What-If 0004: "A Mole of Moles"

[Yakk]

Note that this is strictly greater than 4, not >= 4.

Rounding on orders of magnitude should follow logarithmic rounding: So in base 10, <= 3 is rounded down, and >=4 is rounded up.

4^0.5 < 2.2 and 5^0.5 > 2.2, so you start rounding 2.2 down at base 5.

[sehkzychic]

For any integer base greater than 4, the number of pounds per kilogram can be rounded to 1 for order-of-magnitude approximations.

6.022x10^23.

602,214,129,000,000,000,000,000.

It's ultimately a subjective question of how much uncertainty you're willing to tolerate, but with numbers this big, when the relevant quantities (like gravitational force) are quadratically related to the size of the numbers involved, I think that 1lb = 1kg is an oversimplification. The doubling of mass takes the molemoon from half Luna's mass to 100% of Luna's mass. Certainly important for any mission to to colonize the molemoon. But let's say that this is combined with my earlier post about using bigger moles (the animal). You now have an object with ~20 times the mass of the original. That's 10 times the mass of Earth's moon. Earth itself is only about 6 times more massive than the moon (judging by gravity anyway). So you've got a planet larger than any of the terrestrial planets made (mostly) of water. As should be obvious to anyone reading this, I'm not a physicist, but a 20-fold increase in mass seems like it will have qualitative effects on the molemoonmoleplanet. Basically, my subjective opinion is that the whole 1lb = 1kg approximation fails when mixed with numbers this big.

[gmalivuk]

For any integer base greater than 4, the number of pounds per kilogram can be rounded to 1 for order-of-magnitude approximations.

6.022x10^23.

602,214,129,000,000,000,000,000.

It's ultimately a subjective question of how much uncertainty you're willing to tolerate

Yes, it is. And since this discussion is about order-of-magnitude approximations, we're evidently willing to err half an order of magnitude in either direction. For decimal numbers, this means a factor of 3.16. Since pounds-to-kilograms is only 2.2, that's within acceptable tolerances.

Certainly important for any mission to to colonize the molemoon.

Yes, and obviously you'd want better than order-of-magnitude napkin scratches if you were planning such a mission. Even the approximation that 1kg = 2.2lbs is off by 0.2%, which becomes significant if you want to have a particular orbit or land at a survivable speed.

[MM1]

Some of you may know this already, but the name of the animal in question derives from the fact that - to very high accuracy - it contains a mole of molecules.

[gmalivuk]

Not sure if you're intentionally joking or what, but that is completely and totally false, just fyi for anyone else who happens across this thread.

[sehkzychic]

I'm fairly sure that no one will be left confused. If they have the slightest understanding of what a mole is and how small most molecules are, they'll know that a mole of water molecules has to be ~18g. Since we've established that moles can be bigger than 18 grams, it's pretty obviously not true.

Wait, now that I think about it, considering how much heavier DNA molecules can be and that water is one of the lightest common molecules in an animal (unless you're counting metallic ions), and without doing even the most basic math, it seems like the mole-of-molecules thing could actually be truthy. Not sure about "high degree of accuracy," but for an animal of about 30-70 grams, it could actually be within a couple orders of magnitude.

Okay biomath time:

Spoiler:

60.4% of an average eukaryote is water by mass. For a 30-gram mole, that would be just about 18 grams of water, so let's assume it's a baby mole, or at least dehydrated. The star-nosed mole genome is about 1.8 gigabases, meaning that each copy of the genome is 1.8 picograms. Since a mole is about 1/2000th the size of a person, and 50trillion seems to be an average guess for a person's total cell count, we could guess that a mole could have about 25billion cells. These few hundred billion DNA molecules already add up to about 45mg. So we might be on the right track: not a lot of molecules, but quite a lot of mass. Considering there's proteins in our body like titin (almost 4GDa each) that make up a significant part of our muscles' structure, it's not crazy to think that a very small mole could actually have a mole of molecules in its body.

Of course, etymologically, it's ridiculous, but kind of a neat linguistic coincidence.

[brainboy]

Oh me yarm! "a mole of moles!" Is this a common question found in the math and science geek universe? The reason I ask... forgive me, I'm not a math geek... I was a band geek, back in high school in the 60s. We had a jolly chemistry teacher who came up with questions similar to the ones on this board and in What If? There were no personal calculators back then, kids. We had to learn how to use a slide rule. Which I never quite mastered, but I do remember a test question that my teacher was fond of: "If a mole could dig a mole of holes, how many moles of holes could a mole of moles dig?" Mr. Stone, wherever you are today, I know you're smiling.

[SecondTalon]

Oh me yarm! "a mole of moles!" Is this a common question found in the math and science geek universe?

Having not really taken much math or physics, I can safely say that yes. Yes it is. Incredibly common.

[Himself]

I am resurrecting this thread to pose a new question:
Would the mole moon convect? If we go with purely compressional heating then the temperature profile (molotherm?) would depend solely on pressure. Any parcel of meat moved up or down would change temperature at the same rate as the ambient adiabat and remain neutrally buoyant. This, of course, ignores other sources of heat. I don't know what chemical reactions the meat would undergo at such pressures. As far as I can tell, radioactive decay would not produce that much heat.

[Keybounce]

I'm kinda curious -- how long before decay, pressure, and such made a "proper" moon, solid core, surface, and such?