Category Archives: thermodynamics

Much of the chemistry you learned is wrong

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When you were in school, you probably learned that understanding chemistry involved understanding the bonds between atoms. That all the things of the world were made of molecules, and that these molecules were fixed proportion combinations of the chemical elements held together by one of the 2 or 3 types of electron-sharing bonds. You were taught that water was H2O, that table salt was NaCl, that glass was SIO2, and rust was Fe2O3, and perhaps that the bonds involved an electron transferring between an electron-giver: H, Na, Si, or Fe… to an electron receiver: O or Cl above.

Sorry to say, none of that is true. These are fictions perpetrated by well-meaning, and sometime ignorant teachers. All of the materials mentioned above are grand polymers. Any of them can have extra or fewer atoms of any species, and as a result the stoichiometry isn’t quite fixed. They are not molecules at all in the sense you knew them. Also, ionic bonds hardly exist. Not in any chemical you’re familiar with. There are no common electron compounds. The world works, almost entirely on covalent, shared bonds. If bonds were ionic you could separate most materials by direct electrolysis of the pure compound, but you can not. You can not, for example, make iron by electrolysis of rust, nor can you make silicon by electrolysis of pure SiO2, or titanium by electrolysis of pure TiO. If you could, you’d make a lot of money and titanium would be very cheap. On the other hand, the fact that stoichiometry is rarely fixed allows you to make many useful devices, e.g. solid oxide fuel cells — things that should not work based on the chemistry you were taught.

Iron -zinc forms compounds, but they don't have fixed stoichiometry. As an example the compound at 60 atom % Zn is, I guess Zn3Fe2, but the composition varies quite a bit from there.

Iron -zinc forms compounds, but they don’t have fixed stoichiometry. As an example the compound at 68-80 atom% Zn is, I guess Zn7Fe3 with many substituted atoms, especially at temperatures near 665°C.

Because most bonds are covalent many compounds form that you would not expect. Most metal pairs form compounds with unusual stoicheometric composition. Here, for example, is the phase diagram for zinc and Iron –the materials behind galvanized sheet metal: iron that does not rust readily. The delta phase has a composition between 85 and 92 atom% Zn (8 and 15 a% iron): Perhaps the main compound is Zn5Fe2, not the sort of compound you’d expect, and it has a very variable compositions.

You may now ask why your teachers didn’t tell you this sort of stuff, but instead told you a pack of lies and half-truths. In part it’s because we don’t quite understand this ourselves. We don’t like to admit that. And besides, the lies serve a useful purpose: it gives us something to test you on. That is, a way to tell if you are a good student. The good students are those who memorize well and spit our lies back without asking too many questions of the wrong sort. We give students who do this good grades. I’m going to guess you were a good student (congratulations, so was I). The dullards got confused by our explanations. They asked too many questions, and asked, “can you explain that again? Or why? We get mad at these dullards and give them low grades. Eventually, the dullards feel bad enough about themselves to allow themselves to be ruled by us. We graduates who are confident in our ignorance rule the world, but inventions come from the dullards who don’t feel bad about their ignorance. They survive despite our best efforts. A few more of these folks survive in the west, and especially in America, than survive elsewhere. If you’re one, be happy you live here. In most countries you’d be beheaded.

Back to chemistry. It’s very difficult to know where to start to un-teach someone. Lets start with EMF and ionic bonds. While it is generally easier to remove an electron from a free metal atom than from a free non-metal atom, e.g. from a sodium atom instead of oxygen, removing an electron is always energetically unfavored, for all atoms. Similarly, while oxygen takes an extra electron easier than iron would, adding an electron is energetically unfavored. The figure below shows the classic ion bond, left, and two electron sharing options (center right) One is a bonding option the other anti-bonding. Nature prefers this to electron sharing to ionic bonds, even with blatantly ionic elements like sodium and chlorine.

Bond options in NaCl. Note that covalent is the stronger bond option though it requires less ionization.

Bond options in NaCl. Note that covalent is the stronger bond option though it requires less ionization.

There is a very small degree of ionic bonding in NaCl (left picture), but in virtually every case, covalent bonds (center) are easier to form and stronger when formed. And then there is the key anti-bonding state (right picture). The anti bond is hardly ever mentioned in high school or college chemistry, but it is critical — it’s this bond that keeps all mater from shrinking into nothingness.

