CK-12-Physics-Concepts - Intermediate

23.1. The Bohr Model of the Atom http://www.ck12.org

andfis the frequency. The emission of a photon of light from an atom indicates a change in energy level for an
electron such that

h f=Ehigher−Elower.

The energy of an orbit is related to the inverse of the square of the orbit number. The energy of an electron in a given
energy level of hydrogen is calculated by

En= (− 2. 17 × 10 −^18 J)

( 1

n^2

)

.

The radius of an orbit is related to the square of the orbit number.

Bohr calculated allowed electron energy levels for the hydrogen atom and found the emission spectrum of hydrogen
to match perfectly with particular electron transitions between his suggested energy levels. Other electron transitions
predicted electromagnetic frequencies outside the visible range and when those were looked for, they were present
and also matched precisely with theoretical calculations.

Example Problem:For the hydrogen atom, determine

a. the energy of the innermost energy level(n= 1 ).

b. the energy of the second energy level.

c. the difference between the first and second energy levels.

Solution:

a.En= (− 2. 17 × 10 −^18 J)

( 1

n^2

)

E 1 = (− 2. 17 × 10 −^18 J)

( 1

12

)

=− 2. 17 × 10 −^18 J

b.En= (− 2. 17 × 10 −^18 J)

( 1

n^2

)

E 2 = (− 2. 17 × 10 −^18 J)

( 1

22

)

=− 5. 43 × 10 −^19 J

c.E 2 − 1 = (− 5. 43 × 10 −^19 J)−(− 2. 17 × 10 −^18 J) = 1. 63 × 10 −^18 J

This is the amount of energy that would need to be added to a 1stenergy level electron to raise it to the second energy
level.

Example Problem: According to the Bohr model, how many times larger is the second level hydrogen orbit
compared to the first level hydrogen orbit?

Solution:4 times

Summary

• Rutherford, using Coulomb’s law and Newton’s laws, found that the results of his ’gold foil experiment’ could
  be explained only if all the positive charge of the atom were concentrated in a tiny, central core, now called
  the nucleus.


• The atom is 10,000 times as large as the nucleus and is mostly empty space.


• The energies of light frequencies in the emission spectrum of atoms is much too small to be involved in
  the nucleus of atoms, therefore, any explanation of these wavelengths would have to involve the electron
  arrangement.


• Bohr hypothesized that the electrons were organized into stepwise energy levels within the electron cloud in a
  planetary model and only radiated energy when the electrons were changing from one energy level to another.


• Bohr suggested that the energy levels were quantized, that is, the energy held by the atomic electrons came in
  packages and only whole packages could be absorbed or emitted.


• Einstein’s theory says that each light photon has an energy ofh f, wherehis Planck’s constant, andfis the
  frequency. The emission of a photon of light from an atom indicates a change in energy level for an electron
  such thath f=Ehigher−Elower.


• The energy of an electron in a given energy level of hydrogen is calculated by

En= (− 2. 17 × 10 −^18 J)

( 1

n^2

)

.