Student Name and I.D.: Peter
Lab session: Lab 2B
Submission
Date: 14-3-2001
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1.
Introduction
Quantum
theory applies to atoms predicts the energies of the atomic electrons are
quantized, with a ground energy level and a constellation of excited energy
levels. One of the phenomena that stimulated the development of quantum theory
was the emission of light by atoms only at discrete wavelengths, the atomic
spectrum.
Figure 1: Measuring atomic spectra and the visible part of the hydrogen spectrum
Balmer
proposed a formula to describe the positions of lines in the hydrogen spectrum
occurring in the visible range of wavelengths, which is known as Balmer Series
formula.
This
experiment aims to witness the discreteness of the wavelength of the light
emitted by an atom, and to find the relation between the discrete values of the
wavelength.
2.
Theory
Balmer series formula for positions of lines in the hydrogen spectrum occurring in the visible range of wavelength is given below: ¡K
where
n is an integer greater than or equal to 3 and RH is the Rydberg
constant.
The generalized formula is ¡K
where
m is an integer greater than or equal to 1 and n must be greater than m.
Diffraction
of light by a grating:
Figure 2: Dffraction
of liight of wavelength l by a grating
Light
of a given wavelength £f which is scattered by the grating of spacing d will
interfere constructively when the path difference between scattered waved is
equal to an integral number of wavelengths. This occurs in the direction £c
given by: ¡K
3.
Procedure
Figure 3: Measuring
Hydrogen Spectrum & Calibrating Grating Constant
The
spectrometer was setup and the grating constant was calibrated. The hydrogen
lamp, convex lens and spectrometer were setup as shown in Figure 3. The convex
lens was positioned so that the light is well focused on the collimator slit.
The
nominal 300 lines/mm grating was used to measure the angles of diffraction for
the different orders, n = + 1, + 2, of the different lines of the
mercury lamp. The nominal 600 lines/mm grating was then used to measure the
angles of diffraction for the different orders, n = + 1, of the different
lines of the mercury lamp.
The
above steps were repeated with hydrogen lamp.
4.
Result and Analysis
Uncertainty
of the angle of diffraction £G£c = + 0.0083˚
Part I: For Mercury
Lamp
With
the nominal 300 lines/mm grating,
Consider
the values of the angle of diffraction for n = + 1,
From equation [2], putting n = 1, we have
d sin £c = £f
sin £c = (1/d)£f
Plotting
a graph of sin £c against £f, we have:
Figure 4: 300 lines grating (first order
spectrum) for mercury lamp
Hence,
Grating constant d = 3.33 x 10-6 m
Then consider the values of the angle of diffraction for n = + 2,
For Blue colour, d = 2£f / sin£c = 3.37 x 10-6 m
For
Green colour, d = 2£f / sin£c = 3.56 x 10-6 m
Therefore,
average value of d = 3.42 x 10-6 m
With the nominal 600 lines/mm grating,
Consider
the values of the angle of diffraction,
From equation [2], putting n = 1, we have
d sin £c = £f
sin
£c = (1/d) £f
Plotting
a graph of sin £c against £f, we have:
Figure 4: 600 lines grating (first order
spectrum) for mercury lamp
Hence,
Grating constant d = 1.43 x 10-6 m
Part II: For Hydrogen Lamp
With the nominal 300 lines/mm grating,
Consider the values of the angle of diffraction for n = + 1,
Consider the values of the angle of diffraction for n = + 2,
Hence, average wavelength of blue light = 415.4 nm
average wavelength of green light = 486.2 nm
average wavelength of red light = 679.3 nm
By equation [1],
Hence,
average RH = 1.101 x 10-7 m-1
With the nominal 600 lines/mm grating,
By equation [1],
Hence,
average RH = 1.311 x 10-7 m-1
5.
