Download Emission Spectrum and Planck`s Constant

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Emission Spectrum and Planck’s Constant
Purpose of the Experiment
This experiment enables students to measure emission spectra. To understand the Bohr model
of the hydrogen atom, students must not only familiarize themselves with optical systems for spectrum
measurement and relevant skills but must also determine Planck’s constant according to the experimental
results and data.
Experimental Principle
A spectrometer is employed in this experiment to observe the emission spectra of atoms. The
spectrum to be observed is the Balmer series spectral line in the visible spectrum of light emitted from a
hydrogen discharge tube. The observed wavelengths of spectral lines can be used to derive the value of
h, Planck’s constant.
A.
Model of the Hydrogen Atom
In 1913, Niels Bohr introduced a theoretical model for depicting the internal structure of a
hydrogen atom. His major postulates were as follows:
1. The electron moves in circular orbits around the nucleus under the influence of the Coulomb force.
2. The circumference of the electron orbit is quantized when the standing wave condition is satisfied.
The electron’s orbital angular momentum around the nucleus is L= nh 2π .
3. When transitioning from a higher energy orbital (the initial energy is Ei) to a lower energy orbital
(the final energy level is Ef), the electron releases the energy through electromagnetic waves that
have a frequency of ν = ( Ei − E f ) h .
According to the Bohr model, the centripetal force between the electron and the nucleus is equal to the
Coulomb force between the two.
1 e 2 mv 2
=
(1)
4πε 0 r 2
r
where m is the mass of the electron, v is the speed of the electron, and r is the orbit radius. The electron’s
L = mvr
angular
momentum
satisfies
the
standing
wave
condition;
thus,
Ln = mvr = nh 2π , n = 1,2,3..... . Therefore, the total energy En of the electron at the nth orbital can be
represented using the relationship between kinetic and potential energy.
1
1 e2
En = mv 2 −
(2)
2
4πε 0 rn
Combining Formula (1) with the angular momentum conclusion and simplifying yields an t En of
me 4 1
En = − 2 2 2 , n = 1, 2,3....
(3)
8ε 0 h n
which is the energy level of the electrons in the hydrogen atom. When n = 1, E1 is considered the ground
state energy, and when n = ∞ and E∞ = 0, the electron is removed from the atom or the range of the
nucleus in a process known as ionization. If an electron in the hydrogen atom transits from the
uppermost n = ∞ energy level to the n = 1 level, the potential released energy is
me 4
∆E = E1 − E∞ =
~ 13.6 eV
8ε 0 2 h 2
which is approximate to the energy of ultraviolet light at a wavelength of 91 nm.
In this experiment, we observe the Balmer series in the hydrogen spectrum. This series displays
the electromagnetic wave radiated by the electron that transitions from any n > 2 energy level to the n
= 2 energy level. The emitted waves are visible because their wavelengths are within the visible
spectrum range. Table 1 shows the wavelengths of this series. The relationship of the wavelengths is
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1
λ
=
1
me 4  1
− 2
2 3  2
8cε 0 h  2
n

 , where n > 2 .

(4)
Table 1. The Balmer series wavelengths
Transition of n
3→2
4→2
5→2
6→2
7→2
8→2
9→2
Name
H-α
H-β
H-γ
H-δ
H-ε
H-ζ
H-η
486.1
434.1
410.2
397.0
388.9
383.5
Wavelength (nm) 656.3
B.
Spectrometer
Some of the following content has been adapted from PASCO: “Instruction Manual and
Experiment Guide for the PASCO scientific Model SP-9268A - STUDENT SPECTROMETER.”
Using the dispersion principle of grating or a prism, a spectrometer separates white light into
various colors by refracting light of various wavelengths in different angles.
Fig. 1 is a schematic diagram of the optical path of the spectrometer used in this experiment. The
collimator shapes the incident light emitted by the light source into a parallel beam projected into the
spectrometer. This parallel beam has a comparatively smaller divergence angle that prevents the beam
intensity from decreasing with the propagation distance. When the beam illuminates a dispersion
element, such as diffraction grating (or a prism), it is refracted in a specific angle according to various
wavelengths.
A telescope is positioned behind the dispersion element to collect the collimated parallel beam. An
image is projected onto the retina of the observer through the collimator slit. When measuring the beam
wavelength, the telescope can be rotated to focus on the slit, and the corresponding wavelength can be
calculated based on the angle of the spectrometer. The black vertical crosshair in the eyepiece can be
used as the baseline for focusing the telescope.
