Chutes And Landers

From: Robin Feder <rjfeder@fortleenj.com>
To: Sy Moire <sy@moirestudies.com>
Subj: Questions

Hello again, Mr Moire. Kalif and I have a question. We were talking about falling out of stuff and we wondered how high you have to fall out of to break every bone in your body. We asked our science teacher Mr Higgs and he said it was something that you or Randall Munroe could answer and besides he (Mr Higgs) had to get ready for his next online. Can you tell us? Sincerely, Robin Feder


From: Sy Moire <sy@moirestudies.com>
To: Robin Feder <rjfeder@fortleenj.com>
Subj: Re: Questions

Hello again, Robin. You do take after your Dad, don’t you? Please give my best to him and to Mr Higgs, who has a massive job. Mr Munroe may already have answered your question somewhere, but I’ll give it a shot.

You’ve assumed that the higher the fall, the harder the hit and the more bones broken. It’s not that simple. Suppose, for instance, that your fall is onto the Moon, whose gravity is 1/6 that of Earth. For any amount of impact, however high the fall would have been on Earth, it’d be six times higher on the Moon. So the answer depends where you’re falling.

But the Moon doesn’t have an atmosphere worth paying attention to. That’s important because atmospheres impose a speed limit, technically known as terminal velocity, that depends on a whole collection of things

  • the Mass of the falling object
  • the local strength of Gravity
  • the Density of the atmosphere
  • the object’s cross‑sectional Area in the direction of fall

The first two produce the downward pull of gravity, the others produce the upward push of air resistance. Fun fact — in Galileo’s “All things fall alike” experiments, he always used spheres in order to cancel the effects of air resistance in his comparisons.

Let’s put some numbers to it. Suppose someone’s at Earth’s “edge of space” 100 kilometers up. From the PE=m·g·h formula for gravitational potential energy and dividing out their mass which I don’t know, they have 9.8×105 joules/kilogram of potential energy relative to Earth’s surface. Now suppose they convert that potential to kinetic energy by falling to the surface with no air resistance. Using KE=m·v² I calculate they’d hit at about 1000 meters/second. But in real life, the terminal velocity of a falling human body is about 55 meters/second.

That Area item is why parachutes work. Make a falling object’s area larger and it’ll have to push aside more air molecules on its way down. Anyone wanting to survive a fall wants as much area as they can get. A parachute’s fabric canopy gives them a huge area and a big help. Parachute drops normally hit at about 5 meters/second. Trained people walk away from that all the time. Mostly.

Which gets to the matter of how you land. Parachute training schools and martial arts dojos give you the same advice — don’t try to stop your fall, just tuck in your chin and twist to convert vertical kinetic energy to rolling motion. Rigid limbs lead to bones breaking, ligaments tearing and joints going out of joint.

So let’s talk bones. Adults have about 210 of them, about 90 fewer than when they were a kid. Bones start out as separate bony patches embedded in cartilage. The patches eventually join together as boney tissue and the cartilage proportion decreases with age. Bottom line — kid bones are bendy, old bones snap more easily. For your question, breaking “every bone in your body” is a bigger challenge if you’re young.

But all bones aren’t equal — some are more vulnerable than others. Sesamoid bones, like the ones at the base of your thumb, are millimeter‑sized and embedded in soft tissue that protects them. The tiny “hammer, anvil and stirrup” ear bones are buried deep in hard bony tissue that protects them, too. Thanks to bones and soft tissues that would absorb nearly all the energy of impact, these small bones are almost invulnerable.

To summarize, no matter how high up from Earth you fall from, you can’t fall fast enough to hit hard enough to break every bone in your body. Be careful anyhow.

Regards,
Sy Moire.

~~ Rich Olcott

  • Thanks to Xander and Lucas for their input.

Elementary History

<chirp, chirp> “Moire here.”

“Hi, Sy, it’s Susan.”

“Well, hello. Good to hear from you. What’s up?”

“I’m out here on my back porch, fooling around on my laptop. It’s too nice to work in the lab today.”

“I agree with you. I’m outside, too, enjoying the Springtime. What’s your fooling around?”

“I found a discovery date list for all the chemical elements. Guess which element was the first that humanity worked with in pure form?”

“Mmm, I’d say carbon, in charcoal.”

“Nope, it’s copper.”

“Copper?”

“Mm-hm. Or maybe gold. They both occur as the raw metal but copper’s more common. There was a Copper Age before the Bronze age. The dates are fuzzy because they depend on what the archaeologists find after site scavengers have been there. I’m sending you the first few rows from the list.

Cumulative
Count
Element
(Symbol)
Atomic
number
Estimated
years ago
1Copper (Cu)2910000
2Lead (Pb)826000
3Carbon (C)65750
4Silver (Ag)475000
5Tin (Sn)505000
6Antimony (Sb)515000
7Gold (Au)794500
8Iron (Fe)264000
9Mercury (Hg)803500
10Sulfur (S)162500

“You can win most of them from the right ore with relatively simple processing. It makes sense they’re the ones we got to first.”

