Electronic Structure of Atoms
Bohr's model of the hydrogen atom
In 1913 Niels Bohr developed a theoretical explanation for a phenomenon
known as line spectra.
Bohr's Model of the Hydrogen Atom
Line Spectra
Lasers emit radiation which is composed of a single wavelength.
However, most common sources of emitted radiation (i.e. the sun, a lightbulb)
produce radiation containing many different wavelengths.
When the different wavelengths of radiation are separated from such
a source a spectrum is produced.
A rainbow represents the spectrum of wavelengths of light contained
in the light emitted by the sun

Sun light passing through a prism (or raindrops) is separated
into its component wavelengths

Sunlight is made up of a continuous spectrum of wavelengths
(from red to violet)  there are no gaps
Not all radiation sources emit a continuous spectrum of wavelengths of
light

When high voltage is applied to a glass tube containing various gasses
under low pressure different colored light is emitted

Neon gas produces a redorange glow

Sodium gas produces a yellow glow

When such light is passed through a prism only a few wavelengths
are present in the resulting spectra

These appear as lines separated by dark areas, and thus are called line
spectra
When the spectrum emitted by hydrogen gas was passed through a prism
and separated into its constituent wavelengths four lines
appeared at characteristic wavelengths
In 1885 a Swiss school teacher figured out that the frequencies
of the light corresponding to these wavelengths fit a relatively
simple mathematical formula:
where C = 3.29 x 10^{15} s^{1} (not the 'c' used
for the speed of light)
However, the physical basis for this relationship was unknown.
Bohr's Model

Bohr began with the assumption that electrons were orbiting the nucleus,
much like the earth orbits the sun.

From classical physics, a charge traveling in a circular path should lose
energy by emitting electromagnetic radiation

If the "orbiting" electron loses energy, it should end up spiraling into
the nucleus (which it does not). Therefore, classical physical laws
either don't apply or are inadequate to explain the inner workings of the
atom

Bohr borrowed the idea of quantized energy from Planck

He proposed that only orbits of certain radii, corresponding to defined
energies, are "permitted"

An electron orbiting in one of these "allowed" orbits:

Has a defined energy state

Will not radiate energy

Will not spiral into the nucleus
If the orbits of the electron are restricted, the energies that the electron
can possess are likewise restricted and are defined by the equation:
Where R_{H} is a constant called the Rydberg constant
and has the value
2.18 x 10^{18} J
'n' is an integer, called the principle quantum
number and corresponds to the different allowed orbits for the
electron. Thus, an electron in the first allowed orbit (closest to the
nucleus) has n=1, an electron in the next allowed orbit further
from the nuclei has n=2, and so on.
Thus, the relative energies of these allowed orbits for the electrons
can be diagrammed as follows:
All the relative energies are negative

The lower the energy, the more stable the atom

The lowest energy state (n=1) is called the ground state
of the atom

When an electron is in a higher (less negative) energy orbit (i.e. n=2
or higher) the atom is said to be in an excited state

As n becomes larger, we reach a point at which the electron is completely
separated from the nucleus

E = (2.18 x 10^{18} J)(1/infinity) = 0

Thus, the state in which the electron is separated from the nucleus
is the reference or zero energy state (actually higher
in energy than other states)
Bohr also assumed that the electron can change from one allowed orbit to
another

Energy must be absorbed for an electron to move to a higher state (one
with a higher n value)

Energy is emitted when the electron moves to an orbit of lower energy (one
with a lower n value)

The overall change in energy associated with "orbit jumping" is the difference
in energy levels between the ending (final) and initial orbits:
DE = E_{f}  E_{i}
Substituting in for the previously defined energy equation:
When an electron "falls" from a higher orbit to a lower one the
energy difference is a defined amount and results in emitted electromagnetic
radiation of a defined energy (DE)

Planck had deduced that the energy of the photons comprising EM radiation
is a function of its frequency (E = h)

Therefore, if the emitted radiation from a falling electron had a defined
energy, then it must have a correspondingly defined frequency
Note:

DE is positive when n_{f}
is greater than n_{i}, this occurs when energy is absorbed and
an electron moves up to a higher energy level (i.e. orbit).

When DE is negative, radiant energy
is emitted and an electron has fallen down to a lower energy
state
Revisiting Balmer's equation:
In 1885 a Swiss school teacher figured out that the frequencies
of the light corresponding to these wavelengths fit a relatively
simple mathematical formula:
where C = 3.29 x 10^{15} s^{1} (not the 'c' used
for the speed of light)
Since energy lost by the electrons is energy "gained" by the emitted
EM energy, the EM energy from Bohr's equation would be:
Thus, Balmer's constant 'C' = (R_{H}/h) (Rydberg constant
divided by Planck's constant), and n_{f} = 2. Thus, the only emitted
energies which fall in the visible spectrum are from those electrons which
fell down to the second quantum orbital. Those which fell
down to the first orbital have a higher energy (frequency) than can be
seen in the visible spectrum.
Calculate the wavelength of light that corresponds to the transition
of the electron from the n=4 to the n=2 state of the hydrogen atom. Is
the light absorbed or emitted by the atom?
Since the electron is "falling" from level 4 down to level 2, energy
will be given up and manifested as emitted electromagnetic radiation:
DE = (2.18 x 10^{18} J)((1/16)(1/4))
= 4.09 x 10^{19} J (light is emitted)
4.09 x 1019 J = (6.63 x 10^{34} Js) * (n)
6.17 x 10^{14} s^{1} = n
l = (3.00 x 10^{8} m s^{1})/
(6.17 x 10^{14} s^{1}) = 4.87 x 10^{7}m = 487
nm
Bohr's model of the atom was important because it introduced quantized
energy states for the electrons. However, as a model it was only useful
for predicting the behavior of atoms with a single electron (H, He^{+},
and Li^{2+} ions). Thus, a different model of the atom eventually
replaced Bohr's model. However, we will retain the concept of quantized
energy states
1996 Michael Blaber