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The Bohr Model of the Hydrogen Atom (2)
Nils Bohr proposed a model of the hydrogen atom based on Rutherford’s ideas and the results of the Geiger-Marsden 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 electro-magnetic 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 E1 or energy E2 or E3 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=½mv2 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 En represents the total energy possessed by an electron in level n.    
We see that En is proportional to 1/n2  
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, ninitial to a lower level, nfinal, then the energy of the quantum of electro-magnetic radiation emitted is equal to the difference between the two levels and is therefore given by   
   
 
   
The energy of a quantum of E-M 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 nfinal = 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 H2).   
 
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