Hydrogen Atom Spectrum .
Spectrum of the
Hydrogen Atom
Objective
To calculate the Rydberg constant from the spectrum of atomic hydrogen.
Preparation
1. Read all of this write-up.
2. Understand the experiment, apparatus, and procedures well. You will be
performing many of the operations in the dark.
Overview
Excited hydrogen atoms are produced in an electric discharge which not only
dissociates hydrogen molecules, but excites the atoms as well. These atoms radiate light
at discrete wavelengths, some of which lie in the visible region of the spectrum. This
visible light is dispersed by a diffraction grating in a spectrograph and detected by
photographic film. The wavelength determines the position of the lines in the spectrum,
and the wavelength is calibrated using the Hg spectrum where the lines are intense and
well-known. Measurement of the H atom lines then makes it possible to determine the
Rydberg constant.
Theory
Hydrogen exists mainly as diatomic molecules, but if hydrogen gas is heated very
hot or bombarded by electrons, the molecules can be dissociated into atoms. Thermal
heating will result mainly in ground state atoms, but electron bombardment of the atoms
(as happens in an electric discharge) can produce atoms in excited states. Excited atoms
can undergo transitions to states of lower energy by spontaneously radiating energy
characteristic of the energy difference between the two states, given by the well-known
Einstein relation,
E2 - E1 = ΔE = hν (1)
where h is Planck's constant and ν the frequency of light emitted.
For an atom with one electron, the Schrödinger equation can be solved exactly
and the energy is given by1
En = -
μe4
8h2ε02
1
n2 n = 1, 2, 3, ..... (2)
where e is the charge on the electron and ε0 the permitivity of vacuum. The reduced
mass, μ, of the electron and proton is given in terms of the electron mass, m, and the
proton mass, M, by
μ =
mM
m+M (3)
The negative sign means that a state of principal quantum number, n, is more stable
(lower energy) than one with n = ∞, where the electron and proton are infinitely far apart
and the energy is defined to be zero. For finite values of n the system is thus bound. In
accordance with Eq (2) the ground state is bound by 13.595 eV (the ionization potential)
whereas the first excited state is bound by only 3.4 eV.
Fig 1. A few electronic energy levels of the hydrogen atom. As n increases the energy levels
converge to a limit, and above this limit (the shaded area) there are a continuum of levels
corresponding to complete separation of proton and electron with kinetic energy > 0
shift, but this is much too small to be detected in these experiments. For atoms with more
than one electron, the energy is significantly dependent upon l (!).
Atoms may undergo transitions between any two states of different principal
quantum number, depending on whether they absorb or emit radiation. There are no
"selection rules" for n. An energy level diagram for the H atom is shown schematically
in Fig 1, together with a number of representative transitions in which a photon could be
emitted. (We will detect these photons in this experiment.) These transitions were
historically identified in different spectral regions, with the Lyman series appearing in the
ultraviolet, the Balmer series in the visible, the Paschen in the infrared, and so on. Each
series is characterized by the lowest level, n=1 for the Lyman series, n=2 for the Balmer
series, etc.
Our apparatus will restrict us to observing transitions in the visible region of the
spectrum, the Balmer series. (Roughly speaking, lines from the Lyman series would be
absorbed by the atmosphere and are for us unobservable; lines in the Paschen series lie
the infrared are not recorded by our photographic film.) Combining Eqs (1) and (2) for
the Balmer series (nfinal = 2) we find
ΔE = E2 - En =
μe4
8h2ε02 ⎝
⎛
⎠ ⎞
14
-
1
n2 = hν =
hc
λ (4)
and
1λ
=
μe4
8h3cε02 ⎝
⎛
⎠ ⎞
14
-
1
n2 = R ⎝
⎛
⎠ ⎞
14
-
1
n2 (5)
where R is the Rydberg constant and has the calculated2 value of 109 677.5805 cm-1 for
H. One usually measures the wavelength and connection with theory is made with the
reciprocal wavelength (called the wavenumber, ν ∼ , and measured in cm-1.)
METHOD
Measurement of the wavelengths of the lines in the Balmer series is
experimentally carried out by comparing the hydrogen spectrum with a reference
spectrum for which the wavelengths are already known. The iron or mercury spectra are
frequently used because they have many intense lines which have been measured on an
absolute scale by interferometry. The iron spectrum is very rich (too rich) and we will
use the mercury spectrum which can be more readily identified.
Exposing the Film
Expose the film with the various lamps according to the following procedure:
a. Replace the spectrograph's Plexiglas cover with the film holder.
b. Close the shutter, and then slide the cover of the film holder (the “dark
slide”) out to the mark. If you slide it all the way out, it will be impossible
to replace. Make certain you have not pulled out the film; if you have,
discard the film and start over.
c Cover the film holder with the black felt to minimize light leaks.
d. Make the exposures as described under the section Lamps. Do not close
the outer cover between exposures to avoid moving the film.
e After all exposures have been made, slide the outer cover closed.
f. Replace the film holder with the Plexiglas cover, to keep dust from the
diffraction grating.
g. Develop the film
Lamps
SAFETY: Both arc lamps operate at extremely high voltages. The mercury lamp
emits ultraviolet radiation, which can harm your eyes; do not look directly at the
mercury lamp while it is in use. The sodium lamp needs to warm up for five minutes
before use, and gets extremely hot. The sodium lamp will not focus as narrowly as the
two arc lamps.
PROCEDURE For each exposure, follow the procedure outlined below:
1. Place your light source as shown in Figure 2, at the same height as
the slit assembly. Focus the strip of light onto the paper covering
the shutter by sliding the cylindrical lens along the optical track.
Adjust the position of the strip horizontally by shifting the light
source, and vertically by moving the lens up or down. The strip of
light should pass through all three circles drawn on the shutter.
Remember that any movement of the light source or lens may
change the focus. Once the light is correctly positioned and
focused, tighten the screw holding the lens assembly to the optical
track.
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