“Susan, I’m surprised it took a thousand years to realize you can get sulfur from cinnabar ore at the same time you’re cooking the mercury out of it. I wouldn’t want to be downwind from that process or most of the rest.”

“Sure not. I’ll bet there just wasn’t much interest in sulfur until the alchemists started playing with it. Anyhow, I dumped the element data into a spreadsheet and got some fun facts when I graphed it. Look at this. Eight thousand years for 10 elements through sulfur, then 1800 years of nothing. Arsenic doesn’t show up until the Thirteenth Century when I guess royalty started using it to poison each other. And phosphorus — have you read Neal Stephenson’s Baroque Cycle trilogy?”

“Yes, and I know the episode you’re thinking of, where the hero routed a gang at night by coating himself with glowing phosphorus and bursting out of a cave pretending to be a demon. Stephenson put a lot of words into describing how factories obtained mercury and phosphorus back then.”

“Stephenson puts a lot of words into most everything nerdy. That’s why I enjoy reading him. Oh-ho, now I know how you knew about cinnabar being the source for mercury.”

“Hey, Susan, I don’t only do Physics, but yeah, that was from another Baroque Cycle episode. … Looking at your graph here — things certainly took off at the start of the Eighteenth Century.”

“Yes, indeed. Seventy-four elements, everything that’s not radioactive plus a couple that are. I get a chuckle from cobalt being the first element in that wave after phosphorus. You know the story?”

“What story is that?”

“Seventeenth Century miners kept digging up nasty rocks that emitted poisonous gas when smelted along with the desirable copper and nickel ores. They called the bad stuff kobald Oren, German for ‘goblin ores.’ When a Swedish chemist finally purified the material he simply re-spelled the adjective and called the metal cobalt. I love the linkage with Stephenson’s fictional phosphorus-covered demon.”

“Cute. Why the break between rhenium and technitium?”

“That second wave after 1935 is all radioactives. Funny how the timing paralleled Seaborg’s research career even though he never got involved with technitium, the first artificially-produced element. Imagine being the discoverer of ten different elements.”

“Seaborg practically invented that funny bottom row of the Periodic Table, didn’t he?”

“Oh, yes. Not only did he discover or co-discover more than half of those elements, he was the one who proposed setting off the entire group as Actinides, in parallel with the Lanthanides above them. Oh, that reminds me, I meant to show you the other display I built. You’ve probably never seen one like this.”

“Whoa, you’ve colored each element block by how long we’ve known about it. That’s not the kind of thing you can do with crayons.”

“No, I had to do some programming to get the right tints.”

“What’s the little star for in the middle of the scale?”

“That’s when Mendeleev first proposed the Table, smack in the logarithmic center of my timeline. Don’t you love it?”

~~ Rich Olcott

Rotation, Revolution and The Answer

“Sy, I’m startin’ to think you got nothin’. Al and me, we ask what’s pushing the Moon away from us and you give us angular momentum and energy transfers. C’mon, stop dancin’ around and tell us the answer.”

“Yeah, Sy, gravity pulls things together, right, so how come the Moon doesn’t fall right onto us?”

“Not dancing, Vinnie, just laying some groundwork for you. Newton answered Al’s question — the Moon is falling towards us, but it’s going so fast it overshoots. That’s where momentum comes in, Vinnie. Newton showed that a ball shot from a cannon files further depending on how much momentum it gets from the initial kick. If you give it enough momentum, and set your cannon high enough that the ball doesn’t hit trees or mountains, the ball falls beyond the planet and keeps on falling forever in an elliptical orbit.”

“Forever until it hits the cannon.”

“hahaha, Al. Anyway, the ball achieves orbit by converting its linear momentum to angular momentum with the help of gravity. The angular momentum pretty much defines the orbit. In Newton’s gravity‑determined universe, momentum and position together let you predict everything.”

“Linear and angular momentum work the same way?”

“Mostly. There’s only one kind of linear momentum — straight ahead — but there are two kinds of angular momentum — rotation and revolution.”

“Aw geez, there’s another pair of words I can never keep straight.”

“You and lots of people, Vinnie. They’re synonyms unless you’re talking technicalese. In Physics and Astronomy, rotation with the O gyrates around an object’s own center, like a top or a planet rotating on its axis. Revolution with the E gyrates around some external location, like the planet revolving around its sun. Does that help?”

“Cool, that may come in handy. So Newton’s cannon ball got its umm, revolution angular momentum from linear momentum so where does rotation angular momentum come from?”

“Subtle question, Vinnie, but they’re actually all just momentum. Fair warning, I’m going to avoid a few issues that’d get us too far into the relativity weeds. Let’s just say that momentum is one of those conserved quantities. You can transfer momentum from one object to another and convert between forms of momentum, but you can’t create momentum in an isolated system.”

