Prelab: Rydberg Constant
Read the manual and answer the following questions:
1. The four lines of the hydrogen spectrum correspond to what colors and initial
states?
2. How will the experimental Rydberg constant (Rexpt ) be determined?
3. How will the final energy state, n2 , be determined from the experiment?
Prelab: Rydberg Constant
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EXPERIMENT
Rydberg Constant
Objectives
By the end of this activity, you should be able to:
• obtain the emission spectrum of a hydrogen discharge source
• determine experimentally the Rydberg constant for hydrogen
Introduction
When light is emitted from excited gases or vaporized liquid, or solids, line spectra are
observed. Different substances have characteristic spectra which means that they have a
characteristic set of lines at specific wavelengths. The characteristic color of light from a
gas discharge is indicative of the most intense spectral lines in the visible region. Light
from a hydrogen discharge has a bright pink glow. In the experiment, a diffraction-based
spectrometer will be used to obtain the emission spectrum of hydrogen discharge. The
Rydberg constant for hydrogen will also be determined from the emission measurements.
Theory
According to the classical electromagnetic theory, an electron (with charge −e) moves
under the influence of Coulomb attraction to the positive nucleus (with charge +Ze and
atomic number Z) and follows a circular or elliptical orbit. The energy E of this electron
is
E =K +U =−
1 kZe2
2 r
(1.1)
where K and U are the kinetic and potential energies, respectively, k is the Coulomb
constant, and r is the radius of the orbit. According to classical theory, this atom is
unstable and would eventually collapse as the electron radiates away its energy.
Rydberg Constant
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1st Sem, A.Y. 2015-2016
Physics 73.1
A. The Bohr model of the hydrogen atom
The Danish physicist Neils Bohr addressed the “collapsing atom” problem by postulating
that
1. the electron in the hydrogen atom can move only in certain nonradiating, circular
orbits called stationary states.
2. the frequency f of the emitted radiation is given by
f=
Ei − Ef
h
(1.2)
where h is the Planck’s constant, Ei and Ef are the initial and final energies of the
atom, respectively. In terms of the radii r1 and r2 of the initial and final orbits,
respectively,
1 kZe2 1
1
Ei − Ef
=
−
(1.3)
f=
h
2 h
r2 r1
3. the angular momentum mvr of the electron in a stable orbit is quantized, that is,
mvr = nh̄,
n = 1, 2, . . .
(1.4)
where m is the mass of the electron and h̄ = h/2π.
Using Newton’s second law of motion, the radius of the circular orbit is given by
r=n
2
h̄2
mkZe2
!
2
=n
a◦
Z
(1.5)
where a◦ = h̄2 /mke2 is called the Bohr radius. The frequency of the emitted light is given
by
f =Z
2
mk 2 e4
4πh̄3
!
1
1
− 2
2
n2 n1
!
(1.6)
where n1 and n2 are the initial and final states, respectively. Expressing Equation 1.6 in
terms of the wavelength λ of the emitted light and noting that Z = 1 for the hydrogen
atom, we obtain the Rydberg-Ritz formula
1
1
1
=R
− 2
2
λ
n2 n1
!
(1.7)
where R is the Rydberg constant for the hydrogen atom defined as
R=
mk 2 e4
4πch̄3
(1.8)
with the value R = 1.096776 × 107 m−1 and c is he speed of light.
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Introduction
1st Sem, A.Y. 2015-2016
Physics 73.1
B. The Balmer Series
The Balmer series corresponds to transitions within the visible to ultraviolet spectrum.
It was formulated by the Swiss teacher Johann Balmer in 1884. Balmer suggested that
the formulation is a special case of a more general expression for the emission spectra of
other elements where n2 = 2. In terms of the Rydberg-Ritz formula, we may write the
Balmer series as
!
1
1
1
=R 2 − 2 ,
n1 = 3, 4, 5, . . .
(1.9)
λ
2
n1
Table 1.1 shows the transitions in the Balmer series with the corresponding wavelength
of the emitted light for each transition. Note that only the first four transitions lie in the
visible spectrum.
