The Bohr Model for Hydrogen

By 1913, the state of physics was in some disarray with regard to atomic structure. Rutherford had shone that a nuclear atom existed. But that discovery led to two new questions, namely, 1) what holds the nucleus together?; and 2) What mechanism of atomic structure determines atomic volume? (We will hold off on a discussion of Question 1 to another time.

The query "What mechanism of atomic structure determines atomic volume?" led to other difficult questions. a) If the negatively charged electron is stationary, should it not be attracted to a positive nucleus by Coulomb's Law? So what is the mechanism that keeps them apart? b) If the negatively charged electron were moving, it could be held in place by the Coulomb attractive force of a positive nucleus. But James Clark Maxwell had shown decades before that accelerated charges radiate energy. An electron orbiting a nucleus is undergoing a centripetal acceeration and therefore should radiate energy. But no such energy was found! Thank you Mr. Rutherford!!!!

Enter Neils Bohr, Danish physicist, age 26, stage left. He took the bold approach to suggest that the angular momentum of the electron orbiting the nucleus was quantised (that is, it can only take on certain values) and that this quantisation took precedence over the Maxwell assertion of accelerating charges radiating energy. Please follow the algebra as it unwraps below. As a student you need to know that the Bohr model was subsequently suplanted by a quantun mechanical approach; the Bohr model is but a first order approximation. That's too bad because you will never see a neater solution to a complex problem as what follows

From classical physics
The electrostatic force acts
as a centripetal force

From quantum mechanics
the angular momentum of the
electron is quantized

Solve for v and R


Evaluate these equations for v and r at n=1to n = 5

Soooooo, because the angular momentum is quantized, it turns out that
the space around the nucleus is quantized and so is the speed of the electron.


Electron Energy Levels

for v & R
from above.

Watch the two
terms come

This means that the
electron energy, shown
in line 5 at right,
is quantized

Prove that the
lowest energy
(ground state)
occurs when
n = 1


Let's call this equation #1

Evaluate the energy for n = 2 & n = 3.
What is the difference in energies?
What wavelength is tied to this energy?



(in eV)


 change in
(in eV)

(in nm)
 color of
 6  -.37  from 6 to 2

 410.1 violet 2
 5 -.54  from 5 to 2

 434.0  violet 1
 4 -.85     from 4 to 2

 486.1  blue-green
 3   -1.51   from 3 to 2

 656.2  Red
 2  -3.40  x  x  x x
 1  -13.6  x  x  x

Remember that these energy levels exist only if Bohr's speculation about angular momentum is valid. A hint of validity was supplied some twenty - eight years earlier, in 1885, by Johannes Balmer, a German high school physics teacher and sometimes numerologistt (someone who studies for numbers sake). Of course, no one knew at the time thatr Balmer's discovery would have such import.
Balmer was curious about the wavelength of the four lines lines of the hydrogen spectrum. His query "Are these four numbers random or are they connected in some way?"His answer suggested a connection among the numbers, although Balmer had no idea what it meant. With the equation immediately below he could generate the wavelengths for hydrogen as long as n was an integer greater than 2. Why was it 2 and not another integer , say 3, 4, or 5 , he had no clue. But putting in place of 2 the integers 1, 3, 4, 5, produced other wavelengths in the infrared or ultraviolet that had never been seen before. And when spectroscopists went loking for these new lines they found them exactly where Balmer had predicted.It is one thing to find a relationship among a few numbers; it is something much more important when the new equationcan predict things that are unknown.

The Balmer Equation

J. Balmer found that
this formula could
generate wavelengths

Let n = 3, 4, 5, 6 and

calculate the wavelength

Show that if we Rearrange terms and
Multiply by hc we can get Equation #2

Calculate the

EQ #2

EQ #1


EQ #2
 1. If we can show that these two equations are equal, tthen we can use spectroscopy to analyze atoms.  
2. Let delta E = hc/lambda for red light.   
 3.The brachet in eq 1 = bracket in eq 2 if n = 3  
 4. This works if . try it by plugging in the numbers  

   There are two ideas presented on this page. 1) The space around the atomic nucleus is quantized; electrons move from level to level by addition or removal of definite quantities of energy.2) Spectral lines are tied closely to these energy transitions. Studying atomic spectra gives us a window to atomic structure.

This interesting applet shows how electrons populate energy levels.

 Return to Modern Physics

Last edited 12/26/05