“That sounds a lot like energy, Sy.”

“You’re right, Al, the two are closely related. Newton thought that momentum was THE conserved quantity and all motion depended on it. His arch‑enemy Leibniz said THE conserved quantity was kinetic energy, which he called vis viva. That disagreement was just one battle in the Newton‑Leibniz war. It took science 200 years to understand the momentum/kinetic energy/potential energy triad.”

“Wait, Sy, I’ve seen NASA steer a rocketship and give it a whole different momentum. I don’t see no conservation.”

“You missed an important word, Vinnie — isolated. Momentum calculations apply to mechanical systems — no inputs of mass or non‑mechanical energy. Chemical or nuclear fuels break that rule and get you into a different game.”

“Ah-hahh, so if the Earth and Moon are isolated…”

“Exactly, and you’re way ahead of me. Like we said, no significant net forces coming from the Sun or Jupiter, so no change to our angular momentum.”

“Hey, wait, guys. Solar power. I know we’ve got a ton of sunlight coming in every day.”

“Not relevant, Al. Even though sunlight heats the Earth, mass and momentum aren’t affected by temperature. Anyhow, we’re finally at the point where I can answer your question.”

“About time.”

“Hush. OK, here’s the chain. Earth rotates beneath the Moon and gets its insides stirred up by the Moon’s gravity. The stirring is kinetic energy extracted from the energy of the Earth‑Moon system. The Moon’s revolution or the Earth’s rotation or both must slow down. Remember the M=m·r·c/t equation for angular momentum? The Earth‑Moon system is isolated so the angular momentum M can’t change but the angular velocity c/t goes down. Something’s got to compensate. The system’s mass m doesn’t change. The only thing that can increase is distance r. There’s your answer, guys — conservation of angular momentum forces the Moon to drift outward.”

“Long way to the answer.”

“To the Moon and back.”

~~ Rich Olcott

Several Big Sloshes

“I call distraction, Sy. You were going to explain how come the Moon’s drifting away from us but you got us into radians and stuff. What’s that got to do with the Moon flying away by dragging a big wave around the Earth?”

“It’s not dragging a localized bulge of water like you’re thinking, Vinnie, nothing like that wave on Miller’s Planet. For that matter, the Miller’s Planet wave had a sharply‑rising front which also doesn’t look like the textbook tidal bulge.”

“There’s a textbook on this stuff?”

“Many, Al. Heavy-duty people have spent a lot of time on tides. Think about all the military and commercial navies that depend on boats being able to leave port and dock on schedule.”

“And not run aground <heh heh>”

“Well, yes, Vinnie. Anyhow, like a lot of pre‑computer Physics, that work was based on a simplified ideal system — a moon orbiting a smooth planet with a world‑covering ocean. Water’s drawn horizontally towards the sub‑lunar points making an egg‑shaped surface and everything’s neat.”

“Probably nothing like real life.”

“Of course. Here’s a video I built from satellite altimetry data. The grey dot is roughly the point underneath the Moon as that day progressed. The red‑to‑blue height scale’s in meters.. Not as neat as theory, is it?”

“Wow, that’s a mess. Looks like the Moon’s pulling water along the Canada‑Alaska coast okay, and the western Pacific starts to get a dome going. But the water never catches up before the Moon’s gone.”

“Hey, Vinnie, look how the tides just go round and round New Zealand. And what’s that, Hudson Bay, it’s a pinwheel.”

“Yeah, and in between Africa and Madagascar it’s completely out of phase from what it oughta be.”

“What you’re looking at is slosh. Once again, reality overwhelms a pretty theory. Each basin has its own preferred set of oscillations. None of them match up with the Moon. But the other thing — “

“Tiny numbers. Everything’s like less than a couple meters, not not a big bulge at all.”

“Bingo, Vinnie. Against Earth’s 6.4‑million‑meter radius, those small chaotic sloshes don’t make for effective energy transfer driving the Moon away from us. That theory’s toast.”

“So what’s doing it?”

“There’s two theories that I know of, and they’re probably both right. The first one is Earth tides — that bump you think of as traveling around the planet, but the bump is rock instead of water.”

“That can’t be a big effect. Rocks don’t bend.”

“On a planetary scale they’re not as solid as you think, Vinnie. The rock crust is brittle and really thin, less than half a percent of Earth’s radius. It floats on molten outer mantle which has the fluidity of tapioca pudding. When that structure gives under stress the crust layer cracks. The seismologists and GPS techs have measured surface motion all over the world. When they analyze the maps, the lunar component accounts for up to a meter of coordinated vertical daily movement. Figure the whole Earth is continually being squeezed and pulled to that extent and you’ve got a lot of energy being expended every 24 hours.”

