BOHR - RUTHERFORD HYDROGEN ATOM        topics

This model of 1913 improves Rutherford's original nuclear model by placing restrictions on the possible "orbits" of the electron. It is the "Grade 9 - 10" atom that is divied up as a standard representation.

We start by calculating the total energy = sum of potential and kinetic energies of the orbiting electron

and we equate Coulomb's Law with the necessary centripetal force for orbiting.

(1) F = k e² / r² = mv² / r              e = charge on both electron and proton = 1.6 x 10^-19 C

From the above  E_k = 1/2 mv²  = k e² /2 r       (2)

E_p = Work Done in moving charge from infinity to orbital radius =

= -k e² / r

so the total energy E = -k e² / 2r . (3)  (Notice the energy is "negative' - that simply means that infinitely far away is taken as a "sea level" or zero.)

This equation allows any old radius and energy. Rutherford put it up. It does not solve emission lines. It also has a weakness, the electrons are accelerating ( circular motion ) thus should be radiating radio waves! - losing energy - spiralling in - exit atom.

Enter Niels Bohr

LET (for the hell of it)  linear momentum x radius      mvr = nh /2π   where n = 1,2,3,4,....... (positive integers), h = Planck's Constant

Now to fiddle to sus out the consequences - basically we are going to return to total energy again after wiping out speed.

Square the lot, and reorganise      mv² = (  nh /2π )² / mr²    = k e² / r   from (1)

OUT OF THIS COMES A RESTRICTION ON ORBITAL RADII

r = n²h² / 4π²kme^{2      n = 1,2,3,4,.............}            

Stick this in a calculator

for n=1  r = 0.53 x 10^-11 m   

n = 2    4 x this value,

n = 3    9 x this value .......

When we bung this back in the Energy equation (3) to get

E = - 2π²k²me⁴ / n²h^2  (4)

Energies are restricted!