# Rutherford model and the planetary sistem

#### Bulloc90

I cant understand how the Rutherford model would explain changes in the radium of electron orbits. Here is what I think: Moment 1 - An electron suffers an atraction force that is balanced with his kinectic energy, so he stays in orbit. Moment 2 - The electron looses energy and the orbit radium diminishes, with a smaler distance between the positive and negative charges, the atraction force gets bigger. So, in moment 1 the forces would balance, but, why, in moment 2, when the electron has less energy and is atracted with more intensity, it doesnt "falls" into the nucleum?
Does the model explains it?

A similar mechanism would be applied to the orbits of the planets, right? I dont know how different orbits may exist around the sun, would it have to do with the mass of the planets?

All this is really tricking me. Please explain me in details and with formulas if possible (I am starting college and dont remember really well the part of classical physics that goes about circular movement).

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#### ChipB

PHF Helper
First off, we know that the planetary model of electron orbits is not correct. However, to answer your question ...

In a planetary model what happens for a circular orbit is the attractive force (which is proportional to 1/r^2) is balanced by the centripetal acceleration times planetary mass m (this is not the same as kinetic energy). So you have

GMm/r^2 = mv^2/r

You can rearrange to get:

v^2 = GM/r

Note from this that if r decreases then v increases. Hence while the inward force of gravity is increased with smaller values of r, the outward centripetal acceleration is also increased, so the forces remain balanced. Note also that the mass of the planet m is cancelled out in that first equation - hence the positions of stable orbits about the central mass of the sun doesn't depend at all on the mass of the planet m.

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#### Bulloc90

I didnt really understood the rearrangements you made, but I made my own and I understand that **the closer a planet is to the sun, the faster it moves.**

But, when an electron receives energy, the radium of its orbit increases. This seems to be opposed to the idea expressed above.**

I know that this atomic model is not correct and that the classical physics cannot explain the atom. So, is there an explanation for this opposition, or it is caused by the inconsistency of the model?
Also, it isnt possible to donate energy to a planet in the same way it can happen with an electron, but maybe the comparison is valid in trying to understand the physics.

#### MBW

I think I see the source of your confusion,
In the planetary motion a smaller orbit implies higher velocity
which in turn implies higher energy.
But for electrons, a smaller "orbit" implies lower energy.
However as ChipB pointed out, the planetary orbit model for electrons is fundamentally incorrect.
I suspect that even when it was originally put forward as a model, it was recognised as a vast over-simplification of a more complex situation.
Modern electron "orbital" models are much more complicated
(and frankly a bit beyond me).

#### Bulloc90

Yeah haha, the orbital models are really complex, Ive been introduced to them.

Well, the problem was really the inconsistency of Rutherford model.
I see...

Thank both of you for the help, this has been a doubt for some time already.
(Bigsmile)

#### studiot

I can't agree with ChipB's analysis, since centripetal acceleration, and force, is defined to be directed inwards.

The (classical) planetary orbit theory works well for the mechanical part.
It fails on the electrmagnetic part where any accelerating charge is required to emit EM radiation, which an orbiting electron does not.

But no matter.

The electron (in motion) is subject to an inward (towards the nucleus) acting centripetal force.

This is a real force that is necessary to deflect the electron from straight line motion in accordance with Newtons laws.

Bohr's theory does not include gravity, the only force acting is the electrostatic attraction between the electron and the nucleus. This is many orders of magnitude greater than any gravitational force, which is negligable.

If the electron is given more energy the radius of the orbit increases because this extra energy increases the (electrostatic) potential energy of the electron. Note that for an attractive force the potential energy is zero at infinite separation and becomes more negative as the bodies approach closer. So an increase in potential energy is from a large negative number to a smaller negative number. This is, of course, the same with gravitational attraction.

#### ChipB

PHF Helper
I can't agree with ChipB's analysis, since centripetal acceleration, and force, is defined to be directed inwards.
Perhaps I could have stated it better, but what I wrote is fundamentally correct. From F=ma you have an inward force of magnitude GMm/r^2, which induces an inward acceleration which for a circular orbit has magnitude v^2/r. Now perhaps I should have used proper vector notation, which on this site is difficult without LaTeX, but just sticking with magnitudes of force and acceleration you get from F=ma:

GMm/r^2 = mv^2/r

And yes, as the radius of orbit decreases kinetic energy increases, just as potential energy decreases, and the total mechanical energy remains constant (absent any external forces acting on the planet).

Of course all this is fine for planetary motion, but doesn't translate too well to electron orbits. Replacing the gravitation force with electrostatic force yields:

KQq/r^2 = mv^2/r

where Q and q are the charges of the nucleus and electron respectively. This would imply that electrons move in orbits like planets do (elliptical orbits that obey Kepler's laws of planetary motion), which we know is incorrect.

#### Bulloc90

If the electron is given more energy the radius of the orbit increases because this extra energy increases the (electrostatic) potential energy of the electron. Note that for an attractive force the potential energy is zero at infinite separation and becomes more negative as the bodies approach closer. So an increase in potential energy is from a large negative number to a smaller negative number. This is, of course, the same with gravitational attraction.
Ok, but what would be the reason that the energy received by the electron converts into eletrostatic potential energy, making the orbit radius bigger, and not into kinectic energy, causing an increase in the electron speed and therefore (correct me if Im wrong) in the centripetal force, thus making the orbit radius smaller?

#### studiot

Bohr assumed circular orbits as the simplest possible.
These were soon modified by Sommerfield to elliptical ones.

Another difference is that in the solar system planetary orbits are all in one plane.
Atom electron orbits have always been assumed to be scattered over the surface of a ball.

#### studiot

Looking back I see that you have described this as the Rutherford model and we have been discussing the Bohr model.

Rutherford's model didn't actually address the structure or disposition of the electron cloud. His experiments (alpha particle scattering) simply showed that the positive part must be massive and concentrated at the centre of an atom.

http://en.wikipedia.org/wiki/Rutherford_model

Note I said experiments.

All the development of models was a response to vexpermental observations.

Bohr introduced the quantised model of orbits in response to spectrographic analysis.

This introduced a single quantum number that set permitted orbits and therefore permitted EM radiation changes to match observations of spectral lines.

De Broglie later showed that the circumferences of Bohr orbits are integral multiples of the de Broglie wavelength.

Why can an electron not be attracted inwards?

Well inherent in the proposal was the concept of the lowest possible orbit, which of course was the first occupied.

Any closer and the electron/proton pair would not be stable.