I’ve discussed hydrogen bonds before. I find them fascinating since they make water wet and make life possible. I’d mentioned that they are just like regular bonds except that the quantum hydrogen atom (proton) plays the role that the electron plays. I now have to add that this is not a transfer, but a covalent spot. The H atom (proton) divides up like the electron did in the NaCl above. Thus, two water molecules are attracted by having partial bits of a proton half-way between the two oxygen atoms. The proton does not stay put at the center, there, but bobs between them as a quantum cloud. I should also mention that the hydrogen bond has an anti-bond state just like the electron above. We were never “taught” the hydrogen bond in high school or college — fortunately — that’s how I came to understand them. My professors, at Princeton saw hydrogen atoms as solid. It was their ignorance that allowed me to discover new things and get a PhD. One must be thankful for the folly of others: without it, no talented person could succeed.

And now I get to really weird bonds: entropy bonds. Have you ever noticed that meat gets softer when its aged in the freezer? That’s because most of the chemicals of life are held together by a sort of anti-bond called entropy, or randomness. The molecules in meat are unstable energetically, but actually increase the entropy of the water around them by their formation. When you lower the temperature you case the inherent instability of the bonds to cause them to let go. Unfortunately, this happens only slowly at low temperatures so you’ve got to age meat to tenderize it.

A nice thing about the entropy bond is that it is not particularly specific. A consequence of this is that all protein bonds are more-or-less the same strength. This allows proteins to form in a wide variety of compositions, but also means that deuterium oxide (heavy water) is toxic — it has a different entropic profile than regular water.

Robert Buxbaum, March 19, 2015. Unlearning false facts one lie at a time.

The speed of sound, Buxbaum’s correction

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Ernst Mach showed that sound must travel at a particular speed through any material, one determined by the conservation of energy and of entropy. At room temperature and 1 atm, that speed is theoretically predicted to be 343 m/s. For a wave to move at any other speed, either the laws of energy conservation would have to fail, or ∆S ≠ 0 and the wave would die out. This is the only speed where you could say there is a traveling wave, and experimentally, this is found to be the speed of sound in air, to good accuracy.

Still, it strikes me that Mach’s assumptions may have been too restrictive for short-distance sound waves. Perhaps there is room for other sound speeds if you allow ∆S > 0, and consider sound that travels short distances and dies out far from the source. Waves at these, other speeds might affect music appreciation, or headphone design. As these waves were never treated in my thermodynamics textbooks, I wondered if I could derive their speed in any nice way, and if they were faster or slower than the main wave? (If I can’t use this blog to re-think my college studies, what good is it?)

I t can help to think of a shock-wave of sound wave moving down a constant area tube of still are at speed u, with us moving along at the same speed as the wave. In this view, the wave appears stationary, but there is a wind of speed u approaching it from the left.

Imagine the sound-wave moving to the right, down a constant area tube at speed u, with us moving along at the same speed. Thus, the wave appears stationary, with a wind of speed u from the right.

As a first step to trying to re-imagine Mach’s calculation, here is one way to derive the original, for ∆S = 0, speed of sound: I showed in a previous post that the entropy change for compression can be imagines to have two parts, a pressure part at constant temperature: dS/dV at constant T = dP/dT at constant V. This part equals R/V for an ideal gas. There is also a temperature at constant volume part of the entropy change: dS/dT at constant V = Cv/T. Dividing the two equations, we find that, at constant entropy, dT/dV = RT/CvV= P/Cv. For a case where ∆S>0, dT/dV > P/Cv.

Now lets look at the conservation of mechanical energy. A compression wave gives off a certain amount of mechanical energy, or work on expansion, and this work accelerates the gas within the wave. For an ideal gas the internal energy of the gas is stored only in its temperature. Lets now consider a sound wave going down a tube flow left to right, and lets our reference plane along the wave at the same speed so the wave seems to sit still while a flow of gas moves toward it from the right at the speed of the sound wave, u. For this flow system energy is concerned though no heat is removed, and no useful work is done. Thus, any change in enthalpy only results in a change in kinetic energy. dH = -d(u2)/2 = u du, where H here is a per-mass enthalpy (enthalpy per kg).

dH = TdS + VdP. This can be rearranged to read, TdS = dH -VdP = -u du – VdP.

