Geometry of Electron Shells

Davidjayjordan

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In a true debate, both sides present their evidence or theories, rather than just one being hit with questions. So allow me to ask for your opinions or theories on why there are 2, 8, 18, and 32 electrons in the shells going outward from the Nucleus of an atom.

You can of course say it is because of evolutionary chance and luck, or just say Mother Nature made it that way, or that there is no rthyme or reason or logic or matghematics or physics regarding such shells.

Or you can give your scientific reasoning, or mathematical reasoning.

I would hope for the latter rather than the former.

Thanks.


And maybe after you have FIRST been given a chance to stand behind some theory, and answer some questions about your theory.... I might give my thoughts.

But you go FIRST, the floor is yours.
 
Tell ya what, sport. When you can show that you understand the concepts of quantum numbers and orbitals, maybe I'll get back to you.

Otherwise, you're not worth the time.
 
Davidjayjordan said:
In a true debate, both sides present their evidence or theories, rather than just one being hit with questions. So allow me to ask for your opinions or theories on why there are 2, 8, 18, and 32 electrons in the shells going outward from the Nucleus of an atom.

...

But you go FIRST, the floor is yours.
Click to expand...

If the floor is mine, then I would like to use it to accuse you of movng the goalposts.

In a different thread, you argued that the Platonic solids were somehow the building blocks of the universe. You stated:

Davidjayjordan said:
Allow me to suggest that the geometry that the ancients knew and passed on in their Mystery Schools is the very basis that would help you connect up what you can't comprehend about all of creation.

...

The philosopher Plato concluded that they must be the fundamental building blocks – the atoms – of nature, and assigned to them what he believed to be the essential elements of the universe.
Click to expand...

I pointed out that as our ability to observe the universe gets better, these Platonic solids (cube, dodecahedron, icosahedron, octahedron, and tetrahedron) appear to have less and less to do with the natural order of anything. They are not the fundamental building blocks of matter. I pointed out that the shapes of atoms (teardrops, doughnuts, barbells, etc.) have no relation at all to these geometric oddities that the Greeks liked so much.

In answer to that criticism, you did two things: 1) you abandoned the thread; and 2) you moved your argument from talking about the shape of atoms to talking now about the number of electrons in atomic shells.

I can only assume that you intend to dazzle us with numerology showing some connection between the platonic solids and the number of electrons in electron shells. Of course, I have two objections: 1) This was not your original argument; and 2) Being all relatively low numbers, there is a 100% chance that there will be some sort of correspondence between some of them.

So go ahead and demand that we answer your questions. However, I shall not. I have seen it for what it is: a trap designed to get us to admit to certain numbers so that you can draw your conclusions about Greek geometry.

Good luck with your illogical nonsense.
 
Buckaroo said:
Tell ya what, sport. When you can show that you understand the concepts of quantum numbers and orbitals, maybe I'll get back to you.

Otherwise, you're not worth the time.
Click to expand...

Golly, I just realized that I didn't provide you with a *way* to demonstrate your understanding. Here's a good way to do it: solve Schrodinger's equation for the hydrogen atom, and tell us about it. If you can't, then STFU.
 
Davidjayjordan said:
In a true debate, both sides present their evidence or theories, rather than just one being hit with questions. So allow me to ask for your opinions or theories on why there are 2, 8, 18, and 32 electrons in the shells going outward from the Nucleus of an atom.

You can of course say it is because of evolutionary chance and luck, or just say Mother Nature made it that way, or that there is no rthyme or reason or logic or matghematics or physics regarding such shells.

Or you can give your scientific reasoning, or mathematical reasoning.
Click to expand...

This is actually VERY well understood.
http://en.wikipedia.org/wiki/Electron_shell_configuration

I don't feel like duplicating the whole argument, but basically all it is is counting up the number of orthogonal spherical wave states that it's possible to construct (plus a factor of two from the electron spin). Nothing mysterious at all. And really, even a little bit of work on your part would have found this source. But you're not interested in actually finding out what science already knows, are you?
 
