Figuring out the Bol-TZ-mann statistics

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StarDate: 197694.29

The first step to getting off this god forsaken world will be to have a means of propelling us up from the surface and in to orbit, so that's where we'll begin. Our data bank contains information on a man by the name of Bol-TZ-mann who's astonishing work may come in handy for this. He spent his days studying how we can understand complexities of a gas without understanding how every part of it moves; statistics.

However we first need to clarify that the statistics that we are going to look at and use, only apply for idealised gasses. An ideal gas is one where particle to particle interactions are nonexistent, and the gas is viewed as many point-like particles moving around randomly. 

In our case, since we're using hydrogen gas(H2), this approximation is sufficient, because H2 has very very very weak intermolecular attractions, and because it's size is so very very tiny.

As we start using this Bol-TZ-mann's knowledge to construct ourselves a rocket motor to blast us out from this dust ball, we first have to understand his work. He described, among other things, how random things in nature will tend to have a certain value, though few if any will actually exactly that value. Most things have values close to this, but not exactly this value. He even discovered how one could mathematically describe how many would be how close to this expected value of certain phenomena in nature. His formula gives us that \(P(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^{2}}\), where \(P\) is how many percent of all values end up that exact point, with \(\mu\) being around where we'll find them and \(\sigma\) being how far away from \(\mu\) we can expect to find certain numbers of them. This also means that the integral from one point to another on this should give us how likely it is to find something that is between the points we've taken the integral between.

Anyway, back to what matters here. It turns out that how fast all the particles in a gas moves isn't actually described by this. Oh well, no luck there then. No, not so fast. The speed of parts of a gas may not be described by this, but it turns out velocity in every spatial dimension is! And, even better; this Bol-TZ-mann character knew a thing or two about how this distribution would look for gases! Huzza, we're back in business!

What he discovered was that for a temperature \(T\) in a gas where every particle has a mass of \(m\), we can expect to find that the standard deviation \(\sigma\) for the gas will be \(\sqrt{\frac{k_b T}{m}}\). Also, since the particles can move either in a positive direction or a negative one, it makes sense that the average value where we'll expect to find a particles speed is \(\mu = 0 \frac{m}{s}\).

Bildet kan inneholde: tekst, plott, linje, skrift, diagram.
This is how a velocity distribution looks with velocities in the range \(v_{x} = [-2.5,2.5]*10^{4}\)

 

This velocity distribution will look the same in the other two spatial dimensions.  

BUT! So far we've only looked at the velocity distribution of our particles, what about the positional distribution?! When you think about it, there's not really any reason for a particle to be more likely to be in one place than another, since they're already been flying around all over the place long before we start studying them. It's therefore a safe assumption that they'll be uniformly distributed.

OK, so now we know how we can describe our gas. Well, now what? Now, if we assume all particles in a gas are uniformly distributed, we now have what information we need to create our own gas with pencil and paper. And, luckily, we have our tiny robot helpers to create our pencil and paper gas for us. We will be working on this until our next log entry

 

A more detailed derivation can be found here.

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Av Jarl-Robin B. Evensen, Iver Oknes
Publisert 16. sep. 2020 12:05 - Sist endret 16. sep. 2020 13:52