydrogen is by electrolysis: you run some current through water with a little sulfuric acid or KOH added, and for every two electrons transferred, you get a molecule of hydrogen from one electrode and half a molecule of oxygen from the other.

2 OH- –> 2e- + 1/2 O2 +H2O

2H2O + 2e- –>  H2 + 2OH-

The ratio between amps, seconds and mols of electrons (or hydrogen) is called the Faraday constant, F = 96500; 96500 amp-seconds transfers a mol of electrons. For hydrogen production, you need 2 mols of electrons for each mol of hydrogen, n= 2, so

it = 2F where and i is the current in amps, and t is the time in seconds and n is the number electrons per molecule of desired product. For hydrogen, t = 96500*2/i; in general, t = Fn/i.

96500 is a large number, and it takes a fair amount of time to make any substantial amount of hydrogen by electrolysis. At 1 amp, it takes 96500*2 = 193000 seconds, 2 days, to generate one mol of hydrogen (that’s 2 grams H2 or 22.4 liters, enough to fill a garment bag). We can reduce the time by using a higher current, but there are limits. At 25 amps, the maximum current of you can carry with house wiring it takes 2.14 hours to generate 2 grams. (You’ll have to rectify your electricity to DC or you’ll get a nasty H2 /O2 mix called Brown’s gas, While normal H2 isn’t that dangerous, Browns gas is a mix of H2 and O2 and is quite explosive. Here’s an essay I wrote on separating Browns gas).

Electrolysis takes a fair amount of electric energy too; the minimum energy needed to make hydrogen at a given temperature and pressure is called the reversible energy, or the Gibbs free energy ∆G of the reaction. ∆G = ∆H -T∆S, that is, ∆G equals the heat of hydrogen production ∆H – minus an entropy effect, T∆S. Since energy is the product of voltage current and time, Vit = ∆G, where ∆G is the Gibbs free energy measured in Joules and V,i, and t are measured Volts, Amps, and seconds respectively.

Since it = nF, we can rewrite the relationship as: V =∆G/nF for a process that has no energy losses, a reversible process. This is the form found in most thermodynamics textbooks; the value of V calculated this way is the minimum voltage to generate hydrogen, and the maximum voltage you could get in a fuel cell putting water back together.

To calculate this voltage, and the power requirements to make hydrogen, we use the Gibbs free energy for water formation found in Wikipedia, copied below (in my day, we used the CRC Handbook of Chemistry and Physics or a table in out P-chem book). You’ll notice that there are two different values for ∆G depending on whether the water is a gas or a liquid, and you’ll notice a small zero at the upper right (∆G°). This shows that the values are for an imaginary standard state: 20°C and 1 atm pressure. You can’t get 1 atm steam at 20°C, it’s an extrapolation; behavior at typical temperatures, 40°C and above is similar but not identical. I’ll leave it to a reader to send this voltage as a comment.

Liquid H2O formation ∆G° = -237.14
Gaseous H2O formation ∆G° = -228.61

The reversible voltage for creating liquid water in a reversible fuel cell is found to be -237,140/(2 x 96,500) = -1.23V. We find that 1.23 Volts is about the minimum voltage you need to do electrolysis at 0°C because you need liquid water to carry the current; -1.18 V is about the maximum voltage you can get in a fuel cell because they operate at higher temperature with oxygen pressures significantly below 1 atm. (typically). The minus sign is kept for accounting; it differentiates the power out case (fuel cells) from power in (electrolysis). It is typical to find that fuel cells operate at lower voltages, between about .5V and 1.0V depending on the fuel cell and the power load.

Most electrolysis is done at voltages above about 1.48 V. Just as fuel cells always give off heat (they are exothermic), electrolysis will absorb heat if run reversibly. That is, electrolysis can act as a refrigerator if run reversibly. but electrolysis is not a very good refrigerator (the refrigerator ability is tied up in the entropy term mentioned above). To do electrolysis at reasonably fast rates, people give up on refrigeration (sucking heat from the environment) and provide all the entropy needed for electrolysis in the electricity they supply. This is to say, they operate at V’ were nFV’ ≥ ∆H, the enthalpy of water formation. Since ∆H is greater than ∆G, V’ the voltage for electrolysis is higher than V. Based on the enthalpy of liquid water formation,  −285.8 kJ/mol we find V’ = 1.48 V at zero degrees. The figure below shows that, for any reasonably fast rate of hydrogen production, operation must be at 1.48V or above.

Electrolyzer performance; C-Pt catalyst on a thin, nafion membrane

Electrolyzer performance; C-Pt catalyst on a thin, nafion membrane

If you figure out the energy that this voltage and amperage represents (shown below) you’re likely to come to a conclusion I came to several years ago: that it’s far better to generate large amounts of hydrogen chemically, ideally from membrane reactors like my company makes.

