Physics 111: Elementary Physics
Laboratory E
Atomic Spectra
1.
Introduction
Sunlight comes to us through vacuum because it is electromagnetic radiation, oscillating electric
and magnetic fields. Such fields can exist in the absence of matter, and can propagate over enormous
distances until they are finally absorbed by matter. Sunlight is composed of light of many colors, each of
which corresponds to a different frequency, f, of oscillation of the fields. In vacuum, each wave propagates
at a speed, c, of 2.997 x 108 m/s, independent of the frequency of the wave. As with any wave, the distance
the wave travels in a single oscillation, the wavelength λ, the frequency f, and the speed of the wave c, are
related by the simple equation
c = λf .
The hues which we perceive as visible light are but a small portion of the total light emitted by a star.
Violet light has the shortest wavelength, about 400 nm, and red light the longest, about 700 nm. Ultraviolet
(UV) light has shorter wavelengths and infrared
(IR) light longer wavelengths, ranging out to
10,000 nm. All incandescent bodies have in
common that their light is composed of a range of
colors spreading from the infrared through the
visible and on into the ultraviolet spectrum, as
indicated in the figure at right. The relative
intensity is always higher in the midrange than on
the extreme ends of the spectrum. The position of
the maximum relative intensity is inversely
proportional to the absolute temperature, T, of the
incandescent body.
The total intensity,
represented by the area under the curve, is
proportional to the fourth power of the absolute
temperature, however. The continuous nature of
the spectral distribution is a consequence of the
fact that a normal incandescent object composed of many different materials.
The radiation emitted by a pure chemical
substance, whether an element or a compound, differs
in an important way from the radiation from a normal
incandescent object. For a pure substance, the light
spectrum is no longer continuous. Rather, it is said to
be discrete. It consists of light of only a few, widely
spaced spectral components. Moreover, the particular
colors or components which are present are
characteristic of the material which emits them. As in
the case of the normal incandescent object, the
intensities vary as a function of the wavelength, as
shown in the figure at right. The intensities of the
individual colors (or spectral lines) depend not only on the absolute temperature, but on characteristics of
the material as well.
The distinction between continuous and discrete spectra has practical importance for energy
conservation. A normal incandescent light not only emits strongly in the visible region of the spectrum, it
emits substantial amounts of radiation in the infrared region as well. This is perceived as heat. A
fluorescent light, on the other hand, is designed to work by causing emission of light from just a few
chemical substances chosen so that the spectral components which are emitted are located primarily in the
visible region. The fluorescent light, then, is more efficient since a relatively greater amount of the energy
used to operate the bulb appears in the useful part of the spectrum.
2.
Procedure
A spectrometer will be used to separate in space the spectral components which make up the
emissions from a given light source, and to record information about the wavelengths of these components.
In a spectrometer, a device such as a prism or a diffraction grating is used to send spectral components of
different wavelengths off in different directions. Since diffraction gratings disperse the spectral
components more widely than a prism and so make the components easier to see, one will be used in this
experiment.
A diffraction grating is constructed by ruling a series of closely spaced, parallel lines on a suitable
surface. When the grating is illuminated, the lines act as secondary sources of light, and the secondary
wavelets can interfere with each other when they come together at a distance. The wavelets can enhance or
diminish each other. Which of these occurs at a given point
depends on the distances followed by the waves as they
propagate from the source, through the grating, to the point
of observation. If the paths differ by an integral number of
wavelengths, a condition called constructive interference
occurs, and the two waves enhance each other. The original
single beam is separated into several beams, as shown in the
sketch at left. If the beam is monochromatic, i.e. has a
single wavelength, λ, then the angles, θm, at which the
beams appear is given by the formula
mλ = d sinθm
.
Here d is the distance between adjacent lines on the grating,
and the index m is called the order of the beam. Note that
the direction of the incident beam is taken as the reference
direction. The diffracted beam in that same direction is
called the central beam and, consistent with the diffraction
formula, is considered to be of order m = 0.
For a gaseous chemical element, each of the spectral components will be diffracted. The central beams of
each pattern will coincide, and that light will have the same appearance as the original light. As the
diffraction formula shows, the angle at which a beam of wavelength appears in a given order m depends on
λ. The angle θ1 for violet light will be about half that for red light since the wavelength of the violet light is
about half that of the red light. A viewer looking back toward the source will see a series of bright lines of
different colors as the position of observation shifts from one side to the other. If the grating space d is
small, the dispersion of the beam will be great and the separate colored spectral components will be clearly
distinguishable.
The spectrometer uses this dispersion of the beams to permit making precise measurements of the
angles at which the beams appear. The diffraction grating is mounted on a table at the center of the
instrument and is illuminated by an appropriate source. An eyepiece is mounted on a movable arm which
is then rotated to a viewing position where a given spectral component is visible. An indicator on the arm
marks the position on a calibrated scale of angles fixed to the instrument. The scale is calibrated to 0.5°, or
30 min (30’) of arc. A vernier scale is often also available which permits determining changes of arc to
within 1 min., but this refinement will not be used here.
The typical discharge tube will provide a line of polychromatic light. The spectrometer is aligned
with the arm in the center position and the light source opposite the eyepiece. The diffraction grating is
mounted on the support table, perpendicular to the line of sight. The movable arm is rotated until the
central beam appears at the cross hair of the eyepiece. The position of the arm should be noted as the
reference position. The angles θm associated with the order m are found by rotating the arm to the right or
to the left until the hair line rests on the line. An average of the two readings is taken to minimize any
residual error in the initial alignment.
