The Hot Squeeze

A young man’s knock, eager yet a bit hesitant.

“C’mon in, Jeremy, the door’s open.”

“Hi, Mr Moire. How’s your Summer so far? I got an ‘A’ on that black hole paper, thanks to your help. Do you have time to answer a question now that Spring term’s over?”

“Hi, Jeremy. Pretty good, congratulations, and a little. What’s your question?”

“I don’t understand about the gas laws. You squeeze a gas, you raise its temperature, but temperature’s the average kinetic energy of the molecules which is mass times velocity squared but mass doesn’t change so how does the velocity know how big the volume is? And if you let a gas expand it cools and how does that happen?”

“A classic Jeremy question. Let’s take it a step at a time, big-picture view first. The Gas Law says pressure times volume is proportional to the amount of gas times the temperature, or P·V = n·R·T where n measures the amount of gas and R takes care of proportionality and unit conversions. Suppose a kid gets on an airplane with a balloon. The plane starts at sea level pressure but at cruising altitude they maintain cabins at 3/4 of that. Everything stays at room temperature, so the balloon expands by a third –“

Kid drawing of an airplane with a red balloon
Adapted from a drawing by Xander

“Wait … oh, pressure down by 3/4, volume up by 4/3 because temperature and n and R don’t change. OK, I’m with you. Now what?”

“Now the plane lands at some warm beach resort. We’re back at sea level but the temp has gone from 68°F back home to a basky 95°F. How big is the balloon? I’ll make it easy for you — 68°F is 20°C is 293K and 95°F is 35°C is 308K.”

“Volume goes up by 308/293. That’s a change of 15 in about 300, 5% bigger than back home.”

“Nice estimating. One more stop on the way to the molecular level. Were you in the crowd at Change-me Charlie’s dark matter debate?”

“Yeah, but I didn’t get close to the table.”

“Always a good tactic. So you heard the part about pressure being a measure of energy per unit of enclosed volume. What does that make each side of the Gas Law equation?”

“Umm, P·V is energy per volume, times volume, so it’s the energy inside the balloon. Oh! That’s equal to n·R·T but R‘s a constant and n measures the number of molecules so T = P·V/n·R makes T proportional to average kinetic energy. But I still don’t see why the molecules speed up when you squeeze on them. That just packs the same molecules into a smaller volume.”

“You’re muddling cause and effect. Let’s try to tease them apart. What forces determine the size of the balloon?”

“I guess the balance between the outside pressure pushing in, versus the inside molecules pushing out by banging against the skin. Increasing their temperature means they have more energy so they must bang harder.”

“And that increases the outward pressure and the balloon expands until things get back into balance. Fine, but think about individual molecules, and let’s pretend that we’ve got a perfect gas and a perfect balloon membrane — no leaks and no sticky collisions. A helium-filled Mylar balloon is pretty close to that. When things are in balance, molecules headed outward approach the membrane with some velocity v and bounce back inward with the same velocity v though in a different direction. Their kinetic energy before hitting the membrane is ½m·v²; after the collision the energy’s also ½m·v² so the temperature is stable.”

“But that’s at equilibrium.”

“Right, so let’s increase the outside pressure to squeeze the balloon. The membrane closes in at some speed w. Out-bound molecules approach the membrane with velocity v just as before but the membrane’s speed boosts the bounce. The ‘before’ kinetic energy is still ½m·v² but the ‘after’ value is bigger: ½m·(v+w)². The total and average kinetic energy go up with each collision. The temperature boost comes from the energy we put into the squeezing.”

“So the heating actually happens out at the edges.”

“Yup, the molecules in the middle don’t know about it until hotter molecules collide with them.”

“The last to learn, eh?.”

“Always the case.”

~~ Rich Olcott

Thanks to Mitch Slevc for the question that led to this post.

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The Magnificent Seven

“Hey, Sy, you said there’s seven fundamental standards. We’ve talked about the second and the meter and the kilogram and the ampere. What’s left?”

“The mole, the kelvin and the candela, Al. They’re all kinda special-purpose but each has its charms. The mole, for instance, is cute and fuzzy and has its very own calendar date.”

“You’re pulling our legs, Sy. A cute unit of measure? No way.”

“Hear me out, Vinnie. How many shoes in a dozen pairs?”

“Huh? Two dozen, that’s twenty-four.”

“Sure, but it’s easier to work in dozens. How many hydrogen atoms in a dozen H2O molecules?”

“Two dozen, of course. Are we going somewhere here?”

“Next step. A mole is like a dozen on steroids, about 6×1023 whatevers. How many hydrogen atoms in a mole of H2O molecules?”

“Two moles, I suppose, or 12×1023.”

“You got the idea.”

“Cute.”

“A-hah! Gotcha for one.”

“Fair hit. How about the fuzzy part and the date?”

“The fuzzy has to do with isotopes. Every element has an atomic number and an atomic weight. The atomic number counts protons in the nucleus –all atoms of an element have the same atomic number. But different isotopes of an element have different numbers of neutrons. The ‘weight’ is protons plus neutrons, averaged across the isotopes. If you’re holding a mole of an element, you’re holding its atomic weight in grams. The fuzzy happens because samples of an element from different sources can have different mixtures of isotopes. You may have some special diamonds that contain nothing but carbon-12. A mole of those atoms masses exactly 12 grams. My sample is enriched with 10% of carbon-13. Mole-for-mole, my carbon is a tad heavier than yours. In fact, 6×1023 of my atoms mass 12.10 grams. That’s an extreme example but you get the idea.”

“Fuzzy, a little, OK. And the date thing?”

“June 23 is Mole Day, celebrated by Chemistry teachers everywhere.”

“What’s the kelvin about then?”

