Naked Science Forum
General Science => Question of the Week => Topic started by: jamest on 05/04/2024 12:16:46
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David with a physics question: 'What law governs the distance of an electron from the nucleus of an atom?'
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Heisenberg's indeterminacy principle.
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The Pauli exclusion principle governs the distance of electrons.
It says that no two electrons in any atom can occupy the the same quantum state, which means that the cannot have the same two values for all four quantum numbers: principle, azimuthal, magnetic, and spin.
The best example of this is actually a neutron star, which is effectively a huge single atom with an absurdly large nucleus, much larger than the orbitals of the innermost electrons. So there can be but two electrons in the first orbital, six in the second, and so on. Thus almost all the electrons are forced far away from the center and it is the Pauli exclusion principle which prevents further collapse of the neutron star.
Heisenberg's indeterminacy principle.
Heisenberg's indeterminacy limits measurement of both position and momentum of something. It doesn't limit where something actually is, but quantum theory also does not have a concept of 'where something actually is'. It is an empirical theory and only specifies what one is likely to measure.
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Hydrogen only has one electron, so Pauli doesn't explain where it is!
Electrostatics "sucks" the electron towards the proton, but Heisenberg points out that the product of position and momentum is indeterminate (nothing to do with measurement, which would involve a photon that we don't have!) so it can't be stuck to the proton and has to occupy some kind of cloud of probability around the proton. Schrodinger then describes the possible shapes of the cloud for different electron energies and (subject to Pauli) increasing numbers of electrons and protons.
That said, it is remarkable that the rest mass of a neutron is very close to the sum of a proton and an electron, and a free neutron decays into a proton and an electron, so beware of pushing classical analogies!
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Hi.
Why are we arguing over fine details when the basics haven't been esatblished?
'What law governs the distance of an electron from the nucleus of an atom?'
The basic answer is that it depends on the model(s) you are using.
In a very simple model, the electron is attracted by electrostatic forces to the nucleus but the electron is assumed to be in some motion. So the electron orbits the nucleus like a planet orbiting a star. The laws governing everything are similar to Kepler's laws for planetary motion but where Newtonian gravity is replaced by Couloumb's law of electrostatic attraction.
There is now a whole set of more advanced models we can work through to suggest slightly different answers to the question....
Eventually, you reach a level of sophistication in the models where the basic question needs to be completely re-examined. For example, an electron may not be an ordinary particle as we might imagine, it may be a wave in an underlying field and thus "spread out" all over the place. The question becomes impossible to answer because there are no definite positions for any of the things mentioned in the question and hence no way to determine a distance as the difference between these two positions.
I propose that the best answer for the question is to take things only as far as you can so that words like "electron", "nucleus" and the distance between them continue to have the same meaning as that which is suggested in the question.
In practice, a fairly classical model of an atom works quite well and adequately describes the macroscopic or bulk properties of most materials. We can determine the typical distance between atoms by performing experiments involving X-ray crystallography or surface electron scattering (often called LEED = Low Energy Electron Diffraction). We can reasonably assume that in the solid state of this material, the atoms pack together as closely as they can so that they are effectively in contact with each other. Indeed we find that they are only a few crystal lattice structures for solids and simple cubic crystals (such as b.c.c. [body centred cubic] , f.c.c. and just s.c.) are prevalent.
It is found that most atoms have an effective radius in the order of 1 Angstrom = 10-10 metres. For example, Copper has a crystal lattice constant ≈ 3.6 A (Editing: The website won't support an A with a circle over it, A means Angstrom and any ? character is probably an Angstrom that I didn't edit) => atomic radius ≈ 1.8 A (note that there are two atomic radii between the centre of two atoms that are in contact).
Glance through this website ( https://periodictable.com/Properties/A/LatticeConstants.html - I'm not guaranteeing the reliability, it's just the first site that came to hand and readily displays the info we want). Although there is a little variability the lattice constants are all between 2 - 7 Angstrom.