The electric power to make each 2 grams of hydrogen at 1.5 volts is 1.5 V x 193000 Amp-s = 289,500 J = .080 kWh’s, or 0.9¢ at current rates, but filling a car takes 20 kg, or 10,000 times as much. That’s 800 kW-hr, or $90 at current rates. The electricity is twice as expensive as current gasoline and the infrastructure cost is staggering too: a station that fuels ten cars per hour would require 8 MW, far more power than any normal distributor could provide.

By contrast, methanol costs about 2/3 as much as gasoline, and it’s easy to deliver many giga-joules of methanol energy to a gas station by truck. Our company’s membrane reactor hydrogen generators would convert methanol-water to hydrogen efficiently by the reaction CH3OH + H2O –> 3H2 + CO2. This is not to say that electrolysis isn’t worthwhile for lower demand applications: see, e.g.: gas chromatography, and electric generator cooling. Here’s how membrane reactors work.

R. E. Buxbaum July 1, 2013; Those who want to show off, should post the temperature and pressure corrections to my calculations for the reversible voltage of typical fuel cells and electrolysis.

Chemist v Chemical Engineer joke

What’s the difference between a chemist and a chemical engineer?

 

How much they make.

 

I made up this joke up as there were no other chemical engineer jokes I knew. It’s an OK double entente that’s pretty true — both in terms of product produced and the amount of salary (there’s probably a cause-and-effect relation here). Typical of these puns, this joke ignores the internal differences in methodologies and background (see my post, How is Chemical engineering?). If you like, here’s another engineering joke,  a chemistry joke, and a dwarf joke.

R.E. Buxbaum –  June 28, 2013.

Another Quantum Joke, and Schrödinger’s waves derived

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Quantum mechanics joke. from xkcd.

Quantum mechanics joke. from xkcd.

Is funny because … it’s is a double entente on the words grain (as in grainy) and waves, as in Schrödinger waves or “amber waves of grain” in the song America (Oh Beautiful). In Schrödinger’s view of the quantum world everything seems to exist or move as a wave until you observe it, and then it always becomes a particle. The math to solve for the energy of things is simple, and thus the equation is useful, but it’s hard to understand why,  e.g. when you solve for the behavior of a particle (atom) in a double slit experiment you have to imagine that the particle behaves as an insubstantial wave traveling though both slits until it’s observed. And only then behaves as a completely solid particle.

Math equations can always be rewritten, though, and science works in the language of math. The different forms appear to have different meaning but they don’t since they have the same practical predictions. Because of this freedom of meaning (and some other things) science is the opposite of religion. Other mathematical formalisms for quantum mechanics may be more comforting, or less, but most avoid the wave-particle duality.

The first formalism was Heisenberg’s uncertainty. At the end of this post, I show that it is identical mathematically to Schrödinger’s wave view. Heisenberg’s version showed up in two quantum jokes that I explained (beat into the ground), one about a lightbulb  and one about Heisenberg in a car (also explains why water is wet or why hydrogen diffuses through metals so quickly).

Yet another quantum formalism involves Feynman’s little diagrams. One assumes that matter follows every possible path (the multiple universe view) and that time should go backwards. As a result, we expect that antimatter apples should fall up. Experiments are underway at CERN to test if they do fall up, and by next year we should finally know if they do. Even if anti-apples don’t fall up, that won’t mean this formalism is wrong, BTW: all identical math forms are identical, and we don’t understand gravity well in any of them.

Yet another identical formalism (my favorite) involves imagining that matter has a real and an imaginary part. In this formalism, the components move independently by diffusion, and as a result look like waves: exp (-it) = cost t + i sin t. You can’t observe the two parts independently though, only the following product of the real and imaginary part: (the real + imaginary part) x (the real – imaginary part). Slightly different math, same results, different ways of thinking of things.

Because of quantum mechanics, hydrogen diffuses very quickly in metals: in some metals quicker than most anything in water. This is the basis of REB Research metal membrane hydrogen purifiers and also causes hydrogen embrittlement (explained, perhaps in some later post). All other elements go through metals much slower than hydrogen allowing us to make hydrogen purifiers that are effectively 100% selective. Our membranes also separate different hydrogen isotopes from each other by quantum effects (big things tunnel slower). Among the uses for our hydrogen filters is for gas chromatography, dynamo cooling, and to reduce the likelihood of nuclear accidents.

Dr. Robert E. Buxbaum, June 18, 2013.

