A Light in the Darkness
Background:
In atomic physics, the Rutherford -- Bohr model or Bohr model, introduced by Niels Bohr in 1913,
depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in
perfectly circular orbits around the nucleus -- similar in structure to the solar system, but with
attraction
provided
by electrostatic
forces rather
than
gravity.
After
the plum-pudding
model (1904), and the Rutherford model (1911) came the Rutherford–Bohr model or just Bohr
model for short (1913). The improvement to the Rutherford model is mostly a quantum physical
interpretation that takes into account a flaw that Rutherford had not taken into account. The Bohr
model (or at least aspects of the model) has since been superseded by the quantum theory.
The Bohr model's key success lay in explaining the Rydberg formula for the spectral emission
lines of atomic hydrogen. While the Rydberg formula (which we will be exploring in this
investigation) had been known experimentally for some time, it did not gain a theoretical
underpinning until the Bohr model was introduced. Not only did the Bohr model explain the
structure of the Rydberg formula, it also provided a justification for its empirical results in terms
that were consistent with fundamental physical constants.
We now view the Bohr model is a more archaic model of the hydrogen atom compared to the
modern Quantum Mechanical view of the atom. However, there are still elements of the older
theory that remain in force. For that reason and because of its relative simplicity compared the
Quantum Mechanical model of the atom, the Bohr model continues to be part of science
instruction.
In the early 20th century, experiments by Ernest Rutherford established that atoms consisted of
a cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus.
Rutherford naturally considered a planetary-model as the best way to describe the motion of the
electron around a central nucleus. The model made sense because the mechanics of planetary
motion were well understood and there was no reason to believe that electron motion around the
atomic nucleus would be any different. However, according to Bohr the Rutherford view of a
planetary-model atom had technical difficulties. Bohr argued that the laws of classical mechanics
predicted that the electron would release electromagnetic radiation while orbiting a nucleus. The
corresponding loss of energy would lead to a decrease in the electron’s velocity. This, in turn,
would lead to a rapid spiral of the electron into the nucleus.
Whereas the collapse of a planet into a sun would take an uncounted number of years, on the
time scale of an atom, the process would take no longer than of 16 picoseconds (1.6 x 10-16
sec). This would be catastrophic as all the atoms in the universe would become extremely
unstable in an instant after the Big Bang; not good for us the Romulans, ET or anyone else.
Additionally, as the electron spiraled inward toward the nucleus, the electromagnetic emissions it
produced would rapidly increase in frequency (decrease in wavelength). This would produce a
continuous ‘smear’ in frequency and wavelength across the electromagnetic radiation.
Fortunately, this never has nor will ever happen (well, at least, not any time soon).
Late 19th century experiments with electrical discharges have shown that to the contrary, atoms
will only emit light (electromagnetic radiation) at certain discrete frequencies. For example,
consider the following example of an emission spectrum of neon gas. Note the set frequencies
of each of the lines:
The problem, of course, is that the physics says the spectrum is supposed to ‘smear.’ It does
when you look at a rainbow or the light from the sun. To overcome this difficulty, Niels
Bohr proposed (in 1913) what we now call the Bohr model of the atom using the bright lines
produced by the emission spectrum of hydrogen to make has case. As part of his working
hypothesis, he suggested that electrons could only have certain classical motions, that is:
1. Electrons in atoms must orbit the nucleus ( we are still looking at a planetary model)
2. The orbits (called stationary orbits) must be stable and located at certain discrete
distances from the nucleus. These orbits would be associated with definite energies.
Unlike classical orbits (planets orbiting a sun), the electron's in these orbits acceleration
would not lose energy as required by classical electromagnetics.
3. Electrons could gain and lose energy by jumping from one allowed orbit to another
(Quantum Leap). They would absorb or emit electromagnetic radiation with a
frequency determined by the energy difference of the levels according to the Planck
relation:
where h is Planck's constant. The equation describes photons of light being emitted from
electrons as they transfer back from their excited state to their ground state.
This lab exercise will give you the opportunity to use the Rydberg formula, which will allow you to
determine the energies associated with electron transitions or quantum jumps between orbital
energy levels. When the electron transitions from its original energy level to a higher one and
then jumps back each level till it comes to the original position, a photon is emitted. Using the
derived formula for the different energy levels of hydrogen where the electron ‘quantum leaps’,
one may determine the wavelength of light are emitted on the return journey back to the original
ground state..
The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen
energy levels:
where R is known as Rydberg’s energy, nf is the final energy level, and ni is the initial energy level.
Since the energy of a photon is…
…we can further describe the wavelength of the photon given off by an electron as it transfers
from the excited to ground states as:
This expression is known as the Rydberg formula. The Rydberg constant R (1.09737 x 107 m-1)
represents the limiting value of any photon that can be emitted from the hydrogen atom. What this
means is that the spectrum of hydrogen can be expressed simply by using the Rydberg formula to
determine the relationship between the emission lines in the hydrogen spectrum and the quantum
leap of an electron from one energy level to another. In other words, the equation can be used to
identify the specific ground state of an electron (nf) and the excited state (ni) based upon the
wavelength and frequencies of the lines that make up the emission spectrum of hydrogen. This
formula was known to scientists who studied spectroscopy in the Nineteenth century, but Bohr was
finally able to make sense of it and in doing so radically alter our understanding of the atom.