Hell, I'll even make it easy for you:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html

This site will hold your hand all the way through solving the PDE for the hydrogen atom. Go for it. Notice how the quantum numbers fall out of the equation? Oh, and you might want to learn something about the Pauli Exclusion Principle, as well. I can point you in the right direction, but you've got to do the work.
 
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It's all bound up with Schrodinger's wave equations, I believe. It's not so much that the shells are like solid structures of a different shape, but because the probability of finding electrons between the shells is zero. Can't do the maths, I'm afraid.

The issue you're referring to is electron shell closure. As far as I remember, this is tied up with "quantum numbers" and Pauli's exclusion principle, which states that no two electrons can share the same set of four quantum numbers. The quantum numbers determine the electron's shell, subshell, spin (this distinguishes between the two electrons inhabiting the same orbital) and the shape of the orbital.

There's a good explanation at http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html

I guess you could with equal justification ask why all snowflakes are hexagonal, but it's not magic; it's tied up with the crystalline proprties of water.
 
Wow! That's a lot of posts in the time it took to write one! Still, I can rest easy that this thread is in more competent hands than mine.
 
Traps or whatever aside I am interested to find out why these numbers occur. I can't work out what "all it is is counting up the number of orthogonal spherical wave states that it's possible to construct (plus a factor of two from the electron spin" means. Nor can I glean it from the Wikipedia article.

I can see from the diagrams that the electrons occupy a cloud space and those spaces cannot overlap. So I can see that 1 would just be a simple ball and two would be two balls. Is the doughnut between two balls the next natural shape to occupy the least space or something?

Is there a simple(ish) answer. I'm capable of understanding technical subjects but there are a lot of technical words in the article that lead to other words and so on leading me to believe it's going to take me months to understand more than I need to get the general idea.
 
Splossy said:
Is there a simple(ish) answer. I'm capable of understanding technical subjects but there are a lot of technical words in the article that lead to other words and so on leading me to believe it's going to take me months to understand more than I need to get the general idea.
Click to expand...

Unfortunately, there isn't a really simple answer that goes beyond the descriptive and into the explanatory. Basically the quantum numbers and particular shapes of the orbitals that describe the electrons in an atom fall out of solving Schrodingers equation, which is a partial differential equation that gets pretty hairy once you go beyond very simple cases.

One of the upshots of this equation is all the chemical bookkeeping we learned in high-school or college chemistry. I was frustrated as hell with these courses for not being able to tell me why the rules were the way they were, until I took my first quantum mechanics class.
 
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Loss Leader said:
If the floor is mine, then I would like to use it to accuse you of movng the goalposts.

In a different thread, you argued that the Platonic solids were somehow the building blocks of the universe. You stated:



I pointed out that as our ability to observe the universe gets better, these Platonic solids (cube, dodecahedron, icosahedron, octahedron, and tetrahedron) appear to have less and less to do with the natural order of anything. They are not the fundamental building blocks of matter. I pointed out that the shapes of atoms (teardrops, doughnuts, barbells, etc.) have no relation at all to these geometric oddities that the Greeks liked so much.

In answer to that criticism, you did two things: 1) you abandoned the thread; and 2) you moved your argument from talking about the shape of atoms to talking now about the number of electrons in atomic shells.

I can only assume that you intend to dazzle us with numerology showing some connection between the platonic solids and the number of electrons in electron shells. Of course, I have two objections: 1) This was not your original argument; and 2) Being all relatively low numbers, there is a 100% chance that there will be some sort of correspondence between some of them.

So go ahead and demand that we answer your questions. However, I shall not. I have seen it for what it is: a trap designed to get us to admit to certain numbers so that you can draw your conclusions about Greek geometry.

Good luck with your illogical nonsense.
Click to expand...

Leader, just learn to lead rather than follow. State your theory FIRST, then I shall state mine. The floor remains yours.
 
Buckaroo said:
Unfortunately, there isn't really simple answer that goes beyond the descriptive and into the explanatory. But the upshot is that the quantum numbers and particular shapes of the orbitals that describe the electrons in an atom fall out of solving Schrodingers equation, which is a partial differential equation that gets pretty hairy once you go beyond very simple cases.