Figure 1. Schematic diagram of the optical path of the spectrometer
Figure 2 shows the spectrometer. Beams pass through the collimator slit before entering the
spectrometer. This slit, which should be installed vertically, is located at the center of the slit plate. The
width of the slit can be adjusted using the slit-width adjustment screw. A thinner slit width generally
enhances the precision of wavelength measurements; however, it can also increase the difficulty of
observation because of the lower intensity of the light passed through the slit. Because the light source
used for this experiment is a slender luminous body, we recommend repositioning the entire
spectrometer to where it can collect the strongest intensity and highest number of light beams before
making subsequent adjustments.
Passing through the collimator, the beam illuminates the dispersion element, which is the grating
in this experiment. The grating must be positioned where the full beam cross-section can be collected,
and the angle of the grating plane should be perpendicular to the beam propagation direction. The
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spectrometer table below the grating has three knobs for adjusting the grating incline. Inclining the
grating results in a non-level diffraction , which hinders the focusing of the telescope. In addition, the
height of the spectrometer table can be adjusted using the thumbscrew on the rotating base fixed to the
table.
The beam enters the telescope after it passes through the dispersion element. During the initial
spectrometer setup, we recommend that the black vertical crosshair in the eyepiece is aligned with the
transmitted beam that is not refracted by the grating. The unrefracted beam is known as the
“maximum” or the “light spot of the zero-order diffracted beam.” The vernier scale angle reading is
the baseline for all angles involved in subsequent procedures. If the vertical crosshair is found to be
inclined, rotate the eyepiece for adjustment. If the slit image is not sharp, use the collimator or the
focus knob on the telescope to adjust the focal length.
Figure 2. Spectrometer
The vernier scale on the spectrometer has a precision of one minute of arc (1/60°). For example,
Fig. 3 shows a reading of 155°15’ or 155 + 15 × 1/60°.
Figure 3. Vernier scale measurements or divisions
If a certain monochromatic beam measured by the spectrometer is at angle θ, and the spacing
a sin θ
between lines of the grating is a, the wavelength of the beam is λ =
, where m is the order of
m
diffraction. When the grating is 300 lines/mm, a = 3.3 ×10−3 mm .
We recommend measuring the diffraction angle on both sides of the zero-order diffraction location.
As shown in Fig. 4, the instrument functionality can be verified if the measured angles on both sides are
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identical.
Figure 4. Schematic diagram of angles of diffraction
Laboratory Instruments
1.
2.
3.
4.
5.
Hydrogen discharge lamp
Spectrometer
Diffraction grating
Spirit level
Flashlight
Notes
※ The terminals or junctions of the hydrogen discharge lamp are high in voltage when the tube is
switched on. To avoid accidental electric shocks, do not handle the terminals!
※ Touching the hydrogen discharge lamp or tube with your fingers is strictly forbidden.
Experimental Procedure
A.
The Spectrometer and Hydrogen Discharge Tube Lamp
1. Approximately align the slits of the hydrogen discharge lamp with the spectrometer and then
switch on the lamp.
※
※
※
The hydrogen discharge lamp uses a high-voltage power source of several thousand volts
and poses a high risk of electrical shock.
Clear the space around the lamp before switching it on. Once switched on, avoid moving
the lamp.
Remember not to touch the discharge lamp with your hands!
2. The discharge observation position when the discharge lamp is activated is the narrow area
where the discharge occurs; this is located in the middle of the tube or lamp and measures
between 2 and 3 cm in length and 1 mm in width.
The discharge area typically appears purple. When the lamp begins to age, a portion of the
discharge area becomes white, and the area that appears purple decreases.
Therefore, ensure that the purple discharge area of the light is aligned with the slit when making
adjustments.
3. Observe the collimator with your naked eyes before aligning the lamp with the slit. Adjust the
relative position of the collimator and the slit until a maximum intensity is observed.
If necessary, rotate the slit to align it with the lamp .
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Because the discharge lamp used for this experiment has a lower light intensity, observation
with naked eyes will not cause immediate damage.
4. Align the telescope with the collimator, and align the vertical crosshair in the eyepiece with the
slit image. For more accurate operation, we recommend using the lock screw and fine
adjustment knob for the spectrometer. Furthermore, the following two points should be noted:
☉ Again observe whether the slit is inclined, and make any necessary fine-tuning adjustments.
