The Big Chill

Jeremy gets as far as my office door, then turns back. “Wait, Mr Moire, that was only half my question. OK, I get that when you squeeze on a gas, the outermost molecules pick up kinetic energy from the wall moving in and that heats up the gas because temperature measures average kinetic energy. But what about expansion cooling? Those mist sprayers they set up at the park, they don’t have a moving outer wall but the air around them sure is nice and cool on a hot day.”

“Another classic Jeremy question, so many things packed together — Gas Law, molecular energetics, phase change. One at a time. Gas Law’s not much help, is it?”

“Mmm, guess not. Temperature measures average kinetic energy and the Gas Law equation P·V = n·R·T gives the total kinetic energy for the n amount of gas. Cooling the gas decreases T which should reduce P·V. You can lower the pressure but if the volume expands to compensate you don’t get anywhere. You’ve got to suck energy out of there somehow.”

Illustrations adapted from drawings by Trianna

“The Laws of Thermodynamics say you can’t ‘suck’ heat energy out of anything unless you’ve got a good place to put the heat. The rule is, heat energy travels voluntarily only from warm to cold.”

“But, but, refrigerators and air conditioners do their job! Are they cheating?”

“No, they’re the products of phase change and ingenuity. We need to get down to the molecular level for that. Think back to our helium-filled Mylar balloon, but this time we lower the outside pressure and the plastic moves outward at speed w. Helium atoms hit the membrane at speed v but they’re traveling at only (v-w) when they bounce back into the bulk gas. Each collision reduces the atom’s kinetic energy from ½m·v² down to ½m·(v-w)². Temperature goes down, right?”

“That’s just the backwards of compression heating. The compression energy came from outside, so I suppose the expansion energy goes to the outside?”

“Well done. So there has to be something outside that can accept that heat energy. By the rules of Thermodynamics, that something has to be colder than the balloon.”

“Seriously? Then how do they get those microdegree above absolute zero temperatures in the labs? Do they already have an absolute-zero thingy they can dump the heat to?”

“Nope, they get tricky. Suppose a gas in a researcher’s container has a certain temperature. You can work that back to average molecular speed. Would you expect all the molecules to travel at exactly that speed?”

“No, some of them will go faster and some will go slower.”

“Sure. Now suppose the researcher uses laser technology to remove all the fast-moving molecules but leave the slower ones behind. What happens to the average?”

“Goes down, of course. Oh, I see what they did there. Instead of the membrane transmitting the heat away, ejected molecules carry it away.”

“Yup, and that’s the key to many cooling techniques. Those cooling sprays, for instance, but a question first — which has more kinetic energy, a water droplet or the droplet’s molecules when they’re floating around separately as water vapor?”

“Lessee… the droplet has more mass, wait, the molecules total up to the same mass so that’s not the difference, so it’s droplet velocity squared versus lots of little velocity-squareds … I’ll bet on the droplet.”

“Sorry, trick question. I left out something important — the heat of vaporization. Water molecules hold pretty tight to each other, more tightly in fact than most other molecular substances. You have to give each molecule a kick to get it away from its buddies. That kick comes from other molecules’ kinetic energy, right? Oh, and one more thing — the smaller the droplet, the easier for a molecule to escape.”

“Ah, I see where this is going. The mist sprayer’s teeny droplets evaporate easy. The droplets are at air temperature, so when a molecule breaks free some neighbor’s kinetic energy becomes what you’d expect from air temperature, minus break-free energy. That lowers the average for the nearby air molecules. They slow their neighbors. Everything cools down. So that’s how sprays and refrigerators and such work?”

“That’s the basic principle.”

“Cool.”

~ Rich Olcott

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

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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.

Taming The Elephant

Suddenly they were all on the attack.  Anne got in the first lick.  “C’mon, Sy, you’re comparing apples and orange peel.  Your hydrogen sphere would be on the inside of the black hole’s event horizon, and Jeremy’s virtual particles are on the outside.”