To see Schrödinger’s wave equation derived from Heisenberg for non-changing (time independent) items, go here and note that, for a standing wave there is a vibration in time, though no net change. Start with a version of Heisenberg uncertainty: h =  λp where the uncertainty in length = wavelength = λ and the uncertainty in momentum = momentum = p. The kinetic energy, KE = 1/2 p2/m, and KE+U(x) =E where E is the total energy of the particle or atom, and U(x) is the potential energy, some function of position only. Thus, p = √2m(E-PE). Assume that the particle can be described by a standing wave with a physical description, ψ, and an imaginary vibration you can’t ever see, exp(-iωt). And assume this time and space are completely separable — an OK assumption if you ignore gravity and if your potential fields move slowly relative to the speed of light. Now read the section, follow the derivation, and go through the worked problems. Most useful applications of QM can be derived using this time-independent version of Schrödinger’s wave equation.

Hormesis, Sunshine and Radioactivity

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It is often the case that something is good for you in small amounts, but bad in large amounts. As expressed by Paracelsus, an early 16th century doctor, “There is no difference between a poison and a cure: everything depends on dose.”

Aereolis Bombastus von Hoenheim (Paracelcus)

Phillipus Aureolus Theophrastus Bombastus von Hoenheim (Dr. Paracelsus).

Some obvious examples involve foods: an apple a day may keep the doctor away. Fifteen will cause deep physical problems. Alcohol, something bad in high doses, and once banned in the US, tends to promote longevity and health when consumed in moderation, 1/2-2 glasses per day. This is called “hormesis”, where the dose vs benefit curve looks like an upside down U. While it may not apply to all foods, poisons, and insults, a view called “mitridatism,” it has been shown to apply to exercise, chocolate, coffee and (most recently) sunlight.

Up until recently, the advice was to avoid direct sun because of the risk of cancer. More recent studies show that the benefits of small amounts of sunlight outweigh the risks. Health is improved by lowering blood pressure and exciting the immune system, perhaps through release of nitric oxide. At low doses, these benefits far outweigh the small chance of skin cancer. Here’s a New York Times article reviewing the health benefits of 2-6 cups of coffee per day.

A hotly debated issue is whether radiation too has a hormetic dose range. In a previous post, I noted that thyroid cancer rates down-wind of the Chernobyl disaster are lower than in the US as a whole. I thought this was a curious statistical fluke, but apparently it is not. According to a review by The Harvard Medical School, apparent health improvements have been seen among the cleanup workers at Chernobyl, and among those exposed to low levels of radiation from the atomic bombs dropped on Hiroshima and Nagasaki. The health   improvements relative to the general population could be a fluke, but after a while several flukes become a pattern.

Among the comments on my post, came this link to this scholarly summary article of several studies showing that long-term exposure to nuclear radiation below 1 Sv appears to be beneficial. One study involved an incident where a highly radioactive, Co-60 source was accidentally melted into a batch of steel that was subsequently used in the construction of apartments in Taiwan. The mistake was not discovered for over a decade, and by then the tenants had received between 0.4 and 6 Sv (far more than US law would allow). On average, they were healthier than the norm and had significantly lower cancer death rates. Supporting this is the finding, in the US, that lung cancer death rates are 35% lower in the states with the highest average radon radiation levels (Colorado, North Dakota, and Iowa) than in those with the lowest levels (Delaware, Louisiana, and California). Note: SHORT-TERM exposure to 1 Sv is NOT good for you; it will give radiation sickness, and short-term exposure to 4.5 Sv is the 50% death level

Most people in the irradiated Taiwan apartments got .2 Sv/year or less, but the same health benefit has also been shown for people living on radioactive sites in China and India where the levels were as high as .6 Sv/year (normal US background radiation is .0024 Sv/year). Similarly, virtually all animal and plant studies show that radiation appears to improve life expectancy and fecundity (fruit production, number of offspring) at dose rates as high as 1 Sv/month.

I’m not recommending 1 Sv/month for healthy people, it’s a cancer treatment dose, and will make healthy people feel sick. A possible reason it works for plants and some animals is that the radiation may kill proto- cancer, harmful bacteria, and viruses — organisms that lack the repair mechanisms of larger, more sophisticated organisms. Alternately, it could kill non-productive, benign growths allowing the more-healthy growths to do their thing. This explanation is similar to that for the benefits farmers produce by pinching off unwanted leaves and pruning unwanted branches.