Back this up with our favourite model of an atom, the Schrodinger Hydrogen atom solution, where the tpyical distance of the electron from the nucleus is the Bohr radius ≈ 0.5 A and note that this is one of the smallest atoms around.
Overall, it seems that as regards the bulk properties of materials, all atoms tend to have a radius of about 1 Angstrom. At first consideration, this is a bit surprising, e.g. Iodine has 53 electrons so you would have thought they occupy electron shells that are much further out than the 1s orbital of a Hydrogen atom. However, we need to balance this against the idea that the total charges involved are 53 times greater than that in Hydrogen and all the orbitals are effectively pulled in and are of much smaller radius. Indeed we get a fair explanation of the result by just considering the last electron of any atom. In a simple (often called a Hydrogenic model), the last electron is essentially outside all the other electrons and the nucleus, so it "sees" a total charge (in units of e) interior to its orbit that we can express in terms of Z, the atomic number:
(Total charge seen, in units of e) = (charge from the nucleus) + (charge from the other electrons)
= Z + -(Z-1) = 1.
We can approximate the behaviour of the outermost electron by modeling it as though it sees a total charge of just +1e at the centre of the atom. To say that more simply, the outermost electron would behave like an electron around a Hydrogen nucleus and so it's not at all that surprising it always has a typical distance from the centre of the atom equivalent to the distance displayed in the simple Hydrogen atom.
That's where I'd leave the answer. Summary follows:
1. The standard rule of thumb is that atoms have a radius ≈ 1 Angstrom irrespective of the atomic number of that atom, i.e. what that atom actually is. If you want a "simple law", you can't get much simpler than this. The distance between an outer electron and the nucleus is about 1 A.
2. This is what we see when we perform practical experiments on bulk materials (e.g. LEED experiments on crystals). We have the real-world evidence.
3. A simple explanation is that the outermost electron of any atom will behave and therefore take an orbit as if it is orbiting a Hydrogen nucleus.
4. It's possible to estimate the radius of orbit of an electron around a Hydrogen nucleus. This depends on the model you might use but most of them do suggest a radius of about 1 Angstrom. So the theory agrees reasonably well with the evidence. You could use models (of varying sophistication) to estimate the distance of any other (more interior) electrons from the nucleus, if you wish. However, this has negligible relevance for any properties of the bulk material. We could leave all interior electrons as being "some cloud" rather than assuming they must have a definite radius of orbit.
Best Wishes.
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In a very simple model, the electron is attracted by electrostatic forces to the nucleus but the electron is assumed to be in some motion.
This only makes sense if the electron has arrived from afar on a slightly tangential course, like planets and stuff turning up in a solar system.
Problem is that a solar system only has one principal, massive attractor (like a proton) but can have umpteen planets with arbitrary orbits, whereas all hydrogen atoms seem to be identical and to have just one electron apiece. So it's not a usefully predictive analogy.
And then there's the small matter of accelerating charges emitting photons, which atoms don't, in their ground state.
As I've said many times, if you start from a classical, deterministic model, you can't explain a quantum phenomenon, but if you start with the observation of quantised energy levels and accept the obvious truth of Heisenberg, you can derive Schrodinger's wave equations whose solutions look remarkably like atomic orbitals that match the known structure of materials.
"Distance" must retain its conventional meaning because there seems to be a compressibility limit to samples of elements. Therefore any atom has some sort of diameter and as far as we know all atoms of hydrogen (to keep it simple) consist of identical protons and identical electrons, so they all have the same mean diameter.
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I accept what you are saying, Alan, but the problem is we cannot visualise in "quantum mode" as quantum mechanics is counterintuitive. I see Eternal Student's description as quite reasonable if not entirely rigorous. PS not in the best mode myself having ingested more than my normal meagre ration of beer.
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What's more counterintuitive about a stepladder as distinct from a ramp? Or a fuzzy cloud instead of a whizzing blob?
If you look at Bohr atoms with electrons hurtling about in elliptical orbits there's no way they can form a stable molecule, so it can't be considered an intuitive picture of anything, whereas overlapping orbitals gives you a very clear image that coincides with our knowledge of stereochemistry.