[If you’ve not read my prior post, do that now and this’ll make more sense.  Go ahead, I’ll wait here.]white satin and 5 elephantsJennie’s turn — “Didn’t the chemists define away a whole lot of entropy when they said that pure elements have zero entropy at absolute zero temperature?”

Then Vinnie took a shot.  “If you’re counting maybe-particles per square whatever for the surface, shouldn’t you oughta count maybe-atoms or something per cubic whatever for the sphere?”

Jeremy posed the deepest questions. “But Mr Moire, aren’t those two different definitions for entropy?  What does heat capacity have to do with counting, anyhow?”

Al brought over mugs of coffee and a plate of scones.  “This I gotta hear.”

“Whew, but this is good ’cause we’re getting down to the nub.  First to Jennie’s point — Under the covers, Hawking’s evaluation is just as arbitrary as the chemists’.  Vinnie’s ‘whatever’ is the Planck length, lP=1.616×10-35 meter.  It’s the square root of such a simple combination of fundamental constants that many physicists think that lP2=2.611×10-70 m², is the ‘quantum of area.’  But that’s just a convenient assumption with no supporting evidence behind it.”

“Ah, so Hawking’s ABH=4πrs2 and SBH=ABH/4 formulation with rs measured in Planck-lengths, just counts the number of area-quanta on the event horizon’s surface.”

“Exactly, Jennie.  If there really is a least possible area, which a lot of physicists doubt, and if its size doesn’t happen to equal lP2, then the black hole entropy gets recalculated to match.”

“So what’s wrong with cubic those-things?”

“Nothing, Vinnie, except that volumes measured in lP3 don’t apply to a black hole because the interior’s really four-dimensional with time scrambled into the distance formulas.  Besides, Hawking proved that the entropy varies with half-diameter squared, not half-diameter cubed.”

“But you could still measure your hydrogen sphere with them and that’d get rid of that 1033 discrepancy between the two entropies.”

“Not really, Vinnie.  Old Reliable calculated solid hydrogen’s entropy for a certain mass, not a volume.”

“Hawking can make his arbitrary choice, Sy, he’s Hawking, but that doesn’t let the chemists off the scaffold.  How did they get away with arbitrarily defining a zero for entropy?”

“Because it worked, Jennie.  They were only concerned with changes — the difference between a system’s state at the end of a process, versus its state at the beginning.  It was only the entropy difference that counted, not its absolute value.”

“Hey, like altitude differences in potential energy.”

“Absolutely, Vinnie, and that’ll be important when we get to Jeremy’s question.  So, Jennie, if you’re only interested in chemical reactions and if it’s still in the 19th Century and the world doesn’t know about isotopes yet, is there a problem with defining zero entropy to be at a convenient set of conditions?”

“Well, but Vinnie’s Second Law says you can never get down to absolute zero so that’s not convenient.”

“Good point, but the Ideal Gas Law and other tools let scientists extrapolate experimentally measured properties down to extremely low temperatures.  In fact, the very notion of absolute zero temperature came from experiments where the volume of a  hydrogen or helium gas sample appears to decrease linearly towards zero at that temperature, at least until the sample condenses to a liquid.  With properly calibrated thermometers, physical chemists knocked themselves out measuring heat capacities and entropies at different temperatures for every substance they could lay hands on.”

“What about isotopes, Mr Moire?  Isn’t chlorine’s atomic weight something-and-a-half so there’s gotta be several of kinds of chlorine atoms so any sample you’ve got is a mixture and that’s random and that has to have a non-zero entropy even at absolute zero.”

“It’s 35.4, two stable isotopes, Jeremy, but we know how to account for entropy of mixing and anyway, the isotope mix rarely changes in chemical processes.”

“But my apples and orange peels, Sy — what does the entropy elephant do about them?”

~~ Rich Olcott

Titan’s Atmosphere Is A Gas

One year ago I kicked off these weekly posts with some speculations about how Life might exist on Saturn’s moon Titan. My surmises were based on reports from NASA’s Cassini-Huygens mission, plus some Physical Chemistry expectations for Titan’s frigid non-polar mix of liquid ethane and methane. Titan offers way more fun than that.