“How about the other theory?”

“There’s no direct evidence, so far as I know, but it seems reasonable on physical grounds. We’ve got two gyrations going on here, right? The Moon is on a 29½‑day orbit while the Earth rotates about thirty times faster. But the two motions use different frames. The Earth’s spin axis runs through the geometric center of the planet and tilts 23° from its orbit axis. Meanwhile the whole Earth‑Moon system rotates about its barycenter, their common center of gravity, which stays inside the Earth about ¾ of the way moonward from the Earth’s middle. That rotation is about 5° away from Earth’s orbit’s axis. Imagine a molten blob near the barycenter, happily following the Moon in the Earth-Moon frame, but the rest of the planet is saying, ‘No, no, you’re supposed to be moving hundreds of miles an hour in a different direction!‘ If the blob’s the least bit lighter or heavier than its neighbor blobs, inertial forces expend energy to kick it out of there.”

“So we got two ways to transfer energy steady-like.”

“I think so.”

~~ Rich Olcott

Here’s a Different Angle

“OK, Sy, so there’s a bulge on the Moon’s side of the Earth and the Earth rotates but the bulge doesn’t and that makes the Moon’s orbit just a little bigger and you’ve figured out that the energy it took to lift the Moon raised Earth’s temperature by a gazillionth of a degree, I got all that, but you still haven’t told Al and me how the lifting works.”

“You wouldn’t accept it if I just said, ‘The Moon lifts itself by its bootstraps,’ would you?”

“Not for a minute.”

“And you don’t like equations. <sigh> OK, Al, pass over some of those paper napkins.”

“Aw geez, Sy.”

“You guys asked the question and this’ll take diagrams, Al. Ante up. … Thanks. OK, remember the time Cathleen and I caught Vinnie here at Al’s shop playing with a top?”

“Yeah, and he was spraying paper wads all over the place.”

“I wasn’t either, Al, it was the top sending them out with centri–…, some force I can never remember whether it’s centrifugal or centripetal.”

“Centrifugal, Vinnie, –fugal– like fugitive, outward‑escaping force. It’s one of those ‘depends on how you look at itfictitious forces. From where you were sitting, the wads looked like they were flying outward perpendicular to the top’s circle. From a wad’s point of view, it flew in a straight line tangent to the circle. It’s like we have two languages, Room and Rotor. They describe the same phenomena but from different perspectives.”

“Hey, it’s frames again, ain’t it?”

“Newton’s inertial frames? Sort‑of but not quite. Newton’s First Law only holds in the Room frame — no acceleration, motion is measured by distance, objects at rest stay put. Any other object moves in a straight line unless its momentum is changed by a force. You can tackle a problem by considering momentum and force components along separate X and Y axes. Both X and Y components work the same way — push twice as hard in either direction, get twice the acceleration in that direction. Nice rules that the Rotor frame doesn’t play by.”

“I guess not. The middle’s the only place an object can stay put, right?”

“Exactly, Al. Everything else looks like it’s affected by weird, constantly‑varying forces that’re hard to describe in X‑Y terms.”

“So that breaks Newton’s physics?”

“Of course not. We just have to adapt his F=m·a equation (sorry, Vinnie!) to Rotor conditions. For small movements we wind up with two equations. In the strict radial direction it’s still F=m·a where m is mass like we know it, a is acceleration outward or inward, and F is centrifugal or centripetal, depending. Easy. Perpendicular to ‘radial‘ we’ve got ‘angular.’ Things look different there because in that direction motion’s measured by angle but Newton’s Laws are all about distances — speed is distance per time, acceleration is speed change per time and so forth.”

“So what do you do?”

“Use arc length. Distance along an arc is proportional to the angle, and it’s also proportional to the radius of the arc, so just multiply them together.”

“What, like a 45° bend around a 2-foot radius takes 90 feet? That’s just wrong!”

“No question, Al. You have to measure the angle in the right units. Remember the formula for a circle’s circumference?”

“Sure, it’s 2πr.”

“Which tells you that a full turn’s length is times the radius. We can bridge from angle to arc length using rotational units so that a full turn, 360°, is units. We’ll call that unit a radian. Half a circle is π radians. Your 45° angle in radians is π/4 or about ¾ of a radian. You’d need about (¾)×(2) or 1½ feet of whatever to get 45° along that 2-foot arc. Make sense?”

“Gimme a sec … OK, I’m with you.”

“Great. So if angular distance is radius times angle, then angular momentum which is mass times distance per time becomes mass times radius times angle per time.”

“”Hold on, Sy … so if I double the mass I double the momentum just like always, but if something’s spinning I could also double the angular momentum by doubling the radius or spinning it twice as fast?”

“Couldn’t have put it better myself, Vinnie.”