We now use conservation of mass to put du into terms of P,V, and T. By conservation of mass, u/V is constant, or d(u/V)= 0. Taking the derivative of this quotient, du/V -u dV/V2= 0. Rearranging this, we get, du = u dV/V (No assumptions about entropy here). Since dH = -u du, we say that u2 dV/V = -dH = -TdS- VdP. It is now common to say that dS = 0 across the sound wave, and thus find that u2 = -V2 (dP/dV) at const S. For an ideal gas, this last derivative equals, PCp/VCv, so the speed of sound, u= √PVCp/Cv with the volume in terms of mass (kg/m3).

The problem comes in where we say that ∆S>0. At this point, I would say that u2 = -V(dH/dV) = VCp dT/dV > PVCp/Cv. Unless, I’ve made a mistake (always possible), I find that there is a small leading, non-adiabatic sound wave that goes ahead of the ordinary sound wave and is experienced only close to the source caused by mechanical energy that becomes degraded to raising T and gives rise more compression than would be expected for iso-entropic waves.

This should have some relevance to headphone design and speaker design since headphones are heard close to the ear, while speakers are heard further away. Meanwhile the recordings are made by microphones right next to the singers or instruments.

Robert E. Buxbaum, August 26, 2014

Dr. Who’s Quantum reality viewed as diffusion

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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.

Time as a ball of wibblely wobbly 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/dx2                (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.

If hot air rises, why is it cold on mountain-tops?

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This is a child’s question that’s rarely answered to anyone’s satisfaction. To answer it well requires college level science, and by college the child has usually been dissuaded from asking anything scientific that would likely embarrass teacher — which is to say, from asking most anything. By a good answer, I mean here one that provides both a mathematical, checkable prediction of the temperature you’d expect to find on mountain tops, and one that also gives a feel for why it should be so. I’ll try to provide this here, as previously when explaining “why is the sky blue.” A word of warning: real science involves mathematics, something that’s often left behind, perhaps in an effort to build self-esteem. If I do a poor job, please text me back: “if hot air rises, what’s keeping you down?”

As a touchy-feely answer, please note that all materials have internal energy. It’s generally associated with the kinetic energy + potential energy of the molecules. It enters whenever a material is heated or has work done on it, and for gases, to good approximation, it equals the gas heat capacity of the gas times its temperature. For air, this is about 7 cal/mol°K times the temperature in degrees Kelvin. The average air at sea-level is taken to be at 1 atm, or 101,300  Pascals, and 15.02°C, or 288.15 °K; the internal energy of this are is thus 288.15 x 7 = 2017 cal/mol = 8420 J/mol. The internal energy of the air will decrease as the air rises, and the temperature drops for reasons I will explain below. Most diatomic gases have heat capacity of 7 cal/mol°K, a fact that is only explained by quantum mechanics; if not for quantum mechanics, the heat capacities of diatomic gases would be about 9 cal/mol°K.

Lets consider a volume of this air at this standard condition, and imagine that it is held within a weightless balloon, or plastic bag. As we pull that air up, by pulling up the bag, the bag starts to expand because the pressure is lower at high altitude (air pressure is just the weight of the air). No heat is exchanged with the surrounding air because our air will always be about as warm as its surroundings, or if you like you can imagine weightless balloon prevents it. In either case the molecules lose energy as the bag expands because they always collide with an outwardly moving wall. Alternately you can say that the air in the bag is doing work on the exterior air — expansion is work — but we are putting no work into the air as it takes no work to lift this air. The buoyancy of the air in our balloon is always about that of the surrounding air, or so we’ll assume for now.

A classic, difficult way to calculate the temperature change with altitude is to calculate the work being done by the air in the rising balloon. Work done is force times distance: w=  ∫f dz and this work should equal the effective cooling since heat and work are interchangeable. There’s an integral sign here to account for the fact that force is proportional to pressure and the air pressure will decrease as the balloon goes up. We now note that w =  ∫f dz = – ∫P dV because pressure, P = force per unit area. and volume, V is area times distance. The minus sign is because the work is being done by the air, not done on the air — it involves a loss of internal energy. Sorry to say, the temperature and pressure in the air keeps changing with volume and altitude, so it’s hard to solve the integral, but there is a simple approach based on entropy, S.

Les Droites Mountain, in the Alps, at the intersect of France Italy and Switzerland is 4000 m tall. The top is generally snow-covered.

Les Droites Mountain, in the Alps, at the intersect of France Italy and Switzerland is 4000 m tall. The top is generally snow-covered.