Calibration of the Grating Spacing:
The grating spacing is often known from other information. However, it is possible to measure it
directly using light of a known wavelength. Here the green light from mercury (Hg) will be used. Its
wavelength has been measured to be 546.1 nm. The procedure will be to determine the average θm for two
or three orders, and to calculate the spacing from the diffraction formula for each separate order, and then
finally to average the different values of d to obtain the working value.
Hydrogen Spectra:
A discharge tube containing hydrogen produces the light of atomic hydrogen. This light is rich in
spectral lines which can be described by a simple formula. The formula, an empirical relationship; was
first found in the previous century by Rydberg. It was first derived theoretically in 1913 by Neils Bohr.
An atom of hydrogen consists of a proton and an electron which are bound to each other through the
attraction of the electric force. In contrast to what might be expected from macroscopic phenomena, at the
scale of atoms, not all possible energies are available to the electron. Rather only certain energy levels are
available to the electron. According to quantum mechanics, with which Bohr’s theory is consistent, these
levels are given by the formula
En = -(2.18 x 10-18J)/n2, n = 1, 2, 3, …
Light is emitted from the atom when the electron experiences a quantum jump from an initial level, n i, to a
final level, nf. The change in energy of the electron is made apparent by the emission of a quantum of light,
a photon, which has a frequency given by the difference between the energies of the two levels, divided by
Planck’s constant, h. (h has the value 6.626 x 10 –34J – s)
f = c/λ = (Ei – Ef)/h .
It follows from these expressions that the inverse wavelength, 1/λ. is given by
1/λ = R(1/n2f – 1/n2i),
where R, the Rydberg constant, has the value 1.0974 x 10 7m-1 .
The light emitted when the electron reaches the lowest energy, with nf equal to 1, is in the UV, and
cannot be seen. The light emitted when the electron winds up on the second lowest level, with nf equal to
2, contains light in the visible part of the spectrum. It will be investigated here. The light emitted when
electrons finish in levels with nf equal to 3 or more is in the IR, and also is not visible. It is possible to
verify the presence of the invisible UV or IR light with other types of detectors. The visible light is called
collectively the Balmer series. It begins with a red light, for n i equal to 3, and goes on to other colors, with
successively higher values of ni.
The light source used with the spectrometer should be changed to a hydrogen discharge tube, and the
instrument realigned. The angle, left and right, should be observed for each of the visible spectral
components in the visible spectrum in the first two orders, m equal to 1 and 2. The diffraction formula
should then be used to calculate a value of λ for each of the separate colors, and the index ni consistent with
that wavelength determined. The table below should be used to organize the data and to complete the
calculation.
The experiment will be completed by observing the spectrum of an incandescent lamp.
Physics 111: Elementary Physics
Pre-Lab Exercise
Atomic Spectra
Name: ______________________
Section: _____
Calibration of the Grating Spacing:
A group of students has taken the following data by observing the green line of mercury using a
spectrometer. Find the spacing of the diffraction grating n the spectrometer.
order(m)
θright (deg)
θleft (deg)
θav (deg)
d
1
19.10
19.13
_______
_______
2
40.90
40.94
_______
_______
dav _______
Hydrogen Spectrum:
Use the sample data below, taken with a hydrogen lamp, to complete the data table.
color
order
violet
violet
1
2
θright (deg)
15.08
31.40
θleft (deg)
θav (deg)
15.12
31.35
_______
_______
λ
______
______
λav
blue
blue
1
2
17.00
35.68
16.94
35.72
_______
_______
1
2
23.22
51.95
23.17
51.90
_________
________
_______
_______
λav
red
red
n1
_______
_______
_________
________
_______
_______
λav
_________
________
Questions:
1.
State briefly (100 words or less) the purpose of this experiment .
2.
Begin with the Bohr expression for the energy of the electron in level n, and provide the explicit
details needed to derive the Rydberg expression and the value for the Rydberg constant.
3.
The wavelengths of the visible lines of the hydrogen spectrum have been measured in terms of the
wavelength of a particular line of the Hg spectrum. The wavelength of that line was given to four
significant figures. As a result of accumulated errors in determining the grating spacing d and the
angles at which the hydrogen lines were observed, how many significant figures do you estimate are
you entitled to quote for the hydrogenic wavelengths ?
Physics 111: Elementary Physics
Lab Report
Atomic Spectra
Investigators: ________________________ ,
________________________ ,
_______________________
_______________________
________________________
Date: _____________
Procedure: Describe briefly (200 words or less) the procedures used in this experiment.
Data:
Calibration of the Grating Spectrum:
order(m)
θright (deg)
θleft (deg)
θav (deg)
d
_______
________
_______
_______
_______
_______
________
_______
_______
_______
_______
________
_______
_______
_______
dav _______
Hydrogen Spectrum:
color
order
θright (deg)
θleft (deg)
θav (deg)
_____
_____
____
____
________
________
_______
_______
_______
_______
λ
______
______
λav
_____
_____
____
____
________
________
_______
_______
_______
_______
____
____
________
________
_______
_______
_________
________
______
______
λav
_____
_____
n1
_______
_______
_________
________
______
______
λav
_________
________
Incandescent Lamp:
Complete the experimentation by replacing the hydrogen discharge lamp with an incandescent lamp.
Observe the spectrum, and estimate the upper and lower bounds of the visible spectrum, as seen by eye.
order
θright (deg)
θleft (deg)
θav (deg)
red
_______
_________
________
_______
_______
violet
_______
_________
________
_______
_______
color
Discussion: Summarize briefly the results which you have observed.
λ