It is not conclusive radiation improved human health in any of these studies. It is possible that exposed people happened to choose healthier life-styles than non-exposed people, choosing to smoke less, do more exercise, or eat fewer cheeseburgers (that, more-or-less, was my original explanation). Or it may be purely psychological: people who think they have only a few years to live, live healthier. Then again, it’s possible that radiation is healthy in small doses and maybe cheeseburgers and cigarettes are too?! Here’s a scene from “Sleeper” a 1973, science fiction, comedy movie where Woody Allan, asleep for 200 years, finds that deep fat, chocolate, and cigarettes are the best things for your health. You may not want a cigarette or a radium necklace quite yet, but based on these studies, I’m inclined to reconsider the risk/ benefit balance in favor of nuclear power.

Note: my company, REB Research makes (among other things), hydrogen getters (used to reduce the risks of radioactive waste transportation) and hydrogen separation filters (useful for cleanup of tritium from radioactive water, for fusion reactors, and to reduce the likelihood of explosions in nuclear facilities.

by Dr. Robert E. Buxbaum June 9, 2013

Do antimatter apples fall up?

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by Dr. Robert E. Buxbaum,

The normal view of antimatter is that it’s just regular matter moving backwards in time. This view helps explain why antimatter has the same mass as regular matter, but has the opposite charge, spin, etc. An antiproton has the same mass as a proton because it is a proton. In our (forward) time-frame the anti-proton appears to be attracted by a positive plate and repelled by a negative one because, when you are going backward in time, attraction looks like repulsion.

In this view, the reason that antimatter particles annihilate when they come into contact with matter –sometimes– is that the annihilation is nothing more than the matter particle (or antimatter) switching direction in time. In our (forward) time-frame it looks like two particles met and both disappeared leaving nothing but photons (light). But in the time reversal view, shown in the figure below, there is only one normal matter particle. In the figure, this particle (solid line) comes from the left, and meets a photon, a wiggly line who’s arrow shows it traveling backwards in time. The normal proton then reverses in time, giving off a photon, another wiggly line. I’d alluded to this in my recent joke about an antimatter person at a bar, but there is also a famous poem.

proton-antiproton

This time reverse approach is best tested using entropy, the classical “arrow of time.” The best way to tell you can tell you are going forward in time is to drop an ice-cube into a hot cup of coffee and produce a warm cup of diluted coffee. This really only shows that you and the cup are moving in the same direction — both forward or both backward, something we’ll call forward. If you were moving in the opposite direction in time, e.g. you had a cup of anti-coffee that was moving backward in time relative to you, you could pull an anti -ice cube out of it, and produce a steaming cup of stronger anti-coffee.

We can not do the entropy test of time direction yet because it requires too much anti matter, but we can use another approach to test the time-reverse idea: gravity. You can make a very small drop of antimatter using only a few hundred atoms. If the antimatter drop is really going backwards in time, it should not fall on the floor and splatter, but should fly upward off the floor and coalesce. The Laboratory at CERN has just recently started producing enough atoms of anti-hydrogen to allow this test. So far the atoms are too hot but sometime in 2014 they expect to cool the atoms, some 300 atoms of anti hydrogen, into a drop or two. They will then see if the drop falls down or up in gravity. The temperature necessary for this study is about 1/100,000 of a degree K.

The anti-time view of antimatter is still somewhat controversial. For it to work, light must reside outside of time, or must move forward and backward in time with some ease. This makes some sense since light travels “at the speed of light,” and is thus outside of time. In the figure, the backwards moving photon would look like a forward on moving in the other direction (left). In a future post I hope to give instructions for building a simple, quantum time machine that uses the fact that light can move backwards in time to produce an event eraser — a device that erases light events in the present. It’s a somewhat useful device, if only for a science fair demonstration. Making one to work on matter would be much harder, and may be impossible if the CERN experiments don’t work out.

It becomes a little confusing how to deal with entropy in a completely anti-time world, and it’s somewhat hard to see why, in this view of time, there should be so little antimatter in the universe and so much matter: you’d expect equal amounts of both. As I have strong feelings for entropy, I’d posted a thought explanation for this some months ago imagining anti matter as normal forward-time matter, and posits the existence of an undiscovered particle that interacts with its magnetism to make matter more stable than antimatter. To see how it works, recall the brainteaser about a tribe that always speaks lies and another that always speaks truth. (I’m not the first to think of this explanation).

If the anti hydrogen drop at CERN is seen to fall upwards, but entropy still works in the positive direction as in my post (i.e. drops still splatter, and anti coffee cools like normal coffee), it will support a simple explanation for dark energy — the force that prevents the universe from collapsing. Dark energy could be seen to result from the antigravity of antimatter. There would have to be large collections of antimatter somewhere, perhaps anti-galaxies isolated from normal galaxies, that would push away the positive matter galaxies while moving forward in time and entropy. If the antigalaxies were close to normal galaxies they would annihilate at the edges, and we’d see lots of photons, like in the poem. Whatever they find at CERN, the future will be interesting. And if time travel turns out to be the norm, the past will be more interesting than it was.