I discussed entropy last month, and showed it was a property of state, and further, that for any reversible path, ∆S= (Q/T)rev. That is, the entropy change for any reversible process equals the heat that enters divided by the temperature. Now, we expect the balloon rise is reversible, and since we’ve assumed no heat transfer, Q = 0. We thus expect that the entropy of air will be the same at all altitudes. Now entropy has two parts, a temperature part, Cp ln T2/T1 and a pressure part, R ln P2/P1. If the total ∆S=0 these two parts will exactly cancel.

Consider that at 4000m, the height of Les Droites, a mountain in the Mont Blanc range, the typical pressure is 61,660 Pa, about 60.85% of sea level pressure (101325 Pa). If the air were reduced to this pressure at constant temperature (∆S)T = -R ln P2/P1 where R is the gas constant, about 2 cal/mol°K, and P2/P1 = .6085; (∆S)T = -2 ln .6085. Since the total entropy change is zero, this part must equal Cp ln T2/T1 where Cp is the heat capacity of air at constant pressure, about 7 cal/mol°K for all diatomic gases, and T1 and T2 are the temperatures (Kelvin) of the air at sea level and 4000 m. (These equations are derived in most thermodynamics texts. The short version is that the entropy change from compression at constant T equals the work at constant temperature divided by T,  ∫P/TdV=  ∫R/V dV = R ln V2/V1= -R ln P2/P1. Similarly the entropy change at constant pressure = ∫dQ/T where dQ = Cp dT. This component of entropy is thus ∫dQ/T = Cp ∫dT/T = Cp ln T2/T1.) Setting the sum to equal zero, we can say that Cp ln T2/T1 =R ln .6085, or that 

T2 = T1 (.6085)R/Cp

T2 = T1(.6085)2/7   where 0.6065 is the pressure ratio at 4000, and because for air and most diatomic gases, R/Cp = 2/7 to very good approximation, matching the prediction from quantum mechanics.

From the above, we calculate T2 = 288.15 x .8676 = 250.0°K, or -23.15 °C. This is cold enough to provide snow  on Les Droites nearly year round, and it’s pretty accurate. The typical temperature at 4000 m is 262.17 K (-11°C). That’s 26°C colder than at sea-level, and only 12°C warmer than we’d predicted.

There are three weak assumptions behind the 11°C error in our predictions: (1) that the air that rises is no hotter than the air that does not, and (2) that the air’s not heated by radiation from the sun or earth, and (3) that there is no heat exchange with the surrounding air, e.g. from rain or snow formation. The last of these errors is thought to be the largest, but it’s still not large enough to cause serious problems.

The snow cover on Kilimanjaro, 2013. If global warming models were true, it should be gone, or mostly gone.

Snow on Kilimanjaro, Tanzania 2013. If global warming models were true, the ground should be 4°C warmer than 100 years ago, and the air at this altitude, about 7°C (12°F) warmer; and the snow should be gone.

You can use this approach, with different exponents, estimate the temperature at the center of Jupiter, or at the center of neutron stars. This iso-entropic calculation is the model that’s used here, though it’s understood that may be off by a fair percentage. You can also ask questions about global warming: increased CO2 at this level is supposed to cause extreme heating at 4000m, enough to heat the earth below by 4°C/century or more. As it happens, the temperature and snow cover on Les Droites and other Alp ski areas has been studied carefully for many decades; they are not warming as best we can tell (here’s a discussion). By all rights, Mt Blanc should be Mt Green by now; no one knows why. The earth too seems to have stopped warming. My theory: clouds. 

Robert Buxbaum, May 10, 2014. Science requires you check your theory for internal and external weakness. Here’s why the sky is blue, not green.

Getting rid of hydrogen

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Though most of my company’s business is making hydrogen or purifying it, or consulting about it, we also provide sorbers and membranes that allow a customer to get rid of unwanted hydrogen, or remove it from a space where it is not wanted. A common example is a customer who has a battery system for long-term operation under the sea, or in space. The battery or the metal containment is then found to degas hydrogen, perhaps from a corrosion reaction. The hydrogen may interfere with his electronics, or the customer fears it will reach explosive levels. In one case the customer’s system was monitoring deep oil wells and hydrogen from the well was messing up its fiber optic communications.

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Pd-coated niobium screws used to getter hydrogen from electronic packages.