The purpose of this exploration is to measure the wavelengths of the emission lines in the visible
region of the spectrum produced by hydrogen. Using these data, the values for nf, the Rydberg
constant, and the energy associated with the quantum levels of the hydrogen atom (En), it should
be possible to establish the excited states associated with each of the emission lines produced in
the visible region of the spectrum
We know that ni must be greater than nf, so if we assume nf is 1, then ni can be 2, 3, 4, 5, or 6. If
we assume nf is 2, then ni can be 3, 4, 5, 6, or 7. In either case, we will be looking to see which
data is consistent with the Rydberg constant.
The Rydberg equation can be rearranged to help us more easily work with our data. We know
that:
1
1
1
= 𝑅( −
)
2
2
𝜆
𝑛
𝑛
𝑓
𝑖
For the sake of simplifying our exploration we can rework the equation so that it now looks like this:
1
𝑅
𝑅
=
−
2
2
𝜆
𝑛
𝑛
𝑓
𝑖
1
𝑅
𝑅
=−
+
2
2
𝜆
𝑛
𝑛
𝑖
𝑓
If you look closely at the third equation, what you will observe is that this is essentially an equation
1
1
for a straight line regression, y = mx + b, where: y = , m = 𝑅 (the Rydberg constant) and x = 2,
𝜆
𝑛
𝑖
the inverse of the excited state of the electron (nf) that produces the specific emission line at
wavelength, 𝜆.
1
We can then plot 𝜆 vs
1
2
𝑖
𝑛
where the slope and y-intercept tell us about the Rydberg constant and
the x value gives us the excited state of the electron.
If solving for R from both of these values results in the same Rydberg constant, then we know
that we have found the correct value for nf. Also, since the research team will be plotting the same
1
values each time, while only the values on the x-axis change, an incorrect guess for the value
𝜆
of nf will result in a trend line that is not as straight (more error, worse fit).
If, for example, we look at the ultraviolet spectrum of hydrogen we find distinct emission lines at
wavelengths of 121.5668 nm, 102.5722 nm, 97.2537 nm, 94.9743 nm and 93.7803 nm.
We can begin the process by making the assumption that the ground state of the electron (nf) is
energy level, n = 1. This would be the energy level and the electron’s orbit closest to the nucleus.
We can, if we wish, record five readings based upon the assumption that the electron could have
descended from n = {6, 5, 4, 3, 2}. Additionally, we can also assume that the lowest wavelength
reading would be associated with the electron quantum leaping back from n= 2 to n = 1. The next
lowest measured wavelength would be associated with the electron descending from n = 3 down
to n = 1 and so forth. If we were to tabulate the wavelength data and the hypothesized energy
levels, the table might look like this:
Sample table where nf = 1
1
𝑛𝑖 2
2
0.250
121.566
1
λ
0.00822
3
0.111
102.572
0.00974
4
0.063
97.2537
0.01028
5
0.040
94.9743
0.01052
6
0.028
93.7803
0.01066
ni values
For the value n = 1 (1st quantum energy level)
λ (nm)
For the value n = 2 (2nd quantum energy level)
Since the line with the better correlation coefficient has a better y-intercept of .011 for n = 1 and
b=
𝑅
2
𝑛
𝑓
where 𝑛
𝑓
= 1, we can now converted to m-1, Rydberg’s constant becomes 1.1 x 107 m-1.
Using the value associated with the Rydberg constant, it should now be possible to determine the
excited electron state for each of the ultraviolet lines using the observed concentrations. For
example, consider the 121.5668 nm value for one of the UV emission lines. We’ve established
the electron quantum leapt from the ground state of n = 1 to some energy level representing the
excited state. Upon its transition back to the ground state (ni → nf), photons with the frequencies
associated with the particular emission line being observed were produced. Which energy level
was associated with the excited state?
Consider:
We know the ground state, nf = 1 and we have established the value for R (1.1 x 107 m-1). We
also know the wavelength (121.5668).
Therefore all we have to do is plug in the established values to determine the excited state for the
electrons that produce the emission lines at 121.5668 nm
1
121.567𝑛𝑚
(
𝑥
1𝑚
)
1𝑥109 𝑛𝑚
1
1
= 1.10 𝑥 107 𝑚−1 ( 2 −
)=2
(1 ) (𝑛𝑖 2 )
What we now know is that the emission line at 121.567 nm involves electrons that quantum
leap from n = 1 to n =2 and then upon transitioning back to the first energy level emitted photons
at the above wavelength.
Your goal is to observe and record any emission lines produced in the visible range by both
hydrogen and helium discharge tubes.
Assuming the electrons had previously existed in n =
{1,2,3} ground states, determine the Rydberg constant using the appropriate linear regressions.
Also determine excited electronic states for all the observed lines in the UV emission spectrum