~~ Rich Olcott

Two Against One, And It’s Not Even Close


On a brisk walk across campus when I hear Vinnie yell from Al’s coffee shop. “Hey! Sy! Me and Al got this argument going you gotta settle.”

“Happy to be a peacemaker, but it’ll cost you a mug of Al’s coffee and a strawberry scone.”

“Coffee’s no charge, Sy, but the scone goes on Vinnie’s tab. What’s your pleasure?”

“It’s morning, Al, time for black mud. What’s the argument, Vinnie?”

“Al read in one of his astronomy magazines that the Moon’s drifting away from us. Is that true, and if it is, how’s it happen? Al thinks Jupiter’s gravity’s lifting it but I think it’s because of Solar winds pushing it. So which is it?”

“Here you go, Sy, straight from the bottom of the pot.”

“Perfect, Al, thanks. Yes, it’s true. The drift rate is about 1¼ nanometers per second, 1½ inches per year. As to your argument, you’re both wrong.”

“Huh?”
 ”Aw, c’mon!”

“Al, let’s put some numbers to your hypothesis. <pulling out Old Reliable and screen‑tapping> I’m going to compare Jupiter’s pull on the Moon to Earth’s when the two planets are closest together. OK?”

“I suppose.”

“Alright. Newton’s Law tells us the pull is proportional to the mass. Jupiter’s mass is about 320 times Earth, which is pretty impressive, right? But the attraction drops with the square of the distance. The Moon is 1¼ lightseconds from Earth. At closest approach, Jupiter is almost 2100 lightseconds away, 1680 times further than the Moon. We need to divide the 320 mass factor by a 1680‑squared distance factor and that makes <key taps> Jupiter’s pull on the Moon is only 0.011 percent of Earth’s. It’ll be <taps> half that when Jupiter’s on the other side of the Sun. Not much competition, eh?”

“Yeah, but a little bit at a time, it adds up.”

“We’re not done yet. The Moon feels the big guy’s pull on both sides of its orbit around Earth. On the side where the Moon’s moving away from Jupiter, you’re right, Jupiter’s gravity slows the Moon down, a little. But on the moving-toward-Jupiter side, the motion’s sped up. Put it all together, Jupiter’s teeny pull cancels itself out over every month’s orbiting.”

“Gotcha, Al. So what about my theory, Sy?”

“Basically the same logic, Vinnie. The Solar wind varies, thanks to the Sun’s variable activity, but satellite measurements put its pressure somewhere around a nanopascal, a nanonewton per square meter. Multiply that by the Moon’s cross‑sectional area and we get <tap, tap> a bit less than ten thousand newtons of force on the Moon. Meanwhile, Newton’s Law says the Earth’s pull on the Moon comes to <tapping>
  G×(Earth’s mass)×(Moon’s mass)/(Earth-Moon distance)²
and that comes to 2×1011 newtons. Earth wins by a 107‑fold landslide. Anyway, the pressure slows the Moon for only half of each month and speeds it up the other half so we’ve got another cancellation going on.”

“So what is it then?”
 ”So what is it then?”

“Tides. Not just ocean tides, rock tides in Earth’s fluid outer mantle. Earth bulges, just a bit, toward the Moon. But Earth also rotates, so the bulge circles the planet every day.”

“Reminds me of the wave in the Interstellar movie, but why don’t we see it?”

“The movie’s wave was hundreds of times higher than ours, Al. It was water, not rock, and the wave‑raiser was a huge black hole close by the planet. The Moon’s tidal pull on Earth produces only a one‑meter variation on a 6,400,000‑meter radius. Not a big deal to us. Of course, it makes a lot of difference to the material that’s being kneaded up and down. There’s a lot of friction in those layers.”

“Friction makes heat, Sy. Rock tides oughta heat up the planet, right?”

“Sure, Vinnie, the process does generate heat. Force times distance equals energy. Raising the Moon by 1¼ nanometers per second against a force of 2×1021 newtons gives us <taping furiously> an energy transfer rate of 4×10‑23 joules per second per kilogram of Earth’s 6×1024‑kilogram mass. It takes about a thousand joules to heat a kilogram of rock by one kelvin so we’re looking at a temperature rise near 10‑27 kelvins per second. Not significant.”

“No blaming climate change on the Moon, huh?”

~~ Rich Olcott

Moon Shot

<chirp, chirp> “Moire here.”

“Hi, Mr Moire, it’s Jeremy. Hey, I’ve been reading through some old science fiction stories and I ran across some numbers that just don’t look right.”

“Science fiction can be pretty clunky. Some Editors let their authors play fast and loose on purpose, just to generate Letters to The Editor. Which author and what story?”

“This is Heinlein, Mr Moire. I know his ideas about conditions on Mars and Venus were way off but that was before we had robot missions that could go there and look. When he writes about space navigation, though, he’s always so specific it looks like he’d actually done the calculations.”

“OK, which story and what numbers?”