For many of these problems, the simplest solution is an organic hydrogen getter of palladium-catalyst and a labile unsaturated hydrocarbon, e.g. buckminsterfullerene. These hydrogen getters are effective in air or inert gas at temperatures between about -20°C and 150°C. When used in an inert gas the organic is hydrogenated, there is a finite amount of removal per gram of sober. When used in air the catalyst promotes the water-forming reaction, and thus there is a lot more hydrogen removal. Depending on the organic, we can provide gettering to lower temperatures or higher. We’ve a recent patent on an organo-palladium gel to operate to 300°C, suitable for down-well hydrogen removal.

At high temperatures, generally above 100*C, we generally suggest an inorganic hydrogen remover, e.g. our platinum ceria catalyst. This material is suitable for hydrogen removal from air, including from polluted air like that in radioactive waste storage areas. Platinum catalyst works long-term at temperatures between about 0°C and 600°C. The catalyst-sorber also works without air, reducing Ce2O3 to CeO and converting hydrogen irreversibly to water (H2O). As with the organo-Pd getters, there is a finite amount of hydrogen removal per gram when these materials are used in a sealed environment.

Low temperature, Pd-grey coated, Pd-Ag membranes made for the space shuttle to remove hydrogen from the drinking water at room temperature. The water came from the fuel cells.

Low temperature, metal membranes made for NASA to remove H2 from  drinking water at room temperature.

Another high temperature hydrogen removal option is metallic getters, e.g. yttrium or vanadium-titanium alloy. These metals require temperatures in excess of 100°C to be effective, and typically do not work well in air. They are best suited for removing hydrogen a vacuum or inert gas, converting it to metallic hydride. The thermodynamics of hydriding is such that, depending on the material, these getters can extract hydrogen even at temperatures up to 700°C, and at very low hydrogen pressures, below 10-9 torr. For operation in air or at 100-400°C we typically provide these getters coated with palladium to increase the hydrogen sorption rate. A fairly popular product is palladium-coated niobium screws 4-40 x 1/4″. Each screw will remove over 2000 sec of hydrogen at temperatures up to 400°C. We also provide oxygen, nitrogen and water getters. They work on the same principle, but form metallic oxides or nitrides instead of hydrides.

Our last, and highest-end, hydrogen-removal option is to provide metallic membranes. These don’t remove the hydrogen as such, but transfer it elsewhere. We’ve provided these for the space shuttle, and to the nuclear industry so that hydrogen can be vented from nuclear reactors before it has a chance to build up and case damage or interfere with heat transfer. Because nothing is used up, these membranes work, essentially forever. The Fukushima reactor explosions were from corrosion-produced hydrogen that had no acceptable way to vent.

Please contact us for more information, e.g. by phone at 248-545-0155, or check out the various sorbers in our web-site. Robert Buxbaum, May 5, 2014.

Entropy, the most important pattern in life

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One evening at the Princeton grad college a younger fellow (an 18-year-old genius) asked the most simple, elegant question I had ever heard, one I’ve borrowed and used ever since: “tell me”, he asked, “something that’s important and true.” My answer that evening was that the entropy of the universe is always increasing. It’s a fundamentally important pattern in life; one I didn’t discover, but discovered to have a lot of applications and meaning. Let me explain why it’s true here, and then why I find it’s meaningful.

Famous entropy cartoon, Harris

Famous entropy cartoon, Harris

The entropy of the universe is not something you can measure directly, but rather indirectly, from the availability of work in any corner of it. It’s related to randomness and the arrow of time. First off, here’s how you can tell if time is moving forward: put an ice-cube into hot water, if the cube dissolves and the water becomes cooler, time is moving forward — or, at least it’s moving in the same direction as you are. If you can reach into a cup of warm water and pull out an ice-cube while making the water hot, time is moving backwards. — or rather, you are living backwards. Within any closed system, one where you don’t add things or energy (sunlight say), you can tell that time is moving forward because the forward progress of time always leads to the lack of availability of work. In the case above, you could have generated some electricity from the ice-cube and the hot water, but not from the glass of warm water.

You can not extract work from a heat source alone; to extract work some heat must be deposited in a cold sink. At best the entropy of the universe remains unchanged. More typically, it increases.

You can not extract work from a heat source alone; to extract work some heat must be deposited in a cold sink. At best the entropy of the universe remains unchanged.

This observation is about as fundamental as any to understanding the world; it is the basis of entropy and the second law of thermodynamics: you can never extract useful work from a uniform temperature body of water, say, just by making that water cooler. To get useful work, you always need something some other transfer into or out of the system; you always need to make something else hotter, colder, or provide some chemical or altitude changes that can not be reversed without adding more energy back. Thus, so long as time moves forward everything runs down in terms of work availability.