Musical Color and the Well Tempered Scale

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by R. E. Buxbaum, (the author of all these posts)

I first heard J. S. Bach’s Well Tempered Clavier some 35 years ago and was struck by the different colors of the different scales. Some were dark and scary, others were light and enjoyable. All of them worked, but each was distinct, though I could not figure out why. That Bach was able to write in all the keys without retuning was a key innovation of his. In his day, people tuned in fifths, a process that created gaps (called wolf) that prevented useful composition in affected keys.

We don’t know exactly how Bach tuned his instruments as he had no scientific way to describe it; we can guess that it was more uniform than the temper produced by tuning in fifths, but it probably was not quite equally spaced. Nowadays electronic keyboards are tuned to 12 equally spaced frequencies per octave through the use of frequency counters.  Starting with the A below “middle C”, A4, tuned at 440 cycles/second (the note symphonies tune to), each note is programmed to vibrate at a wavelength that is lower or higher than one next to it by a factor of the twelfth root of two, 12√2= 1.05946. After 12 multiples of this size, the wavelength has doubled or halved and there is an octave. This is called equal tempering.

Currently, many non-electric instruments are also tuned this way.  Equally tempering avoids all wolf, but makes each note equally ill-tempered. Any key can be transposed to another, but there are no pure harmonies because 12√2 is an irrational number (see joke). There is also no color or feel to any given key except that which has carried over historically in the listeners’ memory. It’s sad.

I’m going to speculate that J.S. Bach found/ favored a way to tune instruments where all of the keys were usable, and OK sounding, but where some harmonies are more perfect than others. Necessarily this means that some harmonies will be less-perfect. There should be no wolf gaps that would sound so bad that Bach could not compose and transpose in every key, but since there is a difference, each key will retain a distinct color that JS Bach explored in his work — or so I’ll assume.

Pythagoras found that notes sound best together when the vibrating lengths are kept in a ratio of small numbers. Consider the tuning note, A4, the A below middle C; this note vibrates a column of air .784 meters long, about 2.5 feet or half the length of an oboe. The octave notes for A4 are called A3 and A5. They vibrate columns of air 2x as long and 1/2 as long as the original. They’re called octaves because they’re eight white keys away from A4. Keyboards add 4 black notes per octave so octaves are always 12 notes away. Keyboards are generally tuned so octaves are always 12 keys away. Based on Pythagoras, a reasonable presumption is that J.S Bach tuned every non-octave note so that it vibrates an air column similar to the equal tuning ratio, 12√2 = 1.05946, but whose wavelength was adjusted, in some cases to make ratios of small, whole numbers with the wavelength for A4.

Aside from octaves, the most pleasant harmonies are with notes whose wavelength is 3/2 as long as the original, or 2/3 as long. The best harmonies with A4 (0.784 m) will be with notes with wavelengths (3/2)*0.784 m long, or (2/3)*0.784m long. The first of these is called D3 and the other is E4. A4 combines with D3 to make a chord called D-major, the so-called “the key of glory.” The Hallelujah chorus, Beethoven’s 9th (Ode to Joy), and Mahler’s Titan are in this key. Scriabin believed that D-major had a unique color, gold, suggesting that the pure ratios were retained.

A combines with E (plus a black note C#) to make a chord called A major. Songs in this key sound (to my ear) robust, cheerful and somewhat pompous; Here, in A-major is: Dancing Queen by ABBA, Lady Madonna by the Beatles; Prelude and Fugue in A major by JS Bach. Scriabin believed that A-major was green.

A4 also combines with E and a new white note, C3, to make a chord called A minor. Since E4 and E3 vibrate at 2/3 and 4/3 the wavelength of A4 respectively, I’ll speculate that Bach tuned C3 to 5/3 the length of A4; 5/3*.0784m =1.307m long. Tuned his way, the ratio of wavelengths in the A minor chord are 3:4:5. Songs in A minor tend to be edgy and sort-of sad: Stairway to heaven, Für Elise, “Songs in A Minor sung by Alicia Keys, and PDQ Bach’s Fugue in A minor. I’m going to speculate the Bach tuned this to 1.312 m (or thereabouts), roughly half-way between the wavelength for a pure ratio and that of equal temper.

The notes D3 and E3 will not sound particularly good together. In both pure ratios and equal tempers their wavelengths are in a ratio of 3/2 to 4/3, that is a ratio of 9 to 8. This can be a tensional transition, but it does not provide a satisfying resolution to my, western ears.