☉ The width of the slit is adjustable. A thinner slit facilitates the alignment of the crosshair
graticule and provides more accurate angle measurements. However, if the slit is too thin,
the reduced amount and intensity of the light passed through hinders the observation of
darker spectral lines. Therefore, you must determine the slit width appropriate for
subsequent procedures after considering differing influences.
5. After the alignment is complete, record the vernier scale reading as the baseline for all angles.
☉ Subsequent measurements are based on this angle baseline. Ensure that the correct baseline
is recorded.
6. Position the diffraction grating on the spectrometer table at the center of the spectrometer. The
following points should be noted:
☉ Record the grating specifications (lines/ mm).
☉ To facilitate observation within the adjustable angle of the telescope, ensure the beam
incident on the grating is dispersed at the correct angle.
☉ Position the diffraction grating plane perpendicular to the incident beam.
☉ Adjust the height and incline angle of the grating using the thumbscrew on the bottom of
the spectrometer table.
☉ The following image URL shows a spectrum that may be observed in this experiment:
http://en.wikipedia.org/wiki/File:Visible_spectrum_of_hydrogen.jpg
B.
Spectrum Measurement
1.
2.
3.
4.
C.
Rotate the telescope in one direction and check for the appearance of spectral lines.
(Determine whether the vernier scale reading changes with the movement of the telescope.
Under the correct circumstances, only the auxiliary scale should move; the main scale should be
fixed and immovable.)
Align the black crosshair with any spectral line and record the angle and diffraction order m.
Observe an order m as high as possible.
☉ Diffraction order m is determined by the principle of multiple-slit diffraction.
The angle baseline corresponds to the maximum of the grating diffraction, which is the
zero-order diffraction beam (m = 0).
Rotate the telescope from the baseline until the first instance of a certain spectral line
appears; this is the first-order diffraction of the beam (m = 1).
By rotating the telescope further in the same direction to a larger angle, the same spectral
line should appear again yet dimmer. This is the second-order diffraction beam (m = 2).
Higher orders of diffraction occur in the same manner.
The values of m at the two sides of the angle baseline are typically separated by adding a
plus or minus sign. To follow the rules for the relationship formula of grating
a sin θ
refraction λ =
, m is designated as plus or minus after making sure that λ is greater
m
than zero.
☉ Repeat measurements of any angle 3 to 5 times to obtain the average and error values.
☉ Record any relevant characteristics of the observed spectral lines, including their color and
brightness.
☉ The width of the slit can be adjusted before actual measurements. A thinner slit facilitates
alignment with the vertical crosshair and provides superior measurement precision;
however, less light can pass through, resulting in lower brightness, which hinders
observation.
Rotate the telescope in the other direction, and repeat Steps B1 and B2.
Calculate the corresponding wavelengths of the spectral lines.
Calculation of Planck’s Constant
The wavelengths of the spectral lines in the emission spectrum of atomic hydrogen measured in this
experiment should match the assumptions of the Bohr model.
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1.
Determine the orbital n from which the electron transitions to the n = 2 orbital based on the
measured wavelengths of the spectral lines in the Balmer series.
1
1
2. Let 2 be the horizontal axis and
the vertical axis, and plot a graph of the results obtained
n
λ
in Step C1. Calculate Planck’s constant h using the curve fitting results (known parameters:
m = 9.11 × 10 −31 kg ; e = 1.6 ×10−19 C ; and ε 0 = 8.85 × 10 −12 C 2 /N ⋅ m 2 ).
D. Spectrum Observation of Various Light Sources
1.
2.
Place any luminous body, such as a flashlight or cell phone, in front of the slit of the
spectrometer. Observe the spectra.
A computer screen can also be used as the luminous body. We recommend creating slides
with a red, blue, or green background using Microsoft PowerPoint, and then autoplay the
slides sequentially. Observe the change in spectra using the spectrometer.
Observe whether the spectra are continuous or discrete, similar to the spectrum of the
hydrogen atom.
Record your observations.
Questions for Reflection
1.
Estimate the wavelength ranges of the first three spectral series of the hydrogen atom. In addition,
label the ranges according to whether they belong to the ultraviolet spectrum, visible spectrum,
and infrared spectrum.
2.
Consider why some of the observed spectra are continuous and some discrete? Is the spectrum of
sunlight continuous or discrete?
3.
Based on the angle precision of the spectrometer used in this experiment, calculate the possible
error in wavelength measurement.