The existence offline spectra can be explained by means of the first "quantum" model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines, and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If we can obtain the energy levels, we can reproduce the hydrogen atom spectra.
The derivation proceeds with three major elements: first, use force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, then solve for the energies.
Assumption 1: The atom is held together by the Coulomb Force
It is an experimental fact that the force F between two point charges qi and q2 separated by a distance r is given by:
are in units of Coulombs. The distance, r, is in meters, of course. Note that the charges may be positive or negative.
For a single electron atom, we take the charge of the nucleus, where Z is the atomic number of the atom (the number of protons in the nucleus). Z equals 1 for hydrogen. The charge of the electron,is -e. Substituting the values into Eq. A1.1 above we obtain:
The minus sign on the force term means the force is "inward," or attractive.
Assumption 2: The electron moves in an elliptical orbit around the nucleus (as in planetary motion).
Let us assume that the electron moves in a circular orbit around the nucleus.
We apply Newton’s second Law (F = ma) here by setting the Coulomb force equal to the centripetal force. The result can be written as
and we can now solve for the radius vs. velocity.
Figure A1.1 Bohr atom model, Z = 3.
Assumption 3: Quantized angular momentum
Bohr now introduced the first of his two new postulates, namely that the only allowed orbits were those for which the angular momentum L was given by:
where m = electron mass; v = velocity; r = radius of the orbit; n = an integer (1,2,3,..); and
where h is simply Planck’s constant, as before.
(One suggestion for a physical basis for this assumption is that if you view the electron as a wave, with wavelength X = h/p = h/mv, then an integral number of wavelengths have to fit around the circumference defined by the orbit, or n • h/mv = 2nr. Otherwise, the electron "interferes" with itself. This all follows as a corollary to the idea that an electromagnetic wave is a particle with energy E = hf as above, and hence the momentum of a photon is This is sufficient to give us
for the velocity of the electron in its orbit. Note that there is an index n, for the different allowed orbits. It follows that
This only works for one-electron atoms (H and He+ as a practical matter), but within that restriction, it works fairly well. For hydrogen (Z = 1) we get the Bohr radius, r\ = 0.528 X 10 10 m as the radius of the smallest orbit. The radius of the Bohr hydrogen atom is half an Angstrom. What is the radius of the orbit for the sole electron in He+ (singly ionized helium, Z = 2)?
Now we can solve for the energy levels.
The potential energy associated with the Coulomb force is:
A negative potential energy means the electron is in a potential ‘well.’ Given this expression for the potential energy, we need a similar expression for kinetic energy.
The kinetic energy T is easily obtained from Eq. A1.3:
Therefore, the total energy of the electron is obtained:
The total energy is negative—a general characteristic of bound orbits. This equation also tells us that if we know the radius of the orbit (r) we can calculate the energy E of the electron.
Substituting the expression for rn (Eq. A1.7) into Eq. A1.11, we obtain
is the energy of the electron in its lowest or "ground" state in the hydrogen atom.
Assumption 4: Radiation is emitted only from transitions between the discrete energy levels
The second Bohr postulate now defines the nature of the spectrum produced from these energy levels. This postulate declares that when an electron makes a transition from a higher to a lower energy level, a single photon will be emitted. This photon will have an energy equal to the difference in energy of the two levels. Similarly, a photon can only be absorbed if the energy of the photon corresponds to the difference in energy of the initial and final states.