Now for the other white notes. The next white key over from A4 is G3, two half-tones longer that for A4. For equal tuning, we’d expect this note to vibrate a column of air 1.059462 = 1.1225 times longer than A4. The most similar ratio of small whole numbers is 9/8 = 1.1250, and we’d already generated one before between D and E. As a result, we may expect that Bach tuned G3 to a wavelength 9/8*0.784m = .88 meters.

For equal tuning, the next white note, F3, will vibrate an air column 1.059464 = 1.259 times as long as the A4 column. Tuned this way, the wavelength for F3 is 1.259*.784 = .988m. Alternately, since 1.259 is similar to 5/4 = 1.25, it is reasonable to tune F3 as (5/4)*.784 = .980m. I’ll speculate that he split the difference: .984m. F, A, and C combine to make a good harmony called the F major chord. The most popular pieces in F major sound woozy and not-quite settled in my opinion, perhaps because of the oddness of the F tuning. See, e.g. the Jeopardy theme song, “My Sweet Lord,” Come together (Beetles), Beethoven’s Pastoral symphony (Movement 1, “Awakening of cheerful feelings upon arrival in the country”). Scriabin saw F-major as bright blue.

We’ve only one more white note to go in this octave: B4, the other tension note to A4. Since the wavelengths for G3 was 9/8 as long as for A4, we can expect the wavelength for B4 will be 8/9 as long. This will be dissonant to A4, but it will go well with E3 and E4 as these were 2/3 and 4/3 of A4 respectively. Tuned this way, B4 vibrates a column 1.40 m. When B, in any octave, is combined with E it’s called an E chord (E major or E minor); it’s typically combined with a black key, G-sharp (G#). The notes B, E vibrate at a ratio of 4 to 3. J.S. Bach called the G#, “H” allowing him to spell out his name in his music. When he played the sequence BACH, he found B to A created tension; moving to C created harmony with A, but not B, while the final note, G# (H) provided harmony for C and the original B. Here’s how it works on cello; it’s not bad, but there is no grand resolution. The Promenade from “Pictures at an Exhibition” is in E.

The black notes go somewhere between the larger gaps of the white notes, and there is a traditional confusion in how to tune them. One can tune the black notes by equal temper  (multiples of 21/12), or set them exactly in the spaces between the white notes, or tune them to any alternate set of ratios. A popular set of ratios is found in “Just temper.” The black note 6 from A4 (D#) will have wavelength of 0.784*26/12= √2 *0.784 m =1.109m. Since √2 =1.414, and that this is about 1.4= 7/5, the “Just temper” method is to tune D# to 1.4*.784m =1.098m. If one takes this route, other black notes (F#3 and C#3) will be tuned to ratios of 6/5, and 8/5 times 0.784m respectively. It’s possible that J.S. Bach tuned his notes by Just temper, but I suspect not. I suspect that Bach tuned these notes to fall in-between Just Temper and Equal temper, as I’ve shown below. I suspect that his D#3 might vibrated at about 1.104 m, half way between Just and Equal temper. I would not be surprised if Jazz musicians tuned their black notes more closely to the fifths of Just temper: 5/5 6/5, 7/5, 8/5 (and 9/5?) because jazz uses the black notes more, and you generally want your main chords to sound in tune. Then again, maybe not. Jimmy Hendrix picked the harmony D#3 with A (“Diabolus”, the devil harmony) for his Purple Haze; it’s also used for European police sirens.

To my ear, the modified equal temper is more beautiful and interesting than the equal temperament of todays electronic keyboards. In either temper music plays in all keys, but with an un-equal temper each key is distinct and beautiful in its own way. Tuning is engineering, I think, rather than math or art. In math things have to be perfect; in art they have to be interesting, and in engineering they have to work. Engineering tends to be beautiful its way. Generally, though, engineering is not perfect.

Summary of air column wave-lengths, measured in meters, and as a ratio to that for A4. Just Tempering, Equal Tempering, and my best guess of J.S. Bach's Well Tempered scale.

Summary of air column wave-lengths, measured in meters, and as a ratio to that for A4. Just Tempering, Equal Tempering, and my best guess of J.S. Bach’s Well Tempered scale.

R.E. Buxbaum, May 20 2013 (edited Sept 23, 2013) — I’m not very musical, but my children are.

Chaos, Stocks, and Global Warming

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Two weeks ago, I discussed black-body radiation and showed how you calculate the rate of radiative heat transfer from any object. Based on this, I claimed that basal metabolism (the rate of calorie burning for people at rest) was really proportional to surface area, not weight as in most charts. I also claimed that it should be near-impossible to lose weight through exercise, and went on to explain why we cover the hot parts of our hydrogen purifiers and hydrogen generators in aluminum foil.