There is also a first law; it states that energy is conserved. That is, if you want to heat some substance, that change requires that you put in a set amount of work plus heat. Similarly, if you want to cool something, a set amount of heat + work must be taken out. In equation form, we say that, for any change, q +w is constant, where q is heat, and w is work. It’s the sum that’s constant, not the individual values so long as you count every 4.174 Joules of work as if it were 1 calorie of heat. If you input more heat, you have to add less work, and visa versa, but there is always the same sum. When adding heat or work, we say that q or w is positive; when extracting heat or work, we say that q or w are negative quantities. Still, each 4.174 joules counts as if it were 1 calorie.

Now, since for every path between two states, q +w is the same, we say that q + w represents a path-independent quantity for the system, one we call internal energy, U where ∆U = q + w. This is a mathematical form of the first law of thermodynamics: you can’t take q + w out of nothing, or add it to something without making a change in the properties of the thing. The only way to leave things the same is if q + w = 0. We notice also that for any pure thing or mixture, the sum q +w for the change is proportional to the mass of the stuff; we can thus say that internal energy is an intensive quality. q + w = n ∆u where n is the grams of material, and ∆u is the change in internal energy per gram.

We are now ready to put the first and second laws together. We find we can extract work from a system if we take heat from a hot body of water and deliver some of it to something at a lower temperature (the ice-cube say). This can be done with a thermopile, or with a steam engine (Rankine cycle, above), or a stirling engine. That an engine can only extract work when there is a difference of temperatures is similar to the operation of a water wheel. Sadie Carnot noted that a water wheel is able to extract work only when there is a flow of water from a high level to low; similarly in a heat engine, you only get work by taking in heat energy from a hot heat-source and exhausting some of it to a colder heat-sink. The remainder leaves as work. That is, q1 -q2 = w, and energy is conserved. The second law isn’t violated so long as there is no way you could run the engine without the cold sink. Accepting this as reasonable, we can now derive some very interesting, non-obvious truths.

We begin with the famous Carnot cycle. The Carnot cycle is an idealized heat engine with the interesting feature that it can be made to operate reversibly. That is, you can make it run forwards, taking a certain amount of work from a hot source, producing a certain amount of work and delivering a certain amount of heat to the cold sink; and you can run the same process backwards, as a refrigerator, taking in the same about of work and the same amount of heat from the cold sink and delivering the same amount to the hot source. Carnot showed by the following proof that all other reversible engines would have the same efficiency as his cycle and no engine, reversible or not, could be more efficient. The proof: if an engine could be designed that will extract a greater percentage of the heat as work when operating between a given hot source and cold sink it could be used to drive his Carnot cycle backwards. If the pair of engines were now combined so that the less efficient engine removed exactly as much heat from the sink as the more efficient engine deposited, the excess work produced by the more efficient engine would leave with no effect besides cooling the source. This combination would be in violation of the second law, something that we’d said was impossible.

Now let us try to understand the relationship that drives useful energy production. The ratio of heat in to heat out has got to be a function of the in and out temperatures alone. That is, q1/q2 = f(T1, T2). Similarly, q2/q1 = f(T2,T1) Now lets consider what happens when two Carnot cycles are placed in series between T1 and T2, with the middle temperature at Tm. For the first engine, q1/qm = f(T1, Tm), and similarly for the second engine qm/q2 = f(Tm, T2). Combining these we see that q1/q2 = (q1/qm)x(qm/q2) and therefore f(T1, T2) must always equal f(T1, Tm)x f(Tm/T2) =f(T1,Tm)/f(T2, Tm). In this relationship we see that the second term Tm is irrelevant; it is true for any Tm. We thus say that q1/q2 = T1/T2, and this is the limit of what you get at maximum (reversible) efficiency. You can now rearrange this to read q1/T1 = q2/T2 or to say that work, W = q1 – q2 = q2 (T1 – T2)/T2.

A strange result from this is that, since every process can be modeled as either a sum of Carnot engines, or of engines that are less-efficient, and since the Carnot engine will produce this same amount of reversible work when filled with any substance or combination of substances, we can say that this outcome: q1/T1 = q2/T2 is independent of path, and independent of substance so long as the process is reversible. We can thus say that for all substances there is a property of state, S such that the change in this property is ∆S = ∑q/T for all the heat in or out. In a more general sense, we can say, ∆S = ∫dq/T, where this state property, S is called the entropy. Since as before, the amount of heat needed is proportional to mass, we can say that S is an intensive property; S= n s where n is the mass of stuff, and s is the entropy change per mass. 

