Jul
15
2024

To Scale

Confession: I love The Big Bang Theory and thanks to TiVo, I'm working on watching all 279 episodes, in order.

But in just about every episode, there's this stuff that makes my physics major brain hurt a bit:

Reader, as Wolfgang Pauli allegedly said: "That is not only not right; it is not even wrong." Electrons are not shiny balls; they do not orbit atomic nuclei on shiny rails. (They don't make whooshing sounds, either, but that's even more of a quibble.)

Now: at a certain level, attempting to visualize what atoms "really look like" is futile. It's just math down there, solutions to the Schrödinger equation, or some other formulation.

But they could at least try to get the scale right.

Or, more accurately, once you try to get the scale right, you can see why they didn't.

Let's imagine—because we're not going to actually do it—building a scale model of a good old water molecule, H2O: an oxygen atom, with two hydrogen atoms hanging off to one side.

A hydrogen atom nucleus, a single proton, has a radius "root mean square charge radius") 8.4e-16 meters.

Let's say our scale model uses a ping-pong ball to represent a proton. A ping-pong ball's radius is 20 millimeters, or 2.0e-2 meters. (Fascinating fun fact from the link: "the size increased from 38mm to 40mm after the 2000 Olympic Games." I did not know that!)

So for our scale model, we have to multiply atomic/molecular distances by a scale factor of (2.0e-2/8.4e-16) = 2.38e13 (I.e., just under 24 trillion.)

The radius of an oxygen atom's nucleus is generally reported to be 2.8e-15 meters. Multiplying this by our scale factor, gives 2.8e-15 * 2.38e13 = 6.7e-2 meters, or 67 millimeters, about the size of a medium grapefruit.

So: to start building our scale model, gather together a grapefruit and two ping-pong balls. Where do we put them?

This page reports the distance between the oxygen nucleus and the hydrogen nuclei is 0.943 angstroms, or 9.43e-11 meters.

This scales up to 9.43e-11 * 2.38e13 = 2244 meters. Or about 1.4 miles. So:

  1. Put your grapefruit down;
  2. Walk 1.4 miles in a straight line;
  3. Drop one ping-pong ball;
  4. Return to the grapefruit;
  5. Turn approximately 106° from your original direction;
  6. march another 1.4 miles, and drop your other ping-pong ball.

That completes placement of the nuclei. Now we have to consider the electrons (ten of them) that swarm around the nuclei. Where do they go, and how do we represent them?

Reader, the best thing I can think up, visualization-wise, is a fuzzy cloud. That Schrödinger equation thing I referred to above would (if we solved it) give us a probability of finding an electron within a certain hunk of space. That probability is relatively high close to the nuclei, and gets much smaller as you get further away. And, as an added complication, the electrons have a higher probability to flock around the oxygen nucleus than the hydrogen nuclei. Visualize that however you'd like. The page referenced above does it with color.

The page referenced above talks about the "Van der Waals" diameter of the water molecule, which is as good a size estimate as we are likely to get; that's about 2.75e-10 meters. Scaling that distance up gives 2.8e-10 * 2.38e13 = 6664 meters, or about 4.14 miles.

So, to summarize: our scale model water molecule is a fuzzy cloud over 4 miles in diameter, in which is hiding a grapefruit and two ping-pong balls. And I admit, this would be difficult to picture on the TV screen in a way that might appeal to viewers. Still, it would be better than what they did.

Another fun fact: the electrons make up only 0.03% of the mass of the water molecule. For your typical Poland Spring 500 milliliter (16.9 oz) water bottle, that means most of the volume is those fuzzy electrons. Their total mass, however, is a mere 150 milligrams or so; the remaining 499.85 grams resides in those tiny nuclei.

Here's one thing that does not scale well. How fast do water molecules typically move? Much faster than the "atoms" you see on The Big Bang Theory. Googling will tell you that they exhibit a range of speeds (a Maxwell-Boltzmann distribution, approximately) and for water molecules at (roughly) room temperature, the average speed works out to 590 meters/sec (≈1300 mph).

So be glad that the water molecules in that Poland Spring bottle don't suddenly ("at random") decide to start moving in the same direction.

Yeah, that's impossible. Conservation of momentum saves us there.

But scaling that to our model, we get 1.4e16 meters/sec.

Which is (um) 47 million times the speed of light.

Very difficult to visualize!


Last Modified 2024-07-15 5:20 PM EST

Dumbest Thing Seen on Facebook Yesterday

It was this, posted in all serious earnestness:

I was sorely tempted to snark at this on FB. Unfortunately, it was posted by a wonderful lady I went to high school with, and had a major (unrequited) crush on back then, over a half-century ago.

In fact, I still kind of have a crush on her. I don't want to hurt her feelings.

So I'll retreat to snarking here. I'm pretty sure she doesn't read Pun Salad. I'll use my Fisking template: The dumbness is reproduced in its entirety on the left with a lovely #EEFFFF background color; my remarks are on the right.

There's not actually money being spent to wipe away student loans.

True, in a sense. That money was "spent" when it went into the coffers of higher education institutions.

Now, the only question is: who's gonna pay it back?

A student has $20,000 college debt and has paid $250/ month for 10 years. They have paid $30,000, but still have an outstanding balance of $15,000 on the school loans, due to the way the loans are written.

Um, yes. That's the way loans work.

But the numbers caused me to seek out this loan calculator. If you are paying $250/month on an initial balance of $20K, and only manage to work your balance down to $15K after 10 years, that seems to imply an interest rate of 13.83% or so.

That's, um, reality-challenged. Actual student loan interest rates (current and historical) aren't, and never have been, that high. (Data here.)

The loan relief is saying - you've paid $10,000 more than your loan value, so we're discharging that remaining balance.

That may be what they're saying—I have no idea. But what they are doing (or attempting to do) is a transfer of loan debt from the borrower to US taxpayers, present and future. This shouldn't be hard to understand.

The loaner of the $ already received their loan amount, plus interest.

The "loaner" is the buyer of US government debt: t-bonds, t-notes, and t-bills. They would not (and should not be expected to) get less than the promised return on their investment. They expect to be paid back in full, in real money.

In fact, if the government did try to stiff its creditors, that would be a default. There would be headlines. And the whole government financial scheme would teeter. And…

Well, it wouldn't be pretty.

And 18 yr old old bowerers are ill prepared to understand the long term ranifications of these high interest loans.

Gee, I always kind of suspected it was a mistake to let these kids vote. If "bowerers" can't understand "ranifications" of their freely-chosen financial obligations, what are the chances they'll make wise choices in the voting booth? Worse than a coin-flip, I'd bet.

Better to forgive the remaining balance to allow their spending to help the economy.

… which ignores the money taxpayers will be shelling out to cover the "discharged" debts of student borrowers. That's money those taxpayers will not be "spending to help the economy."

Recommended reading: That Which is Seen, and That Which is Not Seen, by Frédéric Bastiat.

Like we did with banks.

Yeah, that was a bad idea too. But at least (at the time) people were making the argument that it was necessary to prevent the US financial system from collapsing. That is not the case with student loan bailouts. It's just naked vote-buying, and nobody's bothering to pretend differently.

And at least the bank bailout was (more or less) OKd by Congress. That's not the case with student loan bailouts either. The shift (both proposed and enacted) of debt burden from borrowers to taxpayers is carried out via executive decree.

I'm currently reading C. Bradley Thompson's America's Revolutionary Mind, an examination of the political theory behind the Declaration of Independence. The Americans of that time would not have hesitated to call this "taxation without representation". And an instance of tyranny.

Sadly, we live in more docile times.