Comment by philzook
1 year ago
I've got a related one I like. Why are the Avogadro's number and Boltzmann's constant inverses of each other N ~ 1/k? The statement doesn't make sense because the units don't work out, but it is true in mks. It's because they multiply to the gas constant which is ~1. They both are numbers to transfer from the microscopic to human scale units and they cancel for the gas constant, which is about human scale experience of gases.
Funnily Avogadro's constant is actually equal to 1: it's defined as Avogadro's number times mol, but mol is itself a dimensionless quantity equal to the inverse of Avogadro's number.
Multiplying by increasingly complicated expressions equivalent to “1” is what I remember doing for almost every problem in Quantum Mechanics.
But it's a coincidence, right? N*k=8.31 is pressure*volume/temperature for a mole of gas. Temperature has a relatively small range (100-1000) and there's no reason why the range of P*V couldn't be far from that range, for example 0.01–0.1, with a different definition of meter, second or kilogram.
Meter, second, and kilogram were all chosen to be approximately the scale of a human, and the combined multiplicative units like Pascal, m^3, and Kelvin/Celsius are also numerically 1 in these units.
> Meter, second, and kilogram were all chosen to be approximately the scale of a human,
“Approximately the scale of a human” leaves so much wiggle room that I don’t see how one can defend that claim.
> and the combined multiplicative units like Pascal
You don’t explicitly claim it, but I wouldn’t say the Pascal is “Approximately the scale of a human”. Atmospheric air pressure is about 10⁵ pascal, human blood pressure about 10⁴ pascal, and humans can very roughly produce about that pressure by blowing.
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Pascal is pretty small and m^3 is pretty large though. That's why we still often use millibar and liter. They just happen to largely cancel out.
A human is ~1 meter, 1e2 kilograms and 1e9 seconds.