24.2. The Bohr Atom http://www.ck12.org
photon would do, of course. A photon of a specific energy had to be absorbed by the electron in order for the electron
to move into a stationary state of higher energy. The electron would then be in an excited stationary state and would
quickly emit the energy it had absorbed and fall back into its previous lower stationary state. Energy could, therefore,
be emitted or absorbed only in going from one stationary state to another within the atom. That idea could explain
the reason for a discrete line spectrum and do away with Maxwell’s prediction that all accelerating charges must
radiate electromagnetic energy.
Bohr made several important assumptions in order to (mathematically) derive a model of the hydrogen atom based
on quantization. We state his assumptions (“postulates”) below:
- Energy of the electrons within an atom is quantized.
- Momentum of the electrons within an atom is quantized.
- The orbits of the electrons are circular.
Using these assumptions, Bohr was able to derive the following:
The allowed radii for the hydrogen atom are
rn=n^2 r 1 ,n= 1 , 2 ,.. ., Equation A,
wherer 1 is the smallest orbital radius of the hydrogen atom, commonly referred to as theBohr radius. The Bohr
radius isr 1 = 0. 529 × 10 −^10 m. The result is in good agreement with the previous estimates of the radius of the atom.
The allowable energy levels, or stationary states, of the Bohr atom are,
E=En 21 ,n= 1 , 2 ,.. ., Equation B,
whereE 1 is called theground stateof the atom. The ground state is the lowest-energy state of the atom. In the
hydrogen atom, the ground state energy is about
E 1 =− 13. 6 eV
The negative sign indicates that the electron must absorb 13.6eVin order to be ejected from the atom. This energy
is calledthe ionization energy of the ground state of the hydrogen atom.
EnergiesE 2 ,E 3 , ... represent higher-energy, or “excited” states of the electron. Notice that the higher the energy
level, the less negative is the corresponding energy. For example,
E 2 >E 1 , whereE 2 =− 3. 40 eV
The difference between allowable energy levels can be expressed as
h f=Eh−El,
whereEhis a higher energy state of the electron andElis a lower energy state of the electron. When an electron
undergoes a change in energy, we often say it has undergone atransition.
Bohr was able to derive the results of the Balmer, Lyman, and Paschen series based on his assumptions. A slightly
more generalized version of these series given by Bohr is
1
λ=R(
1
n^2 l−1
n^2 h)
wherenlis the lower state andnhis the higher state.
http://demonstrations.wolfram.com/AbsorptionAndEmissionOfRadiationByAtoms/
Illustrative Example 25.2.2
What is the wavelength of a photon emitted when a transition occurs betweenn= 4 to n=3 in a hydrogen atom?
Solution:
We will solve this problem two ways:
Method 1