“Temperature. And most solid-state electronics. Zero kelvin is absolute zero, the coldest temperature something can get, when the maximum heat has been sucked out and all its atoms have minimum vibrational energy. From there you heat it up degree by degree until you get to where water can co-exist as liquid, solid and water vapor. It used to be the standard to call that temperature 273.16 K.”

“Used to be? Water doesn’t do that any more?”

“Oh, it still does, but the old standard had problems. It used five different ‘official’ techniques and 16 different calibration checks to cover the range from 3 K up to the melting point of copper. Some of those standards, like the melting pressure of helium-3, are not only inconvenient but expensive. Others led to measured intermediate temperatures that disagree depending on which direction you’re going. The defined standards did nothing for the plasma people who work above 1500 K. It was a mess.”

“So how does the new standard fix that?”

“It exploits new tech, especially in solid-state science. The Boltzmann constant, kB, is sort of the quantum of heat capacity at the microscopic level. The product kBT is a threshold energy. Practically everything that happens at the quantum level depends on the ratio of some process energy divided by kBT. If the ratio’s high the process runs; if it’s low, nothing. In-between, the response is predictably temperature-dependent. Thanks to a plethora of new solid-state thermal sensors that depend on that logic, we now have a handle on the range from microkelvins to kilokelvins and above.”

“Pretty good. What’s the last one?”

“It’s the one I’m least happy about, the candela. It’s a unit for how bright a light source is, sort-of. Take the source’s power output at all optical frequencies and ‘correct’ that by how much each frequency would stimulate a mathematically modeled ‘standard human eye.’ Isolate the ‘corrected’ watts at 555 nanometers, multiply by Kcd=683. It’s a time-hallowed metric that lighting designers depend upon, but it skips over little things like we actually see with rod cells and three kinds of cone cells, none of which match the standard curve. Kcd is just too human-centered to be a universal constant.”

“Humans ain’t universal. We’re not even on Mars.”

“Yet.”

~~ Rich Olcott

Rockin’ Round The Elephant

<continued…>  “That’s what who said?  And why’d he say that?”

“That’s what Hawking said, Al.  He’s the guy who first applied thermodynamic analysis to black holes.  Anyone happen to know the Three Laws of Thermodynamics?”

Vinnie pipes up from his table by the coffee shop door.  “You can’t win.  You can’t even break even.  But you’ll never go broke.”

“Well, that’s one version, Vinnie, but keep in mind all three of those focus on energy.  The First Law is Conservation of Energy—no process can create or destroy energy, only  transform it, so you can’t come out ahead.  The Second Law is really about entropy—”

“Ooo, the elephant!”white satin and black hole 2

“Right, Anne.  You usually see the Second Law stated in terms of energy efficiency—no process can convert energy to another form without wasting some of it. No breaking even.  But an equivalent statement of that same law is that any process must increase the entropy of the Universe.”

“The elephant always gets bigger.”

“Absolutely.  When Bekenstein and Hawking thought about what would happen if a black hole absorbed more matter, worst case another black hole, they realized that the black hole’s surface area had to follow the same ‘Never decrease‘ rule.”

“Oh, that Hawking!  Hawking radiation Hawking!  The part I didn’t understand, well one of the parts, in that “Black Holes” Wikipedia article!  It had to do with entangled particles, didn’t it?”

“Just caught up with us, eh, Jeremy?  Yes, Stephen Hawking.  He and Jacob Bekenstein found parallels between what we can know about black holes on the one hand and thermodynamic quantities on the other.  Surface area and entropy, like we said, and a black hole’s mass acts mathematically like energy in thermodynamics.  The correlations were provocative ”

“Mmm, provocative.”

“You like that word, eh, Anne?  Physicists knew that Bekenstein and Hawking had a good analogy going, but was there a tight linkage in there somewhere?  It seemed doubtful.”

“Nothin’ to count.”

“Wow, Vinnie.  You’ve been reading my posts?”

“Sure, and I remember the no-hair thing.  If the only things the Universe can know about a black hole are its mass, spin and charge, then there’s nothing to figure probabilities on.”

“Exactly.  The logic sequence went, ‘Entropy is proportional to the logarithm of state count, there’s only one state, log(1) equals zero,  so the entropy is zero.’  But that breaks the Third Law.  Vinnie’s energy-oriented Third Law says that no object can cool to absolute zero temperature.  But an equivalent statement is that no object can have zero entropy.”

“So there’s something wrong with black hole theory, huh?”

“Which is where our guys started, Vinnie.  Being physicists, they said, ‘Suppose you were to throw an object into a black hole.  What would change?’

“Its mass, for one.”

“For sure, Jeremy.  Anything else?”

“It might not change the spin, if you throw right.”

“Spoken like a trained baseball pitcher.  Turns out its mass governs pretty much everything about a black hole, including its temperature but not spin or charge.  Once you know the mass you can calculate its entropy, diameter, surface area, surface gravity, maximum spin, all of that.  Weird, though, you can’t easily calculate its volume or density — spatial distortion gets in the way.”

“So what happens to all those things when the mass increases?”

“As you might expect, they change.  What’s interesting is how each of them change and how they’re linked together.  Temperature, for instance, is inversely proportional to the mass and vice-versa.  Suppose, Jeremy, that you threw two big rocks, both the same size, into a black hole.  The first rock is at room temperature and the other’s a really hot one, say at a million degrees.   What would each do?”

“The first one adds mass so from what you said it’d drop the temperature.  The second one has the same mass, so I don’t see, wait, temperature’s average kinetic energy so the hot rock has more energy than the other one and Einstein says that energy and mass are the same thing so the black hole gets more mass from the hot rock than from the cold one so its temperature goes down … more?  Really?”

“Yup.  Weird, huh?”

“How’s that work?”

“That’s what they asked.”

~~ Rich Olcott