Studying Hydrogen using a Spectrometer
Hyder Al Hassani - 0401822
Introduction
Aims
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To introduce an optical spectrometer as a tool for making precise measurements.
Measure the refractive indices of a prism for different wavelengths of light
Calculate Rydberg’s Constant using data extrapolated from a calibration graph
created by examining the emission spectra of helium, hydrogen and mercury.
The optical spectrometer is an instrument used to produce spectral lines from light emitted by
specific sources and examine their wavelengths and other properties.
Below is a diagram of an optical spectrometer from a side perspective
There are 3 essential parts of the optical spectrometer:
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The collimator – this part of the spectrometer makes sure the only light that passes
through the prism is made up of parallel beams. It consists of a tube with an slit at
one end and a lens system. Parallel light is created because the slit is at the focus of
the lens.
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The prism table – this is where the prism sits. It is horizontal to the plane that the light
from the collimator enters and is able to be rotated on a vertical axis. This feature is
especially useful in measuring the angle of minimum deviation. The spectrometer
used in this experiment also has a Vernier scale attached to the prism table which
allows measurement of rotation in respect to the collimator.
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The telescope – this is the part of the spectrometer through which the light emitted
from the source is viewed after is passes through the collimator and the prism. The
telescope can be rotated around the prism table so that it can be put into a position
where it receives the most amount of light from the prism. The eyepiece in the
telescope has cross hairs in it so that the telescope can be positioned correctly.
There is another set of Vernier scales attached to the telescope which again can be
used to measure the rotation.
These parts of the optical spectrometer can be seen again in the following diagram:
In this diagram you can see the two Vernier scales; the outside scale is the one attached to
the telescope and the only one used in this experiment. It is a main feature in making the
Optical spectrometer a very precise method of measurement.
‘A Vernier has two scales, an indicating scale and a data scale. These move past each other,
usually on a slide. When the measurement is taken, the zero point of the indicating scale lies
at the true datum of the measurement. This will usually be between two gradations of the data
scale and the indicator scale is used to ascertain where between the two it lies.
The Vernier's indicating scale has a series of gradations at a slightly different spacing to those
on the data scale. One of these will align exactly with a gradation on the data scale. The
indicator scale number at this aligned point will be the extra digit of the measurement.
In instruments using decimal measure the indicating scale would have nine gradations
covering the same length as the ten on the data scale. Only one will align with a mark on the
data scale and it is the number of this one which indicates the next decimal place. 1
1
Definition of a Vernier scale – Wikipedia : http://en.wikipedia.org/wiki/Vernier_scale
The first part (Part A) of the experiment was to calculate the refractive indices of a prism at
different wavelengths. To do so the following equation was used:
INSERT FORUMLA HERE
‘n’ is the refractive index to be calculated.
‘A’ is the prism angle.
‘d’ is the angle of minimum deviation. The angle of minimum deviation is defined2 as the corresponding angle
between the direction of the incident wave and that transmitted by the body
The methods for obtaining the prism angle and the angle of minimum deviation are explained
in the Method section of this report.
The second part (Part B) of the experiment was to calculate Rydberg’s constant from the
Hydrogen atomic spectrum.
When electrons have collided inelastically with atoms in a discharge, the atoms are left in an
excited state. When these atoms de-excite photons are emitted. The wavelength of the
emitted light from an atomic discharge is given by the Bohr model of the atom3:
INSERT FORMULA HERE
Rh is Rydberg’s constant.
‘nf’ and ‘ni’ are constants representing principal quantum numbers.
(lambda) is the wavelength of light
The wavelengths used are calculated using a calibration graph using results obtained from
examining the emission spectra of mercury and helium, and to a lesser degree, the hydrogen
emission spectrum. The details surrounding the creation of the calibration graph and the data
extrapolated from it are covered in the Method and Analysis sections of this report
respectively.
2
American Meteorology Society Glossary http://amsglossary.allenpress.com/glossary/search?id=minimum-deviation1
3
Paragraph extracted from page 32 of Physics 1X lab manual.
Method
The apparatus used in all the parts of the experiment was the same:
Optical Spectrometer
Prism
Helium, mercury, hydrogen and filament lamps
Magnifying glass (to help with reading the Vernier scales)
Before discussing the specific method used in calculating parts A and B of the experiment a
few preliminary procedures will be explained. In particular the method of measuring the angle
of minimum deviation is explained here because it is used very often in the experiment and it
would be tedious to repeat it every time it is relevant.
Focusing the telescope:
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Point the telescope at a fairly distant object.
Adjust the lens until the image through the telescope is clear and focused.
Focus the cross wires by adjusting the eyepiece of the telescope.
Measuring the straight through position:
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Remove the prism from the prism table
Place a lamp which will produce a line spectrum at the slit end of the collimator.
Read and record the telescope position on both of the Vernier Scales.
Measuring the angle of minimum deviation:
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Place the prism on the prism table with the ground face of the prism against the
support bracket.
Place a filament lamp at the slit end of the collimator
Set the spectrometer so that light from the collimator hits the prism is the way shown
the figure below. Then turn the telescope so that a continuous spectrum is visible
through it.
The angle between the direction the light first leaves the collimator and the direction
at which it is seen through the telescope is the angle of deviation.
Replace the filament lamp with a lamp that has line spectrum (for example: helium).