I’d previously discussed chaos and posted a chart of the earth’s temperature over the last 600,000 years. I’d now like to combine these discussions to give some personal (R. E. Buxbaum) thoughts on global warming.

Black-body radiation differs from normal heat transfer in that the rate is proportional to emissivity and is very sensitive to temperature. We can expect the rate of heat transfer from the sun to earth will follow these rules, and that the rate from the earth will behave similarly.

That the earth is getting warmer is seen as proof that the carbon dioxide we produce is considered proof that we are changing the earth’s emissivity so that we absorb more of the sun’s radiation while emitting less (relatively), but things are not so simple. Carbon dioxide should, indeed promote terrestrial heating, but a hotter earth should have more clouds and these clouds should reflect solar radiation, while allowing the earth’s heat to radiate into space. Also, this model would suggest slow, gradual heating beginning, perhaps in 1850, but the earth’s climate is chaotic with a fractal temperature rise that has been going on for the last 15,000 years (see figure).

Recent temperature variation as measured from the Greenland Ice. A previous post had the temperature variation over the past 600,000 years.

Recent temperature variation as measured from the Greenland Ice. Like the stock market, it shows aspects of chaos.

Over a larger time scale, the earth’s temperature looks, chaotic and cyclical (see the graph of global temperature in this post) with ice ages every 120,000 years, and chaotic, fractal variation at times spans of 100 -1000 years. The earth’s temperature is self-similar too; that is, its variation looks the same if one scales time and temperature. This is something that is seen whenever a system possess feedback and complexity. It’s seen also in the economy (below), a system with complexity and feedback.

Manufacturing Profit is typically chaotic -- something that makes it exciting.

Manufacturing Profit is typically chaotic — and seems to have cold spells very similar to the ice ages seen above.

The economy of any city is complex, and the world economy even more so. No one part changes independent of the others, and as a result we can expect to see chaotic, self-similar stock and commodity prices for the foreseeable future. As with global temperature, the economic data over a 10 year scale looks like economic data over a 100 year scale. Surprisingly,  the economic data looks similar to the earth temperature data over a 100 year or 1000 year scale. It takes a strange person to guess either consistently as both are chaotic and fractal.

gomez3

It takes a rather chaotic person to really enjoy stock trading (Seen here, Gomez Addams of the Addams Family TV show).

Clouds and ice play roles in the earth’s feedback mechanisms. Clouds tend to increase when more of the sun’s light heats the oceans, but the more clouds, the less heat gets through to the oceans. Thus clouds tend to stabilize our temperature. The effect of ice is to destabilize: the more heat that gets to the ice, the more melts and the less of the suns heat is reflected to space. There is time-delay too, caused by the melting flow of ice and ocean currents as driven by temperature differences among the ocean layers, and (it seems) by salinity. The net result, instability and chaos.

The sun has chaotic weather too. The rate of the solar reactions that heat the earth increases with temperature and density in the sun’s interior: when a volume of the sun gets hotter, the reaction rates pick up making the volume yet-hotter. The temperature keeps rising, and the heat radiated to the earth keeps increasing, until a density current develops in the sun. The hot area is then cooled by moving to the surface and the rate of solar output decreases. It is quite likely that some part of our global temperature rise derives from this chaotic variation in solar output. The ice caps of Mars are receding.

The change in martian ice could be from the sun, or it might be from Martian dust in the air. If so, it suggests yet another feedback system for the earth. When economic times age good we have more money to spend on agriculture and air pollution control. For all we know, the main feedback loops involve dust and smog in the air. Perhaps, the earth is getting warmer because we’ve got no reflective cloud of dust as in the dust-bowl days, and our cities are no longer covered by a layer of thick, black (reflective) smog. If so, we should be happy to have the extra warmth.

Religion vs Philosophy joke

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“A philosopher is a blind man in a dark room looking for a black cat that isn’t there. A theologian is the man who finds it.” ~ H. L. Mencken

The distinction joke here is more sad than funny, I would say. It speaks to the inability of people to grapple with the big questions of their life in any really rational way. We’d like to be able to communicate directly with God, and have him speak back, but we can’t quite, and at some level we’d be too small for the interaction. We’d like to be able to stop evil with our religion, by holding up a cross, say, or by squirting holy water, but we can’t. I suspect it’s better that way, but sad. We’d like to know how and why the universe came to be, and what happens after death, but our best rational efforts are helpless. All of this is as they should be, says the philosopher, and he’s right, but it’s sad that it is and that he is. And then the theologian (rabbi, priest, imam) says he’s got all the answers and all the powers too. It’s too sad for words.

