I vaguely recall a professor in university stating that the octet rule was a general guideline and that deeper principles guide the number of electrons in each shell of an atom.

Is there a simple way to explain this?

The octet rule is only useful for elements where the S and P orbitals are the valence orbitals. The S orbital has room for two electrons and the P orbital has room for six electrons. Atoms like to have their valence orbitals filled so at least for the right side of the periodic table, they will try and share electrons with other atoms to make it up.

A neutral nitrogen atom has five electrons. When it forms the molecule of Nitrogen, it shares three of these electrons with the neighboring nitrogen. This makes six electrons, plus the two left over giving both nitrogen atoms eight electrons.

It is only a “guideline” because numerous examples exist where it doesn’t apply. If you try to draw a lewis structure for carbon monoxide, you either have to make the carbon negative and the oxygen positive, or leave the carbon with only six electrons. Technically the true nature is a hybrid of both structures and this plays out in its chemistry.

Atoms to the left of the periodic table tend to just lose electrons to get a full valence set, so Sodium is almost allways a cation and Magnesium is almost always a dication.

Hydrogen (and Helium if it bonded), has a two electron rule.

Transition metals have D orbitals to contend with. D orbitals have room for ten electrons. Add that to the S and P orbitals, and transitions have a tendency to follow the eighteen electron rule. There are even more exceptions to this than the eight electron rule though.

Elements towards the bottom of the periodic table also have D orbitals to content with, so even non-metals can be really wishy washy on the octet rule. You can see this in the formation of hypervalent molecules such as SF6.

Its this part that I meant in the OP. Is there a simple way to explain the overall principles that guide how many electrons are in each shell. For example, why does the first shell only hold two electrons. Why not 5 or 50?

Allright, I’m going to have to strain my brain to go much deeper than this.

This has to do with the quantum numbers associated with each molecular orbital. The most important quantum number is the principle quantum number (N). The lowest number is 1.

For a principle quantum number of one, the only available angular momentum(l) number is zero. The angular momentum always has an absolute value less than the principle quantum number.

An angular momentum of 0 is an S orbital. The pauli exclusion principle says that you can’t have more than one electron in the same state at the same time. Since an electron can either have a spin (m[sub]s[/sub]) 1/2 or -1/2, there is room for only two electrons in this orbital.

If you have N = 2, then l can be 1 or 0. For l = 0 you have an S orbital with room for two electrons. For l = 1, one more quantum number has to be introduced called the magnetic moment (m[sub]l[/sub]). For l = 1 then m[sub]l[/sub] can be either 1, 0 , or -1. That accounts for p[sub]x[/sub], p[sub]y[/sub] and p[sub]z[/sub]. Each electron can have m[sub]s[/sub] = 1/2 or -1/2. That makes room for six electrons.

For D orbitals:

N = 3

l = 2

m[sub]l[/sub] = -2 , -1, 0, 1, 2

m[sub]s[/sub] = -1/2, 1/2

That makes room for ten electrons.

These seem like man made rules. What aspects of the natural world restrict the inner shell to hold only two electrons? Maybe this goes beyond my mental ability.

Fundamentally, it all goes back to the Schrodinger equation and the Heisenberg uncertainty principle. They in turn were derived from a combination of keen mathematical insight and many kinds of experimental results. For example, spectral analysis.

Ed

I probably could have explained this when I was taking Physical Chemistry. That class was so traumatic though that I erased most of the memories of it with alcohol.

The Pauli Exclusion Principle explains why a given orbital can hold only two electrons, if you can wrap your head around Wave Functions

Basically, electrons occur in the configurations they do because that’s the way the wave functions work out.

Right, and the wave function is the Schrodinger equation.

Ed

It sure looks like man made rules when all you get is the final answer.

Let me try to explain the process it took to get that answer.

The physical/mathematical problem to solve is one where a “light weight” charged particle interacts with a “heavy weight” particle of equal sized charge, but an opposite sign charge. One charge is positive and the other is negative, although thy are both “one unit” of charge. The masses of both particles are small enough that we don’t have to worry about their gravitational interactions. The sizes and distances of the particles are such that we can treat them as geometric points.

The solution, or solutions, we’re looking for are the steady state ones. So we only have to worry about the time-independent Schrodinger equation, not the time-dependent one. One less thing for us to worry about.

The only force between the two particles is the Coulomb force. The force in this case is attractive, since the two particles are of opposite charge. The Coulomb force depends only on the charge of the particles, and the size of the distance between them.

The most natural way to treat this system is to set up a sphere, with the nucleus (the heavy particle) at the center. We can locate the electron (the light particle) on the shell of a sphere (whose diameter is the distance between the electron and nucleus), and at a certain “longitude” and “latitude”. If the electron moves closer to the nucleus, we’ll place it on a smaller sphere; if the electron moves away, it’s on a larger sphere.

Note that the Coulomb force doesn’t “care” where the electron sits on the sphere, only on the radius of the sphere.

Now plug that Coulomb force into the Schrodinger equation and solve - the version of the Schrodinger equation that is independent of time. We’re trying to solve the three dimensional Schrodinger equation.

The standard technique when solving a three-dimensional differential equation, and the Schrodinger equation is in that category, is to separate it into three equations - one equations for each of the dimensions. If you’ve never gotten to that level of math, you would not have heard of this. We’re left with three equations to solve.

When one tries to solve these three equations, there’s a problem. The solutions only converge - come to a steady form - if certain constants that come up while solving the equations have integer values.

That’s where the magic numbers come from! If you don’t get these “quantum numbers”, the equations cannot be solved.

It also turns out that the three quantum numbers - you get one for each equation - are related to each other. The quantum number for the radial equation cannot be less than one. It turns out the radial equation sets the energy of the electron (the force is only related to distance from the nucleus, after all). The quantum number for one of the angles (I forget whether it’s the “north/south” angle or the “east/west” one) is greater than or equal to zero, but strictly less than the principal (radial) quantum number. The quantum number for the other angle ranges from the negative of the second quantum number to the positive value.

With these three quantum numbers we can describe any state of the electron in a hydrogen atom. The quantum numbers come from nature, and the nature of mathematics. It is surprising that nature would pick integers for this, but that’s the way it worked out.

Did that make any sense? It’s been a long day for me.

Yes, when I said I was going to have to strain my brain to get much deeper than this, this question is precisely the type of thing I was thinking about. In order to explain, why quantum numbers are meaningful, I have to dive into Hamiltonian mechanics that are well outside my field. At one time, I knew it, but I passed that cum years ago.

There is nothing man made about it. It is pure math. Based on the questions you are asking, it is way outside your understanding. Don’t feel bad, nobody here has answered it any better than I could either.

Sometimes your better off cutting your losses and taking everyones word for it.

Once upon a time a bunch of smart guys were trying to understand atomic structure. One specific smart guy named Erwin Schrodinger picked a bunch of physical principles that he felt must hold even for the very weird micro world.

He put the math that represents these principles together in a differential equation which we now call the Schrodinger equation. Miraculously, solving this equation gave us the answers to the questions you are now asking. (Almost, and sorta,)

Thanks so much for your effort. This is what I was looking for. I don’t understand it but I get the basic idea.