One of the end results of this equation is all the chemical bookkeeping we learned in high-school or college chemistry. I was frustrated as hell with these courses for not being able to tell me why the rules were the way they were, until I took my first quantum mechanics class.
Click to expand...

Buckaroo, may I suggest that truths are simplier than you can imagine. Just learn to discern what is important and then formulate an opinion based on important truths.

Lifes mysteries are not just for the so called intellectual, for they sometimes are totally blind and bias. True skeptics are limited more by lack of insight as most of us surely have the God given brains to figure out what has been created.

Don;t look to others to SOLVE problems, when we personnally can figure out much much more than what we think, if we THINK.

So give it a go, and see what you come up with. Thanks
 
Honestly....do you really think you are likely to have a theory that competes with the quantum theory?

Some unbelievably smart people dedicated every waking minute of their lives to come up with these theories. Maybe I'm wrong but I have a feeling your theory won't be quite so well thought out...

Buckaroo: would I be right in saying that these shapes are purely mathematical contructs anyway? Do the theories suggest those shapes actually exist in 3d space in a meaningful way?
 
Splossy said:
Traps or whatever aside I am interested to find out why these numbers occur. I can't work out what "all it is is counting up the number of orthogonal spherical wave states that it's possible to construct (plus a factor of two from the electron spin" means. Nor can I glean it from the Wikipedia article.

I can see from the diagrams that the electrons occupy a cloud space and those spaces cannot overlap. So I can see that 1 would just be a simple ball and two would be two balls. Is the doughnut between two balls the next natural shape to occupy the least space or something?

Is there a simple(ish) answer. I'm capable of understanding technical subjects but there are a lot of technical words in the article that lead to other words and so on leading me to believe it's going to take me months to understand more than I need to get the general idea.
Click to expand...

Good thought processes, Slossy as it is obvious you are truly trying to figure it out. Good on ya.
 
Djj,

the theory has already been expressed.

I would say that before you're even challenged to solve the Hydrogen atom, you demonstrate to us your knowledge of spherical coordinates, and demonstrate transformations between cartesian and spherical coordinates, and demonstrate basic proficiency with the calculus used to solve problems in spherical coordinates.

Then you can tell us about Schroedinger's equation, the case of the time-independent Schroedinger equation, then solve it for simple potentials in the one dimensional case... AND THEN... in spherical coordinates.

That's the sequence (more or less) that it's done in 'weak, useless, sheep-to-the-slaughter' physics programs. I should know.

...

Or you can accept that the Pauli Exclusion Principle has been verified by experiment.

...

ETA: Your computer uses transistors. If you don't understand, or worse, deny the relationship between Quantum Mechanics and the technology you're using, you shouldn't be allowed to use it. By the way, from engineers everywhere... for the ability to express your idiotic opinions on forums like this... you're welcome.
 
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Big Al said:
It's all bound up with Schrodinger's wave equations, I believe. It's not so much that the shells are like solid structures of a different shape, but because the probability of finding electrons between the shells is zero. Can't do the maths, I'm afraid.

The issue you're referring to is electron shell closure. As far as I remember, this is tied up with "quantum numbers" and Pauli's exclusion principle, which states that no two electrons can share the same set of four quantum numbers. The quantum numbers determine the electron's shell, subshell, spin (this distinguishes between the two electrons inhabiting the same orbital) and the shape of the orbital.

There's a good explanation at http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html

I guess you could with equal justification ask why all snowflakes are hexagonal, but it's not magic; it's tied up with the crystalline proprties of water.
Click to expand...


Big Al, a little complicated.... but GOOD possible Research and a good contribution worth studying.

As there is a difference between shells, a space between shells which literally establishes shells. But why are they in these shells, and why does it take 2, then 8, then 18, then 32 electrons to fill up those shells.

Is the answer simplier....
 
Splossy said:
Buckaroo: would I be right in saying that these shapes are purely mathematical contructs anyway? Do the theories suggest those shapes actually exist in 3d space in a meaningful way?
Click to expand...

Yes and no! The shapes are mathematical constructs, describing the probability (after a little more manipulation) that an electron will occupy the space they encompass. So they also correspond to real 3D space. These kinds of things become tricky once we start talking about wave-particle duality!
 
