Nils Bohr
proposed a model of the hydrogen atom based on Rutherford’s ideas
and the results of the GeigerMarsden experiments. 

He assumed that the electron orbits the nucleus (a single
proton) in a circular path. 





mass of the electron, m 

charge on one electron, e (same magnitude as the charge on one
proton) 

speed of the electron in its orbit, v 



This simple picture presents a problem. 



It is observed that when electric charges are accelerated, they
emit electromagnetic radiation. 

If the electron moves in a circle, it is accelerating, therefore
it should emit radiation, lose energy, slow down and crash into the
nucleus in a very short time. 

In other words, the model seems to suggest that hydrogen atoms
cannot exist… not a very satisfactory state of affairs. 



To get round this problem he suggested that electrons can exist
in atoms only in certain stable (allowed) orbits having a discrete
set of energies; that is, an electron can have energy E_{1}
or energy E_{2} or E_{3} etc but not somewhere in
between these values. 

Note that a free electron (not part of an atom) can have any
amount of energy that it feels like having; it is just when it is trapped in an atom that its behaviour is restricted. 



Bohr calculated values for the radii and energy of these allowed
orbits and found that he could predict the wavelengths of the atomic
hydrogen spectrum. 



It turned out that these orbits are such that the angular
momentum of the electron is always an integral multiple of the
quantity h/2π where h is
Planck's constant. 

In other words, the angular momentum of the electron is
quantized. 



In deriving the details of the energy levels, it is convenient
to start from this idea. 

Putting this idea as an equation, we have 



where n = 1, 2, etc 

The lowest amount of angular momentum that an electron in the
hydrogen atom can possess is therefore h/2π. 

When the electron has this value for L the atom is said to be in
the ground state. 



The angular momentum possessed by a point mass is given by 



so for the electron in the atom, we can write 



This can be used to define the radii of the allowed orbits
in this model 



equation 1 


The force acting on the electron is given by Coulomb's law 



If the electron orbits in a circular path, the force is also
given by 



so, the speed, v, of the electron is given by 





and, substituting this into equation 1, above, allows us to
calculate the values of the radii of the allowed orbits 



equation 2 


The kinetic energy of the electron will, as usual, be given by
K=½mv^{2} therefore 





The potential, V, at a point a distance r away from a proton is
given by 





Therefore the potential energy possessed by an electron
at a distance r away from a proton is given by 



The total energy (K + P) is therefore 



In this equation for the total energy we will now substitute for
r from equation 2 above to give 





where E_{n} represents the total energy possessed by an
electron in level n. 

We see that E_{n} is proportional to 1/n^{2} 

Notice that, as all these energy levels are negative
values, increasing n increases the total energy of
the electron (it goes to smaller negative values). 



If an electron falls from one energy level, n_{initial}
to a lower level, n_{final}, then the energy of the quantum
of electromagnetic radiation emitted is equal to the difference
between the two levels and is therefore given by 







The energy of a quantum of EM radiation is given by Planck's
formula E=hc/λ 

Therefore 



and so we have (ignoring the minus sign... 1/λ
can't be negative) 





which has the same form as the Balmer series if we put n_{final}
= 2 and 





We therefore see that Bohr's model does
correctly predict the wavelengths in the spectrum of hydrogen
atoms. 

However, it cannot be extended to other
atom and molecules (nor even H_{2}). 