Table 1.1: Transitions in the Balmer series
ni Color of emitted light λ (nm)
3
red
656.3
4
blue-green
486.1
5
blue-violet
434.1
6
violet
410.2
7
ultraviolet
346.6
Figure 1.1: Energy level diagram for the hydrogen atom
C. Energy levels of the hydrogen atom
The total energy of the electron in the hydrogen atom (Z = 1) in terms of the quantized
orbit radius r is
En = −
Rydberg Constant
1 ke2
E◦
=− 2
2 r
n
(1.10)
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1st Sem, A.Y. 2015-2016
Physics 73.1
where E◦ = hcR is the ground state energy of the electron, and hc = 1239.8eV ·nm. Since
the energies are quantized, the frequencies and wavelengths of the radiation emitted by
the hydrogen atoms are also quantized.
Figure 1.1 shows the energy-level diagram for hydrogen showing different possible
transitions from the ultraviolet to the infrared range of the electromagnetic spectrum.
Materials
The following materials are required for the experiment:
• Student spectrometer
• Diffraction grating (100 lines/mm)
• Hydrogen discharge tube
• High voltage power supply
• Magnifying lens
• Desk lamp
Procedure
CAUTION:
• Do not touch the surface of the grating. Hold it on the sides.
• Be careful when mounting or unmounting lamps because the springs might
snap even if you did not continue mounting or unmounting them. Seek
advice from your instructor or the lab technician in case of doubt.
1. Insert the hydrogen discharge tube into the high voltage power supply then switch
on the power supply.
2. Align the student spectrometer using the brightest emission line of hydrogen and a
diffraction grating of 100 lines/mm (similar procedure with the Light Spectroscopy
experiment). Complete Table 1.2.
3. Locate the angular displacement of the first-order visible spectral lines of hydrogen
by rotating the telescope of the spectrometer. Tabulate the colors and angular
displacement of the visible spectral lines in Table 1.3.
4. Calculate the wavelength of the line spectra then complete Table 1.3.
5. Repeat steps 3 and 4 for the 2nd order diffraction.
6. Plot 1/λexpt vs 1/n21 for the first and second order colors if Figure 1.2 and 1.3,
respectively.
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Materials
Physics 73.1
1st Sem, A.Y. 2015-2016
7. Use linear regression to obtain the slope and y-intercept of the plots.
8. Calculate Rexpt from the slope and n2,expt from the y-intercept of the best fit line.
Record the calculated data in Table 1.4. Compare with theoretical values.
Rydberg Constant
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1st Sem, A.Y. 2015-2016
Physics 73.1
Name:
Group members:
Date:
Instructor:
Data Sheet
A. Alignment
Table 1.2: Angular displacement of a hydrogen emission line separated using a grating
Direct Image
Color Line
∆θcw
∆θcw
Absolute Difference
Trial
θ◦cw
θ◦ccw
θcw
θccw | θ◦cw − θcw | | θ◦ccw − θccw |
| ∆θcw − ∆θccw |
Grating Constant:
Color:
B. Hydrogen Diffraction Line Spectrum
ni
Table 1.3: Balmer series for the 1st and 2nd order diffraction of Hydrogen
First Order
Second Order
−1
−2
∆θ
λexpt
(λexpt )
(ni )
∆θ
λexpt
(λexpt )−1 (ni )−2
3
4
5
6
Figure 1.2. λ−1 vs n−2
plot for the first
i
order diffraction
Rydberg Constant
Figure 1.3. λ−1 vs n−2
plot for the second
i
order diffraction
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1st Sem, A.Y. 2015-2016
Order
1
2
Table 1.4: Calculation of Rydberg Constant
Rtheo
Rexpt
% Error n2,theo n2,expt
Physics 73.1
% Error
Questions
1. Calculate the ground state energy (in eV) of the hydrogen atom, using your Rexpt .
2. Compare Rexpt between the first and second order diffraction. How do you account
for the discrepancies?
3. How close is your Rexpt from Rtheo . What are the possible sources of errors (except
human errors)?
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Procedure