Reading the links provided I am again reminded of the wonderful individual and collective intelligence of scientists in this field. The depth of thought and rigor is really quite amazing.
 
Splossy said:
Traps or whatever aside I am interested to find out why these numbers occur. I can't work out what "all it is is counting up the number of orthogonal spherical wave states that it's possible to construct (plus a factor of two from the electron spin" means. Nor can I glean it from the Wikipedia article.
Click to expand...

First, let's establish what we mean by orthogonal. Well, with vectors in Euclidean space, the concept is pretty simple: it means that they're 90 degrees apart. One test for orthogonality is that their dot product is zero. Well, it turns out that this concept is generalizable, and can be applied to much more than just vectors in 3D euclidean space. It can also be applied to functions. So what's the equivalent of taking a dot product of a function?

First, let's review dot products for vectors. Let's say we have two vectors, V and W. We can label their components as Vx, Vy, and Vz, and Wx, Wy, and Wz. Now the dot product is
V dot W = Vx Wx + Vy Wy + Vz Wz
We just multiply each component, then add them up.
Now for functions. Let's say we have functions f(x) and g(x). The equivalent to multiplying each component is to multiply the values of the function at the same point, or f(x)g(x). Now we just add all that up, or in other words, integrate the functions. So
[latex]f(x) \cdot g(x) = \int{f(x) g(x) dx}[/latex]
Let me give you an example of two functions that are orthogonal: sin(x) and sin(x+pi/2). Those two functions are orthogonal, and ANY function of the form sin(x+c) (where c is an arbitrary constant) can be expressed as some unique linear combination of those first two orthogonal functions. This is equivalent to picking out an x and a y axis in a 2D plane, and expressing any vector as some linear combination of unit x and y vectors. We could have picked other orthogonal functions: sin(x+pi/8) and sin(x-3pi/8) also work, for example. But we can only pick two functions of this form which are mutually orthogonal, any third function cannot be orthogonal to both of them.

This is only a first step, but have you followed so far?
 
Davidjayjordan said:
Buckaroo, may I suggest that truths are simplier than you can imagine. Just learn to discern what is important and then formulate an opinion based on important truths.

Lifes mysteries are not just for the so called intellectual, for they sometimes are totally blind and bias. True skeptics are limited more by lack of insight as most of us surely have the God given brains to figure out what has been created.

Don;t look to others to SOLVE problems, when we personnally can figure out much much more than what we think, if we THINK.

So give it a go, and see what you come up with. Thanks
Click to expand...

See, here's the thing -- we KNOW THE ANSWER. There are no mysteries in this particular subject. Furthermore, the answer is simply and elegantly expressed in mathematics. There's nothing "intellectual" about it -- like any foreign language, you just have to study it. I have personally solved these problems using this language, and I understand them. You have not, and you do not. You are profoundly ignorant on this subject. So don't presume to lecture ME.
 
Ziggurat said:
First, let's establish what we mean by orthogonal. Well, with vectors in Euclidean space, the concept is pretty simple: it means that they're 90 degrees apart. One test for orthogonality is that their dot product is zero. Well, it turns out that this concept is generalizable, and can be applied to much more than just vectors in 3D euclidean space. It can also be applied to functions. So what's the equivalent of taking a dot product of a function?

First, let's review dot products for vectors. Let's say we have two vectors, V and W. We can label their components as Vx, Vy, and Vz, and Wx, Wy, and Wz. Now the dot product is
V dot W = Vx Wx + Vy Wy + Vz Wz
We just multiply each component, then add them up.
Now for functions. Let's say we have functions f(x) and g(x). The equivalent to multiplying each component is to multiply the values of the function at the same point, or f(x)g(x). Now we just add all that up, or in other words, integrate the functions. So
[latex]f(x) \cdot g(x) = \int{f(x) g(x) dx}[/latex]
Let me give you an example of two functions that are orthogonal: sin(x) and sin(x+pi/2). Those two functions are orthogonal, and ANY function of the form sin(x+c) (where c is an arbitrary constant) can be expressed as some unique linear combination of those first two orthogonal functions. This is equivalent to picking out an x and a y axis in a 2D plane, and expressing any vector as some linear combination of unit x and y vectors. We could have picked other orthogonal functions: sin(x+pi/8) and sin(x-3pi/8) also work, for example. But we can only pick two functions of this form which are mutually orthogonal, any third function cannot be orthogonal to both of them.