“This one’s called, let me check, Gentlemen, Be Seated. It’s about these guys who get trapped in a tunnel on the Moon and there’s a leak letting air out of the tunnel so they seal the leak when one of the guys —”

“I know the story, Jeremy. I’ve always wondered if it was Heinlein or his Editor who got cute with the title. Anyway, which numbers bothered you?”

“I kinda thought the title came first. Anyway, everybody knows that the Earth’s gravity is six times the Moon’s, but he says that the Earth’s mass is eighty times the Moon’s and that’s why the Earth raises tides on the Moon except they’re rock tides, not water tides, and the movement makes moonquakes and one of them might have caused the leak. So why isn’t the Earth’s gravity eighty times the Moon’s, not six?”

“Read me the sentence about eighty.”

“Umm … here it is, ‘Remember, the Earth is eighty times the mass of the Moon, so the tidal stresses here are eighty times as great as the Moon’s effect on Earth tides.‘ I checked the masses in Wikipedia and eighty is about right.”

“I hadn’t realized the ratio was that large, I mean that the Moon is that small. One point for Heinlein. Anyway, you’re comparing north and east. The eighty and the six both have to do with gravity but they’re pointing in different directions.”

“Huh? I thought gravity’s pull was always toward the center.”

“It is, but it makes a difference where you are and which center you’re thinking about. You’re standing on the Earth so the closest center to you is Earth’s and most of the gravity you feel is the one-gravity pull from there. Suppose you’re standing on the Moon —”

“One-sixth, I know, Mr Moire, but why isn’t it one‑eightieth?”

“Because on the Moon you’re a lot closer to the center of the Moon than you were to the center of the Earth back on Earth. Let’s put some numbers to it. Got a calculator handy?”

“Got my cellphone.”

“Duh. OK, Newton showed us that an object’s gravitational force is proportional to the object’s mass divided by the square of the distance to the center. Earth’s radius is about 4000 miles and the Moon’s is about a quarter of that, so take the mass as 1/80 and divide by 1/4 squared. What do you get?”

“Uhh … 0.2 gravities.”

“One-fifth g. Close enough to one-sixth. If we used accurate numbers we’d be even closer. See how distance makes a difference?”

“Mm-hm. What about Heinlein’s tidal stuff?”

“Ah, now that’s looking in the other direction, where the distance is a lot bigger. Earth-to-Moon is about 250,000 miles. Standing on the Moon, you’d feel Earth’s one‑g gravity diminished by a factor of 4000/250000 squared. What’s that come to?”

“Umm… the distance factor is (4000/250000)² … I get 250 microgravities. Not much. Heinlein made a good bet with his characters deciding that the leak was caused by a nearby rocket crash instead of a moonquake.”

“How about Heinlein’s remark about the Moon’s effect on Earth?”

“Same distance but one eightieth the mass so I divide by 80 — three microgravities. Wow! That can’t possibly be strong enough to raise tides here.”

“It isn’t, though that’s the popular idea. What really happens is that the Moon’s field pulls water sideways from all directions towards the sub‑Lunar point. Sideways motion doesn’t fight Earth’s gravity, it just makes the water pile up in the center.”

“Hah, piled-up water. Weird. Well, I feel better about Heinlein now.”

~~ Rich Olcott

Getting over The Hill

“You guys want refills? You look like you’re gonna be here a while.”

“Yes, thanks, Al. Your lattes are sooo good. And can we have some more paper napkins?”

“Sure, but don’t let ’em blow away or nothin’, OK? I hate havin’ to pick up the place.”

“They’ll stay put. Just a half-cup of mud for me, thanks.”

The Spring breeze has picked up a little so we hitch our chairs closer together. Susan reaches for a paper napkin, draws a curve. “Here’s another pattern you haven’t featured yet, Sy. It’s in every chemist’s mind when they think about reactions.”

“OK, I suppose this is molecules A and B on one side of some sort of wall and molecule C on the other.”

“It’ll be clearer if I label the axes. It’s a reaction between A and B to make C. The horizontal axis isn’t a distance, it’s a measure of the reaction’s progress toward completion. Beginning molecules to the left, completed reaction to the right, transition in the middle, see? The vertical axis is energy. We say the reaction is energetically favored because C is lower than A and B separately.”

“Then what’s the wall?”

“We call it the barrier. It’s some additional dollop of energy that allows the reaction happen. Maybe A or B have to be reconfigured before they can form an A~B transition state. That’s common in carbon chemistry, for instance. Carbon usually has four bonds, but you can get five‑bonded transition states. They usually don’t last very long, though.”

“Right, carbon and its neighbors prefer the tetrahedral shape. Five‑bonded carbon distorts the stable electron clouds. Heat energy shoves things into position, I suppose.”

“Often but hardy always. Especially for large molecules, heat’s more likely to jostle things out of position than put them together. That’d what cooking does.”