The environment on Titan is different from everything we’re used to on Earth. For instance, the atmosphere’s weird.earth-vx-titanTitan’s atmosphere is heavy-duty compared with Earth’s — 6 times deeper and about 1½ times the surface pressure. When I read those numbers I thought, “Huh? But Titan’s diameter is only 40% as big as Earth’s and its surface gravity is only 10% of ours. How come it’s got such a heavy atmosphere?”

Wait, what’s gravity got to do with air pressure? (I’m gonna use “air pressure” instead of “surface atmospheric pressure” because typing.) Earth-standard sea level air pressure is 14.7 pounds of force per square inch. That 14.7 pounds is the total weight of the air molecules above each square inch of surface, all the way out to space.

(Fortunately, air’s a hydraulic fluid so its pressure acts on sides as well as tops. Otherwise, a football’s shape would be even stranger than it is.)

Newton showed us that weight (force) is mass times the the acceleration of gravity. Gravity on Titan is 1/10 as strong as Earth’s, so an Earth-height column of air on Titan should weigh about 1½ pounds.

But Titan’s atmosphere (measured to the top of each stratosphere) goes out 6 times further than Earth’s. If we built out that square-inch column 6 times taller, it’d weigh only 9 pounds on Titan, well shy of the 22 pounds the Huygens lander measured. Where does the extra weight come from?

My first guess was, heavy molecules. If gas A has molecules that are twice as heavy as gas B’s, then a given volume of A would weigh twice as much as the same volume of B. An atmosphere composed of A will press down on a planet’s surface twice as hard as an atmosphere composed of B.

Good guess, but doesn’t apply. Earth’s atmosphere is 78% N2 (molecular weight 28) and 21% O2 (molecular weight 32) plus a teeny bit of a few other things. Their average molecular weight is about 29. Titan’s atmosphere is 98% N2 so its average molecular weight (28) is virtually equal to Earth’s. So no, those tarry brown molecules that block our view of Titan’s surface aren’t numerous enough to account for the high pressure.

My second guess is closer to the mark, I think. I remembered the Ideal Gas Law, the one that says, “pressure times volume equals the number of molecules times a constant times the absolute temperature.” In symbols, P·V=n·R·T.

Visualize one gas molecule, Fred, bouncing around in a cube sized to match the average volume per molecule, V/n=R·T/P. If Fred goes outside his cube in any direction he’s likely to bang into an adjacent molecule. If Fred has too much contact with his neighbors they’ll all stick together and become a liquid or solid.

The equation tells us that if the pressure doesn’t change, the size of Fred’s cube rises with the temperature. Just for grins I calculated the cube’s size for standard Earth conditions: (22.4 liters/mole)×(1 cubic meter/1000 liters)×(1 mole/6.02×1023 molecules)=37.2×10-27 cubic meter/molecule. The cube root of that is the length of the cube’s edge — 3.3 nanometers, about 8.3 times Fred’s 0.40-nanometer diameter.

titan-boxes
Fred and neighbors

Earth-standard surface temperature is about 300°K (absolute temperatures are measured in Kelvins). Titan’s surface temperature is only 94°K. On Titan that cube-edge would be 8.3*(94/300)=2.6 times Fred’s diameter — if air pressure were Earth-standard.

But really Titan’s air pressure is 1.5 times higher because its column is so tall and contains so much gas. The additional pressure squeezes Fred’s cube-edge down to 2.6*(1/1.5)=1.8 times his diameter. Still room enough for Fred to feel well-separated from his neighbors and continue acting like a proper gas.

The primary reason Titan’s atmosphere is so dense is that it’s chilly up there. Also, there’s a lot of Freds.

~~ Rich Olcott

– For the technorati… The cube-root of the Van der Waals volume for N2. And yeah, I know I’m almost writing about Mean Free Path but I think the development’s simpler this way.