This is only a first step, but have you followed so far?
Click to expand...


Hmm...I'd LIKE to say yes but I can only honestly answer sort of. I don't know what a dot product is and looking that up didn't really help at all.

I appear to have forgotton almost all my A-level maths and physics and some of the earlier stuff too. I'm having to look up what a vector is and work from there.

I think you are probably wasting your time trying to help understand this.
 
Dear researchers,

Can I suggest that if you just look at the relationship between the numbers 2, 8, 18 and 32. ...and its expansion. You will find a progression pattern between them, that will be the START to the solution. Its not very complicated at all, easy enough for any of us to understand.
 
Now this is more like it! So I don't need to bother with all that pesky maths and quantum theory afterall...
 
Buckaroo said:
See, here's the thing -- we KNOW THE ANSWER. There are no mysteries in this particular subject. Furthermore, the answer is simply and elegantly expressed in mathematics. There's nothing "intellectual" about it -- like any foreign language, you just have to study it. I have personally solved these problems using this language, and I understand them. You have not, and you do not. You are profoundly ignorant on this subject. So don't presume to lecture ME.
Click to expand...

If you know the answer, do state the answer, so others can know
 
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Davidjayjordan said:
If you know the answer, to state the answer, so others can know
Click to expand...

I already gave a link with the complete explanation. And yes, the explanation INCLUDES the regular progression of the numbers:

2*(1) = 2
2*(1+3) = 8
2*(1+3+5) = 18
2*(1+3+5+7) = 32
and so on.

Furthermore, that link gives a physical explanation as to WHY that progression of numbers (and not some other random number progression) is the correct one.
 
Splossy said:
Now this is more like it! So I don't need to bother with all that pesky maths and quantum theory afterall...
Click to expand...

Splossy, after thinking a bit, I remembered that you can make a rough analogy by considering a length of taut string that has been plucked so that standing waves are set up along it. The "quantum numbers" of this system would correspond to the integer count of the number of peaks in the standing wave. The string in this case would represent the probability amplitude of the quantum wavefunction describing the electron.

Not exactly the same as an atom, but it gives you a flavor of where the solution comes from, with no math.
 
Davidjayjordan said:
Dear researchers,

Can I suggest that if you just look at the relationship between the numbers 2, 8, 18 and 32. ...and its expansion. You will find a progression pattern between them, that will be the START to the solution. Its not very complicated at all, easy enough for any of us to understand.
Click to expand...
Actually, the non-mathematic/simplistic explanation is as follows:

Electrons in an atom have to be in an orbital, which is a mathematically-defined volume of space. Each orbital is described by three quantum numbers- n (which determines the size of the orbital, and the number of nodes = areas which have no electron density), l (which determines the shape of the orbital, and the number of planar nodes), and ml (which determines where in the x,y,z directions the orbital is pointing). No two orbitals can have the same set of three quantum numbers; no two orbitals can be identical, and each orbital can hold two electrons.
In the first shell, n = 1, and there are no nodes (how does it smell? Terrible!), so there is no planar nodes, and no variation in shape or direction. There can only be one orbital in the first shell. It is spherical and is called the 1s orbital (s doesn't actually stand for sphere, but it's an easy way to remember it). Therefore the first shell can have two and only two electrons.

In the second shell, n=2, and there is one node. This could be a radial node, which makes another spherical orbital (the 2s orbital), which has no variation in direction. We could also have a planar node, and this plane could lie in the xy plane, the xz plane, or the yz plane. We have three orbitals differing in their orientation in space, so we have the 2px, the 2py, and the 2pz orbitals. Grand total of 4 orbitals, so eight electrons.

Why do the 2p orbitals appear as dumbell shapes? If you take a spherical balloon and twist it so that there's no air in the centre, you now have the basic shape of something with a planar node. A dumbell.