The philosopher in this joke is (I imagine) a PhD scientist, like me. While rational thought is great, and a PhD scientist can actually predict quite a lot that will happen in some cases, we have no real clue as to why things happen — except in terms of other things that we can’t explain: forces, gravity, electrons. It seems clear that the answer to the big-issue questions can not be found in science or rational philosophy. Nor can science deal well with one-time events like the creation of the universe, or unmeasurable items like where the apparent zero-point energy of quantum mechanics comes from. Untestable, one time events are the basis of religion and not science: science is the opposite of religion.

We thus turn to the theologian. In a sense, he has the answer: it’s God, Jesus, Jihad, prayer… Perhaps these words mean the same thing, or perhaps something different. A theologian can talk about this for hours. He has all the answers, but when he’s done, he’s left them as incomprehensible as before. Likely he is as confused as we are, but he doesn’t know it, or show it. While something like God does seem to underly the concept of time, or creation (the big bang), a one-word answer, like “God” isn’t really an answer. Even though there appears to be a God, God doesn’t seem contained within the word — he’s not there. And calling “God” doesn’t give us the power we’d want: it does not save the drowning, or cure disease.

Though the theologian will likely tell you miracle stories, and show you a pretty picture: long-haired Jesus, seated Zeus, or a dancing woman with the head of an elephant, that’s God and it isn’t. The reason people believe the theologian, is optimism: we hope he knows, though we know he doesn’t. Besides, the theologian has a costume and an audience, and that helps. He keeps on talking till he wears the audience down. Eventually we believe he sees the black cat in the dark room called God. Eventually we don’t care that he can’t do anything on the physical plane. Theologians work in pairs to increase their believability: one tells you the other is much smarter and holier than you; the other one tells you the same about the first. Eventually, you believe them both — or at least you believe you are stupider and eviler than they are.

A wise and good philosopher or theologian is very hard to find. He doesn’t talk too much, and instead lets his fine example do the teaching. He does charity and justice (Gen. 18:18) and makes good lemonade from the lemons life gives him. He will admit that he doesn’t really know which set of words and bows actually open up God’s warehouse (or if any are particularly effective) “God speaks within a cloud” (Ex. 40:34, etc.); “[His] thoughts are not our thoughts,” (Is. 55:8, etc.). “No man can see my face and live” (Ex. 33:20).

What percentage of leaders are like this? “In a thousand, I have found one leader of men”, says Solomon (Eccles 7:28). “The other 999 follow after the women” (Groucho Marx).

My hope with this blog post is not to diminish the good of rabbis, priests, or other theologians, but rather that you will not finish reading the post thinking you are stupid or evil for not understanding your theologian’s many words. Also, I can hope that you will seek justice, help the downtrodden, and make yourself into something of value. Then again, you might be tempted to run off to a bad theologian — to someone who will encourage you to pray long and hard, and who will get you to pay him for a picture of God that only he can provide — that is, for his special picture of the black cat, in the dark room, that can never be photographed.

Robert E. Buxbaum; Amateur philosopher, and maker of a good glass of lemonade.

My steam-operated, high pressure pump

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Here’s a miniature version of a duplex pump that we made 2-3 years ago at REB Research as a way to pump fuel into hydrogen generators for use with fuel cells. The design is from the 1800s. It was used on tank locomotives and steamboats to pump water into the boiler using only the pressure in the boiler itself. This seems like magic, but isn’t. There is no rotation, but linear motion in a steam piston of larger diameter pushes a liquid pump piston with a smaller diameter. Each piston travels the same distance, but there is more volume in the steam cylinder. The work from the steam piston is greater: W = ∫PdV; energy is conserved, and the liquid is pumped to higher pressure than the driving steam (neat!).

The following is a still photo. Click on the YouTube link to see the steam pump in action. It has over 4000 views!

Mini duplex pump. Provides high pressure water from steam power. Amini version of a classic of the 1800s Coffee cup and pen shown for scale.

Mini duplex pump. Provides high pressure water from steam power. A mini version of a classic of the 1800s Coffee cup and pen shown for scale.

You can get the bronze casting and the plans for this pump from Stanley co (England). Any talented machinist should be able to do the rest. I hired an Amish craftsman in Ohio. Maurice Perlman did the final fit work in our shop.

Our standard line of hydrogen generators still use electricity to pump the methanol-water. Even our latest generators are meant for nom-mobile applications where electricity is awfully convenient and cheap. This pump was intended for a future customer who would need to generate hydrogen to make electricity for remote and mobile applications. Even our non-mobile hydrogen is a better way to power cars than batteries, but making it mobile has advantages. Another advance would be to heat the reactors by burning the waste gas (I’ve been working on that too, and have filed a patent). Sometimes you have to build things ahead of finding a customer — and this pump was awfully cool.