Error Analysis
For mercury lamp,
Using Excel,
In 300 lines grating (first order):
£G d / d = £G slope / slope = 0.0000211 / 0.000326
Hence £G d = 0.216 x 10-6 m
In 600 lines grating (first order):
£G d / d = £G slope / slope = 0.0000342 / 0.000695
Hence
£G d = 0.07 x 10-6 m
For mercury lamp,
In 300 grating (first order):
Uncertainty of l of blue light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 402.9 x 10-9[(0.216 x 10-9/ 3.42 x 10-6)2 + (0.0083 / 6.97)2]1/2
= 0.48 nm
The % error of the wavelength of the blue light
= [(435.8 ¡V 402.9) / 435.8] x 100%
= 7.55%
Uncertainty of l of green light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 492.8 x 10-9[(0.216 x 10-9/ 3.42 x 10-6)2 + (0.0083 / 8.5)2]1/2
= 0.48 nm
The % error of the wavelength of the green light
= (546.1 ¡V 492.8) / 546.1] x 100%
= 9.76%
Uncertainty of l of red light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 679.3 x 10-9[(0.216 x 10-9/ 3.42 x 10-6)2 + (0.0083 / 11.77)2]1/2
= 0.51 nm
The % error of the wavelength of the red light
= [(679.3 - 656) / 656] x 100%
= 3.55%
In 300 lines grating (second order):
Uncertainty of l of blue light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 427.9 x 10-9[(0.216 x 10-9/ 3.42 x 10-6)2 + (0.0083 / 14.92)2]1/2
= 0.24 nm
The % error of the wavelength of the blue light
= [(435.8 - 427.9) / 435.8] x 100%
= 1.81%
Uncertainty of l of green light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 479.5x10-9[(0.216 x 10-9/ 3.42 x 10-6)2 + (0.0083 / 16.77)2]1/2
= 0.26 nm
The % error of the wavelength of the green light
= [(546.1 - 479.5) / 546.1] x 100%
= 12.2%
For RH, % difference = 0.38%
In
600 lines grating (first order):
Uncertainty of l of blue light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 363.2 x 10-9[(0.216 x 10-9/ 1.43 x 10-6)2 + (0.0083 / 14.71)2]1/2
= 0.212 nm
The % error of the wavelength of the blue light
= [(435.8 ¡V 363.2) / 435.8] x 100%
= 16.66%
Uncertainty of l of green light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 406.1 x 10-9[(0.216 x 10-9/ 1.43 x 10-6)2 + (0.0083 / 16.5)2]1/2
= 0.214 nm
The % error of the wavelength of the green light
= [(546.1-406.1 / 546.1) x 100%
= 25.64%
Uncertainty of l of red light
= l [(Dd/d)2 + (£G£c/£c)2]1/2
= 550.6 x 10-9[(0.216 x 10-9/ 1.43 x 10-6)2 + (0.0083 / 22.66)2]1/2
= 0.218 nm
The % error of the wavelength of the red light
= [(656 ¡V 550.6) / 656] x 100%
=
16.07%
For
RH, % difference = 19.53%
6.
Discussion
1)
From the above analysis, we can find that the experimentally measured for
the hydrogen lamp agree with the known values to within experimental
uncertainty.
2)
The Balmer series formula explains the experimentally measured
wavelength. The radiations identified as the Balmer series correspond to
transitions from higher levels to the n = 2 level.
3)
The known value of RH is 1.0968 x 10-7 m-1.
The experimentally determined value is close to the known value. The accuracy
can be increased by recording more sets of data.
4)
600 lines gratings would give a better result. This is because by using
600 lines gratings, the angular width is smaller and hence more easily to
observer the discrete colour light.
7. Error Sources
Some of the errors include:
a) Errors in measuring the angular width of the line spectrum: some lines are relatively think and blurred to be observed.
b) Dispersion of light from the sources
c)
Approximation in the calculation
8. Suggestion and improvement
Precaution:
a)
In order to prolong the lifetime of
the hydrogen lamp, the lamp should be operated intermittently.
b)
After calibrating
the telescope, prevent any change to the focus of the telescope doing the
experiment.
Improvement:
a) Do more set of data to minimize the uncertainty of the data.
b)
Make sure that the discrete light spectrum is on the reference light
before recording.
9.
Conclusion
From the experiment, we can see that the emission of light by atoms occurs only at discrete wavelengths. Also, the Rydberg constant is found to be 1.101 x 10-7 m-1 (for 300 lines grating) and 1.311 x 10-7 m-1 (for 600 lines grating).
¡@