In the third shell, n = 3, so we have two nodes.
Two radial nodes gives us the 3s orbital.
One radial and one planar gives us three 3p orbitals.
Two planar nodes give us the d orbitals. The shape of most of these is given by taking your balloon, twisting it in the centre to give a planar node, and then twisting it again along another axis (you can do it if your balloon isn't inflated very much)- you get a cloverleaf shape. The odd one out (the donut + dumbell shape) is made by taking the two nodal planes and mathematically adding them to give a nodal double-cone. The odd shape is simply what's left of a spheroid when you subtract a double cone from it.

Why are there 5 d orbitals in a shell? It's because of the possible values of ml. l can be anywhere from zero to one less than n; ml can be from -l to +1 (integer values only). So if n = 3, l can be 0 (3s), 1 (3p), or 2 (3d). When l = 2, ml can be -2, -1, 0, +1, +2. Five possible values = 5 solutions to the Schroedinger Eq'n = 5 orbitals.

So the third shell has 1 + 3 + 5 = 9 orbitals, and thus can hold 18 electrons.

The fourth shell has one 4s orbital, three 4p orbitals, five 4d orbitals, and seven of the next set, the 4f orbitals (which look like you've twisted any of the d orbitals to give another planar node in the balloon). 1 + 3 + 5 + 7 = 16, so 32 electrons can fit in the fourth shell.

The number of orbitals in a shell is always equal to n^2, so the number of electrons that can fit in the shell is always twice a perfect square, which is the amazing pattern you were pointing out earlier.

If you want to see the shapes of orbitals, I recommend the Orbitron Gallery:
http://winter.group.shef.ac.uk/orbitron/index.html

Any more chemistry-related numerology?
 
Davidjayjordan said:
In a true debate, both sides present their evidence or theories, rather than just one being hit with questions. So allow me to ask for your opinions or theories on why there are 2, 8, 18, and 32 electrons in the shells going outward from the Nucleus of an atom.
Click to expand...

There aren't

The real numbers are 2,2,6,2,6 then it gets mesy.
 
Ziggurat said:
I already gave a link with the complete explanation. And yes, the explanation INCLUDES the regular progression of the numbers:

2*(1) = 2
2*(1+3) = 8
2*(1+3+5) = 18
2*(1+3+5+7) = 32
and so on.

Furthermore, that link gives a physical explanation as to WHY that progression of numbers (and not some other random number progression) is the correct one.
Click to expand...

I'm almost out of popcorn.
 
Davidjayjordan said:
Dear researchers,

Can I suggest that if you just look at the relationship between the numbers 2, 8, 18 and 32. ...and its expansion. You will find a progression pattern between them, that will be the START to the solution. Its not very complicated at all, easy enough for any of us to understand.
Click to expand...

Here's a thought -- instead of being all coy and beating around the bush, why don't you TELL US WHAT YOU BELIEVE. What, are we just supposed to take your word for it that you've come up with a solution that is superior to what we know from quantum mechanics?
 
Davidjayjordan said:
Leader, just learn to lead rather than follow. State your theory FIRST, then I shall state mine. The floor remains yours.
Click to expand...

In accordance with your wishes, my theory is that you are clumsily attempting to set a trap while avoiding answering the points raised in your Platonic solids thread.

Now you state your theory.
 
You MAY know quantum MECHANICS.
YOU may know CHEMISTRY.
You may KNOW maths.
but you DO not KNOW phi.

BEHOLD.

I apologise if this is a DIFFICULT topic to introduce. Behold FIBONACCI numbers;

1, 1, 2, 3, 5, 8, 13, 21,

DO you see KNOW?

Take

the 1st and 2nd number = 2

the 4th and 5th number = 8

the 3rd, 4th, 5th and 6th number = 18

the 2nd, 3rd, 4th, 5th, 6th, 7th number = 32

Do YOU see know?

Do you SEE know?

Do you see KNOW?
 
Maybe it would be more fun to guess what DJJ's theory actually is. There's mine. I've a funny feeling it might not be too far wrong :)
 
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