Lab #7 – Understanding the Emission Spectrum of the H atom
Background
The interaction of light energy with atoms provides some of the strongest evidence for understanding the
arrangement of electrons in the atom. Through the work of Bohr, Einstein, DeBroglie, and many other brilliant
scientists, we have developed quantum mechanics, the branch of physics which helps us understand this
interaction and allows us to infer a great deal about electron configurations.
We now believe that electron behavior is quantized, specifically that electrons can only exist in restricted, noncontinuous energy levels. When an atom absorbs light energy, its electron is jumping instantly, not
incrementally, from one allowed energy to another. This is referred to as a quantum leap. A good analogy is to
consider that atoms exist on a staircase, as opposed to a ramp. On this staircase, an electron has to live on one
stairstep or another, but is not allowed to exist suspended between steps. The steps represent allowed energy
levels, whereas the space between steps represents energies that are not allowed. When an electron absorbs
energy, it can jump up one or more steps; when an electron drops down one or more steps, it gives off energy.
Normal objects, like baseballs and cars, exist on ramps; there are no restrictions on the allowed energies, and
cars and baseballs speed up gradually, not in quantum leaps.
For the simplest atom, the H atom with only 2 particles, determining the allowed energies is easy;
(1)
EH atom = -1312.04 kJ/mol
n2
where n is an integer from 1 to infinity.
The energy absorbed by an atom as its electron is excited to a higher energy level can be in several forms – light
energy and electrical energy are the most commonly absorbed forms. The energy given off as an electron
relaxes to a lower state is in the form of electromagnetic radiation, sometimes in the human visible range. The
energy of a photon of light is given by
(2)
Elight = hν = hc,
λ
where h = Planck’s constant = 6.62618 x 10-34 J s
and c = the speed of light = 2.997925 x 108 m/s
This corresponds to the difference in energy between the higher and lower energy levels,
(3)
ΔEelectron (kJ/mol) = Ehigh – Elow = -1312(1/n2high – 1/n2low)
Putting Equations (2) and (3) together (substituting for h and c, and adjusting for n = 6.02205 x 1023) yields
Eq’n (4), which directly relates the energy of the electron transition to the wavelength of light emitted for the
transition.
(4)
ΔEelectron (kJ/mole) = 1.19627 x 105
λphoton (nm)
Procedure
1. Calculate the allowed energies for the 1st eight energy levels in the hydrogen atom. How many sig figs?
n
1
2
3
4
5
6
7
8
Ee (kJ/mol)
2. Calculate the ΔEelectron and wavelength of light that corresponds to each of the following e- transitions.
Transition
ΔEelectron (kJ/mol)
λphoton (nm)
Transition
87
64
86
63
85
62
84
83
61
54
82
81
53
52
76
51
75
43
74
42
73
41
72
32
71
31
65
21
ΔEelectron (kJ/mol)
λphoton (nm)
3. Assign an electron transition to account for each of the observed emission wavelengths.
Wavelength
(nm)
97.25
Transition
nhigh  nlow
Wavelength
(nm)
410.29
Transition
nhigh  nlow
Wavelength
(nm)
1005.2
102.57
434.17
1094.1
121.57
486.27
1282.2
389.02
656.47
1875.6
397.12
954.86
4052.3
Transition
nhigh  nlow
4. Draw a potential energy well showing the first six energy levels on the next page. Use a sharp pencil to
draw a horizontal line for each energy level. Label each level on the right side with its n value; label on the left
side with the corresponding energy.
5. Draw every transition by using arrows. Group the transitions by nhigh; for example, draw all of the
transitions starting at the 6th energy level in one group. Label each arrow at the top with the wavelength that
would be emitted as a result of the transition.
Potential Energy Well for the Hydrogen atom
Energy (kJ/mole)
n
Questions (All questions relate to the hydrogen atom)
1. Define ground state:
2. The human visible range is from about 400 nm to about 700 nm. Which transitions fall within this range?
3. What do the transitions in Question #2 have in common?
4. What is the name of this series of transitions?
5. Why do transitions to the 1st energy level never produce visible light?
6. What is the name of the series of transitions to the 1st energy level?
7. Why do transitions to the 3rd energy level never produce visible light?
8. What is the name of the series of transitions to the 3rd energy level?
9. Calculate the energy of light you would expect to be produced by the transition 10  3. Show all work.
10. What is the largest value possible for n?
value of n?
What would the energy be for an electron with that
How tightly bound is that electron?
11. There are transitions in the Balmer series in addition to the four that produce visible light. Calculate the
shortest wavelength of light in the Balmer series. (Would short wavelength be high energy of low energy?)
12. The ionization energy of an atom is the energy needed to remove an electron, which is to say, add energy
until the electron has an energy of 0 kJ/mol. What is the ionization energy of the ground state hydrogen atom,
in kJ/mole?
What is the ionization energy in J/atom?
What is the ionization energy in eV/atom? (1 electron volt = 1 eV = 1.602 x 10-19 J)
Prelab Questions
The helium ion, He+, is a simple two-body problem like the hydrogen atom, so its quantum mechanics are
simple, too. For the He+ ion, the allowed energies are given by
En = -5248.16 (kJ/mol)
n2
1. Calculate the first five energy levels in the He+ ion:
E1 =
E2 =
E3 =
E4 =
E5 =
2. Calculate the wavelength of light that would be emitted as the result of a electron transition from n=5 to n=2.
3. Three spectral lines in the spectrum of the He+ ion occur at (i) 121.57 nm, (ii) 164.12 nm, and (iii) 25.64 nm.
Determine the electron transition that accounts for each emission.
(i)
(ii)
(iii)