Why do electrons arrange into different “shells” around protons?

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Like why are there two shells sometimes and three shells other times? Why are there different numbers of electrons on one shell than others? And what’s in between the protons and electrons?

I’m a little stoney and just can’t comprehend how something at this small of a scale has these “laws of nature.” Idk… hard to explain why this is fascinating me but it is. Especially the space in between the protons and electrons. Like are we all just empty space floating around?

In: Chemistry
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They essentially want to find the constellation in which all forces acting upon them are equalized. Additionally electrons can only exist in a limited number of energy levels, which rises with the radius to the core. So electrons will always try to get near the core, but can only exist at very specific steps, and not at every possible distance. Think of this like groves in a wooden board around the middle. The electrons can only sit in these groves, and there can only be a limited number of electrons per grove.

From my understanding, there is nothing between the proton and electron, these shells are just energy levels and there are different number of electrons on the “shells” bc each have different amounts of energy.

Electrons are not little “balls of negative-ness”. They have wave-like properties. They can only “orbit” (that’s not how it really works) when there are an integral number of wavelengths in the path. Because they have spin (another property particles can have) it’s possible for up to exactly two electrons to share an orbital path.

The orbital shells aren’t all spheres, many of them have a figure 8 shape, and since three figure eights can be aligned along independent dimensions of out 3D space, layers with 6 electrons can also be built. These structures combine to force electrons into specific “slots”, defined by geometry, rather than any-old-arrangement-you-like.

The shells actually come from something called spherical harmonics.

You can probably imagine how a stretched string can ripple. You could have the middle bulging up and down. Or, you could have the middle staying still, and the sections on either side going up and down (opposite to each other). Or you could have 3 sections wobbling. These are called harmonics, and the more sections there are, the higher the frequency (and so the energy) of the wobbles. But, there is only one kind of vibration at each level.

Then we can move on to a 2D version. Imagine a circular drum skin. Again, the whole thing could go up and down, of your could have concentric rings (like a target) going up and down. These are like the simple wobbles of a string. But, you could also have more complicated patterns, where the left and right halves of the circle are doing different things too. There are some cool GIFs of it on [wiki](https://en.m.wikipedia.org/wiki/Vibrations_of_a_circular_membrane). Crucially, this lets us have multiple different versions in different levels, where there are different rings and different numbers of segments. And, the order of lowest to highest energy sort of jumps around between numbers of circles and numbers of segments.

Now, things get tricky (or trippy)! The 3D version is spherical harmonics. You could maybe think of something like air pressure as the thing that “wobbles”, as this is what sound is. The simplest shape would be a sphere, where the air pressure of the whole thing increases and decreases. But then we can have more complicated shapes. You could have a sort of dumbbell shape, or like two teardrops connected at the top, where the opposite sides increase or decrease in air pressure at opposite times. There are ones where there are 4 of these lobes connected. There are even some with a sort of doughnut-shaped lobe. But these are the different ways that vibrations around a central point can happen. And again, the order is a mix of increasing size and increasing complexity.

But what’s this got to do with electrons? Well, electrons don’t live around atoms like planets around the sun. They’re more like waves. Waves or probability. (I did say it gets trippy.) And because they are waves, they like to exist in these wave patterns of spherical harmonics. The number of electrons in each shell depends on how many arrangements of lobes there are at each level of complexity. For each possible arrangement, we can get two electrons because of something else called spin (think clockwise vs anticlockwise).

So, the first level is just a plain sphere. There’s only one shape, so we get 2•1=8 electrons. This gets us Hydrogen and Helium.

On the next level, we get another sphere, and then 3 shapes with pairs of lobes, pointing left/right, up/down, front/back. This is 2•(1+3)=8 electrons, and gets us Lithium to Neon.

Then we get another sphere, and another set of 3 double-lobes, so another 8 electrons and from Sodium to Argon.

The next layer gets tricky. We get the sphere and the 3 doubles, but we get an extra 5 more complicated shapes. These actually fall *between* the sphere and the doubles, giving 2•(1+5+3)=18 elements on the next row of the table.

Then, going further, we get even more complicated patterns so even more different shapes, and so more electrons per layer.

Negative electrons are dying to get to the positive protons in the atom nucleus. Typically there are an equal number of protons and electrons. Once the first electrons get in there, subsequent electrons can’t cozy up to the nucleus because they’re repelled by the existing ones. Each shell of electrons describe the equilibrium point where any closer, and they’d be repelled by other electrons and any further they’d be attracted by the nucleus.