“The curve reminds me of particle accelerator physics, except it takes way more energy to overcome nuclear forces when you mash sub‑atomic thingies together.”

“Oh, yes, very similar in terms of that general picture — except that the C side could be multiple emitted particles.”

“So your sketch covers a processes everywhere, not just Chemistry. They all have different barrier profiles, then?”

“Of course. My drawing was just to give you the idea. Some barriers are high, some are low, either side may rise or fall exponentially or by some power of the distance, some are lumpy, it all depends. Some are even flat.”

“Flat, like no resistance at all?”

“Oh, yes. Hypergolic rocket fuel pairs ignite spontaneously when they mix. Water and alkali metals make flames — have you seen that video of metallic sodium dumped into a lake and exploding like mad? Awesome!”

“I can imagine, or maybe not. If heat energy doesn’t get molecules over that barrier, what does?”

“Catalysts, mostly. Some do their thing by capturing the reactants in adjacent sites, maybe doing a little geometry jiggling while they’re at it. Some play games with the electron states of one or both reactants. Anyhow, what they accomplish is speeding up a reaction by replacing the original barrier with one or more lower ones.”

“Wait, reaction speed depends on the barrier height? I’d expect either go or no‑go.”

“No, it’s usually more complicated than that. Umm … visualize tossing a Slinky toy into the air. Your toss gives it energy. Part of the energy goes into lifting it against Earth’s gravity, part into spinning motion and part into crazy wiggles and jangling, right? But if you toss just right, maybe half of the energy goes into just stretching it out. Now suppose there’s a weak spot somewhere along the spring. Most of your tosses won’t mess with the spot, but a pure stretch toss might have enough energy to break it apart.”

“Gotcha, the transition barrier might be a probability thing depending on how the energy’s distributed within A and B. Betcha tunneling can play a part, too.”

“Mm? Oh, of course, you’re a Physics guy so you know quantum. Yes, some reactions depend upon electrons or hydrogen atoms tunneling through a barrier, but hardly ever anything larger than that. Whoops, I’m due back at the lab. See ya.”

<inaudible> “Oh, I hope so.”

~~ Rich Olcott

The Edge of Pinkness

Susan Kim takes a sip of her mocha latte. eyes me over the rim. “That’s quite a set of patterns you’ve gathered together, Sy, but you’ve left out a few important ones.”

“Patterns?”

A log-linear plot

“Regularities we’ve discovered in Nature. You’ve written about linear and exponential growth, the Logistic Curve that describes density‑limited growth, sine waves that wobble up and down, maybe a couple of others down‑stack, but Chemistry has a couple I haven’t seen featured in your blog.”

“Such as?”

“Log-linear relationships are a biggie. We techies use them a lot to handle phenomena with a wide range. Rather than write 1,000,000,000 or 109, we sometimes just write 9, the base‑10 logarithm. The pH scale for acid concentration is my favorite example. It goes from one mole per liter down to ten micro‑nanomoles per liter. That’s 100 to 10-14. We just drop the minus sign and use numbers between 0 and 14. Fifteen powers of ten. Does Physics have any measurements that cover a range like that?”

“A handful, maybe, in theory. The limitation is in confirming the theory across a billion-fold range or wider. Atomic clocks that are good down to the nanosecond are our standards for precision, but they aren’t set up to count years. Mmmm … the Stefan‑Boltzmann Law that links an object’s electromagnetic radiation curve to its temperature — our measurements cover maybe six or seven powers of ten and that’s considered pretty good.”

“Pikers.” <but I like the way she grins when she says it>

“I took those Chemistry labs long ago. All I remember was acids were colorless and bases were pink. Or maybe the other way around.”

“You’ve got it right for the classic phenolphthalein indicator, but there are dozens of other indicators that have different colors at different acidities. I’ll tell you a secret — phenolphthalein doesn’t kick over right at pH 7, the neutral point. It doesn’t turn pink until the solution’s about ten times less acidic, near pH 8.”

Adapted from this file by Damitr, CC BY-SA 4.0

“So all my titrations were off by a factor of ten?”

“Oh, no, that’s not how it works. I’m going to use round numbers here, and I’ll skip a couple of things like the distinction between concentration and activity. Student lab exercises generally use acid and base concentrations on the order of one molar. For most organic acids, that’d give a starting pH near 1 or 2, way over on the sour side. In your titration you’d add base, drop by drop, until the indicator flips color. At that point you conclude the amounts of acid and base are equivalent, not by weight but by moles. If you know the base concentration you can calculate the acid.”

“That’s about what I recall, right.”

“Now consider that last drop. One drop is about 50 microliters. With a one‑molar base solution, that drop holds 50 nanomoles. OK?”

<I scribble on a paper napkin> “Mm-hm, that looks right.”

