Download Activity 2 - The Bohr Atom

Activity 2 - The Bohr Atom
Experimental observations show that the energies of photons emitted from hydrogen atoms
showed that the energies of were always described by the equation:
for integer
Experiments by Geiger and Marsden, the ‘Gold-foil’ or ‘Rutherford’ experiment showed that
almost all the mass and positive charge of the nucleus was concentrated into a tiny sphere
at the centre of the atom, which prompted Rutherford’s planetary model, of electrons orbiting
the nucleus, like planets orbiting the Sun.
This model has a fatal flaw: we know that for a body to follow a circular path, there must be a
centripetal force accelerating it towards the centre of the circle, but classical physics showed
that an accelerating electron emits photons, and loses energy. This would mean that
eventually (the maths shows it’s actually very quickly!) all electrons would spiral down into
the nucleus: something was very wrong!
This lead to Bohr developing a new planetary model for the atom. We now know that this
model is not good enough to accurately describe the atom, but it is was a major stepping
stone in the path to a full quantum theory, and gives a good explanation of the observed
photon energies.
Bohr made a very important prediction: that the angular momentum of the electron orbiting
the nucleus is quantised: i.e. it can only take specific values:
where
is an integer and
is Planck’s constant that you should be familiar with.
This quantisation is one of the major differences between quantum and classical physics.
Classically we say that variables can take a continuous spectrum of values, this spectrum is
reduced to a set of discrete values in quantum physics.
So what are the results of having quantised angular momentum?
From mechanics we know that the electron in a circular orbit must have a centripetal force:
Here, this is provided by the Coulomb force of attraction between the electron and the
nucleus,
where this force, F, is directed towards the nucleus, and Z is the atomic number:
hydrogen.
for
Because is an integer we have found that the radius at which the electron orbits is
restricted to a set of discrete values by the quantisation of angular momentum.
If we now look at the energy of an electron in the orbit, we know it must be made up of its
kinetic energy and its electric potential energy:
We use the force equations as we did with the Bohr radius to eliminate
,
Substitute the Bohr radius:
(
)
So we have a new model for the energy levels of the atom. But how does this help get
around the original problem: that the electrons should radiate energy as they are
accelerated? It is because they can only exist in one of the levels. Therefore if the electron
was in the
level, there is no lower level for it to go to if it were to give out a photon, so
it can’t give out a photon and doesn’t lose any more energy!
This model, which displays the desired relationship with respect to n, is called a “semiclassical” model, since we have used classical mechanics (of circular orbits), mixed with the
quantum idea of quantisation, to derive it. A full quantum description of the hydrogen atom
requires a new way of describing energy and momentum and we will touch upon these ideas
next.
Aside:
You might wonder why an electron should have to move in a circle: why not an ellipse like
the planets? This was solved by Sommerfeld and Wilson, who realised that the angular
momentum condition is actually:
∮
where is the momentum of the coordinate
elliptical as well as circular orbits.
e.g.
and . This more general form allows