Velocity of Orbiting Electron

It is possible to find the velocity of an electron in its orbit using:

  1. The Rutherford atom model and Bohr’s assumption
  2. The centripetal force equation
  3. Coulomb’s equation
  4. De Broglie’s equation
  5. The equation for the circumference of a circle (or in this case c wavelength) wavelength = 2πr

To shorten the calculations, the words in the equations will be converted to symbols:

The following calculations are based on de Broglie’s speculative equation:

(1)

(2)

(3)

(A)

 

 

 

 

 

(1)

 

(2)

 

(3)

 

(1)

 

(3)

 

(A )

 

 

(4)

(2)  

  

 

(B)

(3)

 

(5)

(3, 4)

 

 

(C)

(A, B)

 

(6)

(1, 5)

  

 

(D)

(C)

(7)

(6)

  

(E)

(1, D)

(8)

(7)

  

 

(F)

(E)
   

(G)

(F)
   

 

(H)

(G)

(9)

(7, G)

(10)

(9)

(11)

(10)

 

v is the velocity of the electron in the orbital λ = 1.

 

 

Size of a Hydrogen Atom

From the electron velocity you can determine the size of a hydrogen atom.

 

 

 

meters

 

Ionization Energy of Hydrogen

You can also determine the ionization energy of hydrogen.

 

 

 

This gives a voltage of:

(Experimental 13.595 electron volts)

and a frequency of

 

 

 

Second Ionization Constant of Helium

The second ionization constant of helium may also be calculated from  

 

=

 

(Experimental 54.403 ev)

 

Third Ionization Constant for Lithium

You can determine the third ionization constant for lithium in the same way as above.

 

=  

 

(Experimental 122.418 ev)

 

Frequency of Light Emitted When a Hydrogen Atom Forms

The frequency of light emitted when a hydrogen atom forms (from ) is

 

This is just half the frequency of the hydrogen atom’s electron in its orbit

 

 

The same is true of the ion.

 

 

 

 

Frequencies of Light in the Hydrogen Emission Spectrum

You can also determine the frequencies of light in the hydrogen emission spectrum from the ionization energy of hydrogen. Niels Bohr hypothesized (1913) that electrons only occupied orbits where λ was an integer.

 

 

 

 

 

 

 

 

 

 

 

 

The frequencies of light in the hydrogen spectrum correspond to the differences between these energies as the electron goes from one orgit to another. If an electron goes from orbit 2 to orgit 1, a photon with the energy of 13.6 ev 3.40 ev = 10.20 ev is emitted. You can construct a chart showing the energies and frequencies of light emitted in the hydrogen spectrum:

Initial Orbit

Final Orbit

ev

ν

∞

1

13.6

 

4

1

13.6 - .85 = 12.75

 

3

1

13.6 1.51 = 12.09

 

2

1

13.6 3.4 = 10.2

 

∞

2

3.40

 

4

2

3.40 - .85 = 2.55

 

3

2

3.40 1.51 = 1.89

 

These correspond closely with the observed frequencies.



Return to Home Page

Content of this paper may be freely used so long as credit is given to the author, Brian Stedjee

First publication date: September, 2003