“Suppose there’s about 50 milliliters of solution in the flask. Because we’re considering the last drop, the solution in the flask must have become nearly neutral, say pH 6. That means the un‑neutralized acid concentration was 10-6 moles per liter, or one micromolar. Fifty milliliters at one micromolar concentration is, guess what, 50 nanomoles. Your final drop neutralizes the last of the acid sample.”

“So the acid concentration goes to zero?”

“Water’s not that cooperative. Water molecules themselves act like acids and bases. An H2O molecule can snag a hydrogen from another H2O giving an H3O+ and an OH. Doesn’t happen often, but with 55½ moles of water per liter and 6×1023 molecules per mole there’s always a few of those guys hanging around. Neutral water runs 10-7 moles per liter of each, which is why neutral pH is 7. Better yet, the product of H3O+ and OH concentrations is always 10-14 so if you find one you can calculate the other. Take our titration for example. One additional drop adds 50 nanomoles more base. In 50 milliliters of solution that’s roughly 10-6+10-7 molar OH. Call it 1.1×10-6, which implies 0.9×10-8 molar H3O+. Log of that and drop the minus sign, you’re a bit beyond pH 8 which sends phenolphthalein into the pink side. Your titration’s good.”

I eye her over my mug of black mud. “A gratifying indication.”

~~ Rich Olcott

The Latte Connection

An early taste of Spring’s in the air so Al’s set out tables in front of his coffee shop. I’m enjoying my usual black mud when the Chemistry Department’s Susan Kim passes by carrying her usual mocha latte. “Hi, Sy, mind if I take the socially distant chair at your table?”

“Be my guest, Susan. What’s going on in your world?”

“I’ve been enjoying your hysteresis series. It took me back to Physical Chemistry class. I’m intrigued by how you connected it to entropy.”

“How so?”

“I think of hysteresis as a process, but entropy is a fixed property of matter. If I’m holding twelve grams of carbon at room temperature, I know what its entropy is.”

“Mmm, sorta. Doesn’t it make a difference whether the carbon’s a 60‑carat diamond or just a pile of soot?”

“OK, I’ll give you that, the soot’s a lot more random than the diamond so its entropy is higher. The point remains, I could in principle measure a soot sample’s heat capacity at some convenient temperature and divide that by the temperature. I could repeat that at lower and lower temperatures down to near absolute zero. When I sum all those measurements I’ll have the entropy content of the sample at my starting temperature.”

“A classical definition, just what I’d expect from a chemist. But suppose your soot spills out of its test tube and the breeze spreads it all over the neighborhood. More randomness, higher entropy than what you measured, right?”

“Well, yes. I wouldn’t have a clue how to calculate it, but that goes way beyond Carnot’s and Clausius’ original concept.”

“So entropy has at least a thin linkage with history and hysteresis. To you chemists, though, an element or compound is timeless — lead or water have always been lead or water, and their physical constants are, well, constant.”

“Not quite true, Sy. Not with really big molecules like proteins and DNA and rubber and some plastics. Squirt a huge protein like catalase through a small orifice and its properties change drastically. It might not promote any reaction, much less the one Nature designed it for. Which makes me think — Chemistry is all about reactions and they take time and studying what makes reactions run fast or slow is a big part of the field. So we do pay attention to time.”

“Nice play, Susan! You’re saying small molecules aren’t complex enough to retain memories but big ones are. I’ll bet big molecules probably exhibit hysteresis.”

“Sure they do. Rubber molecules are long-chain polymers. Quickly stretch a rubber band to its limit, hold it there a few seconds then let go. Some of the molecular strands lock into the stretched configuration so the band won’t immediately shrink all the way down to its original size. There’s your molecular memory.”

“And a good example it is — classic linear Physics. How much force you exert, times the distance you applied it through, equals the energy you expended. Energy’s stored in the rubber’s elasticity when you stretch it, and the energy comes back out on release.”

“Mostly right, Sy. You actually have to put in more energy than you get out — Second Law of Thermodynamics, of course — and the relationship’s not linear. <rummaging into purse> Thought I had a good fat rubber band somewhere … ah‑hah! Here, stretch this out while you hold it against your forehead. Feel it heat up briefly? Now keep checking for heat while you relax the band.”

“Hey, it got cold for a second!”

“Yep. The stretched-out configuration is less random so its entropy and heat capacity are lower than the relaxed configuration’s. The stretched band had the same amount of heat energy but with less heat required per degree of temperature, that amount of energy made the band hotter. Relaxing the band let its molecules get less orderly. Heat capacity went back up. temperature went back down.”

“Mmm-HM. My hysteresis diagram’s upward branch is stretch energy input and the downward branch is elastic energy output. The energy difference is the area inside the hysteresis curve, which is what’s lost to entropy in each cycle and there we have your intriguing entropy‑hysteresis connection. Still intrigued?”

“Enough for another latte.”

~~ Rich Olcott