Another strange result comes from the efficiency equation. Since, for any engine or process that is less efficient than the reversible one, we get less work out for the same amount of q1, we must have more heat rejected than q2. Thus, for an irreversible engine or process, q1-q2 < q2(T1-T2)/T2, and q2/T2 is greater than -q1/T1. As a result, the total change in entropy, S = q1/T1 + q2/T2 >0: the entropy of the universe always goes up or stays constant. It never goes down. Another final observation is that there must be a zero temperature that nothing can go below or both q1 and q2 could be positive and energy would not be conserved. Our observations of time and energy conservation leaves us to expect to find that there must be a minimum temperature, T = 0 that nothing can be colder than. We find this temperature at -273.15 °C. It is called absolute zero; nothing has ever been cooled to be colder than this, and now we see that, so long as time moves forward and energy is conserved, nothing will ever will be found colder.

Typically we either say that S is zero at absolute zero, or at room temperature.

We’re nearly there. We can define the entropy of the universe as the sum of the entropies of everything in it. From the above treatment of work cycles, we see that this total of entropy always goes up, never down. A fundamental fact of nature, and (in my world view) a fundamental view into how God views us and the universe. First, that the entropy of the universe goes up only, and not down (in our time-forward framework) suggests there is a creator for our universe — a source of negative entropy at the start of all things, or a reverser of time (it’s the same thing in our framework). Another observation, God likes entropy a lot, and that means randomness. It’s his working principle, it seems.

But before you take me now for a total libertine and say that since science shows that everything runs down the only moral take-home is to teach: “Let us eat and drink,”… “for tomorrow we die!” (Isaiah 22:13), I should note that his randomness only applies to the universe as a whole. The individual parts (planets, laboratories, beakers of coffee) does not maximize entropy, but leads to a minimization of available work, and this is different. You can show that the maximization of S, the entropy of the universe, does not lead to the maximization of s, the entropy per gram of your particular closed space but rather to the minimization of a related quantity µ, the free energy, or usable work per gram of your stuff. You can show that, for any closed system at constant temperature, µ = h -Ts where s is entropy per gram as before, and h is called enthalpy. h is basically the potential energy of the molecules; it is lowest at low temperature and high order. For a closed system we find there is a balance between s, something that increases with increased randomness, and h, something that decreases with increased randomness. Put water and air in a bottle, and you find that the water is mostly on the bottom of the bottle, the air is mostly on the top, and the amount of mixing in each phase is not the maximum disorder, but rather the one you’d calculate will minimize µ.

As the protein folds its randomness and entropy decrease, but its enthalpy decreases too; the net effect is one precise fold that minimizes µ.

As a protein folds its randomness and entropy decrease, but its enthalpy decreases too; the net effect is one precise fold that minimizes µ.

This is the principle that God applies to everything, including us, I’d guess: a balance. Take protein folding; some patterns have big disorder, and high h; some have low disorder and very low h. The result is a temperature-dependent  balance. If I were to take a moral imperative from this balance, I’d say it matches better with the sayings of Solomon the wise: “there is nothing better for a person under the sun than to eat, drink and be merry. Then joy will accompany them in their toil all the days of the life God has given them under the sun.” (Ecclesiastes 8:15). There is toil here as well as pleasure; directed activity balanced against personal pleasures. This is the µ = h -Ts minimization where, perhaps, T is economic wealth. Thus, the richer a society, the less toil is ideal and the more freedom. Of necessity, poor societies are repressive. 

Dr. Robert E. Buxbaum, Mar 18, 2014. My previous thermodynamic post concerned the thermodynamics of hydrogen production. It’s not clear that all matter goes forward in time, by the way; antimatter may go backwards, so it’s possible that anti matter apples may fall up. On microscopic scale, time becomes flexible so it seems you can make a time machine. Religious leaders tend to be anti-science, I’ve noticed, perhaps because scientific miracles can be done by anyone, available even those who think “wrong,” or say the wrong words. And that’s that, all being heard, do what’s right and enjoy life too: as important a pattern in life as you’ll find, I think. The relationship between free-energy and societal organization is from my thesis advisor, Dr. Ernest F. Johnson.

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.