Observe the line spectrum and slowly rotate the prism table.
It will be noticed that the spectrum deviates. It will also be noticed that at a certain
point the deviation will stop and then as the prism table is rotated more the spectrum
starts to deviate in the other direction.
The point at which the deviation stops is the point of minimum deviation.
Record the position of the telescope from both of the Vernier scales.
The difference between this position and the straight through position is the angle of
minimum deviation.
Method specific to Part A
Measuring the prism angle
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Set up the collimator and prism in the position shown in the above diagram.
Place a filament lamp at the slit of the collimator
Rotate telescope one way until you see the spectrum reflected from one face of the
prism. Record the position of the telescope using both Vernier scales.
Rotate the telescope until the spectrum reflected from the other face of the prism is
visible. Record the telescope position.
The difference between these two positions is twice the prism angle. So divide this
angle in half to get the prism angle.
Method specific to Part B
Creating the calibration graph
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Measure the angles of minimum deviation for the main spectral lines of helium.
Using the appendices supplied, match the angles to wavelengths. And tabulate the
results.
Plot these results onto graph of minimum deviation against wavelength.
Repeat the first three steps this time for mercury.
Hydrogen Spectrum
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Place the hydrogen lamp and the slit of the collimator and adjust the slit until the
brightest image possible is obtained.
Measure the angles of deviation for the main spectral lines of hydrogen.
Record these for use in the Analysis.
Results
Straight through position: 1st Vernier - 285.75 degrees
2nd Vernier – 195.73 degrees
These results were used to calculate every minimum deviation in both parts of this
experiment.
Part A
Angle of minimum deviation for the red part of the helium spectrum – 47.7 degrees
Angle of minimum deviation for the violet part of the helium spectrum – 50.3 degrees
Value of prism angle – 59.85 degrees
Part B
Helium Lamp
Colour
Red
Yellow
Green
Green – Blue
Blue
Blue – Violet
Angle of minimum
deviation (degrees)
47.70
48.22
49.20
49.28
49.66
50.61
Wavelength (nm)
667.8
587.6
501.6
492.2
471.3
447.2
Mercury Lamp
Colour
Yellow
Green
Blue – Violet
Violet
Angle of minimum
deviation (degrees)
48.28
48.61
50.27
51.97
Wavelength (nm)
578.5
546.1
435.8
404.7
Hydrogen Lamp
Colour
Red
Green – Blue
Violet
Angle of minimum deviation (degrees)
47.99
48.39
50.15
Calibration Graph
Minimum deviation (degrees)
Minimum deviation against Wavelength
52.5
52
51.5
51
50.5
50
49.5
49
48.5
48
47.5
47
400
450
500
550
600
650
Wavelength (nm)
Minimum deviation against Wavelength
700
Analysis
Part A
Using the equation that relates refractive index, prism angle and angle of minimum deviation,
INSERT FORMULA HERE
the refractive indices of the prism for the red(d_r) and violet(d_v) light were calculated.
The refractive index does change a great deal with wavelength. Therefore the results were
calculated to three decimal places.
Using the above formula the following results were obtained:
n_violet = 1.646
n_red = 1.617
I would estimate an uncertainty of 0.0005
Part B
Using it’s equation,
INSERT FORMULA HERE
Rydberg’s constant was calculated using wavelengths of Hydrogen. But first these
wavelengths had to be obtained from the calibration graph.
Using a best fit line drawn on the graph the following wavelengths were matched to lines in
the Hydrogen spectrum:
Colour
Red
Green-Blue
Violet
Angle of Minimum
deviation (degrees)
47.99
48.39
50.15
Wavelength (nm)
614
568
443
In the above equation n_F and n_i are both constants. N_f is always 2 while for red light n_i is
3 and for green-blue it is 4.
So substituting the relevant values into the equation for Rydberg’s constant we get the
following values:
Using red light – 1.17*10^7 m^-1
Using violet light – 0.94*10^7 m^-1
Average – 1.05*10^7 m^-1
I would estimate an uncertainty of 0.03 * 10^7 in the results.
Group Questions
Group Values of R_H (*10^7)
1.31
1.10
1.19
1.24
1.14
1.11
1.08
1.07
1.16
1.17
0.94
1.04
1.04
1.09
1.27
1.10
The group decided that the top value was too high and was not omitted from any subsequent
calculations. The average of these results was calculated to be: 1.12*10^7 m^-1
We calculated the uncertainty using the following equation:
(Max-min)/No. Of values
Substituting the correct values in we found the uncertainty to be 0.024.Therefore, our final
result for Rydberg’s constant was 1.12 +_ 0.024*10^7 m^-1
We compared this result to the established accepted value of Rydberg’s constant:
1.097*10^7m^-1
The difference between the values is 0.023 and within the bounds of our uncertainty.
The group discussed the different areas in which there were possible sources of error. From
out discussion we decided that the area in which error is most prevalent is the best fit line on
the calibration graph.
The line is a curve therefore making it tricky to draw by hand, and even in the best
circumstances no human can be totally accurate with their drawing of the line. This thought is
backed up with the fact that people who used the same equipment and shared results got
different values for Rydberg’s constant. This is due to differences in the best fit line.
Conclusion
In this experiment the aim’s were to learn how to use the optical spectrometer and using this
knowledge to calculate the refractive indices of a prism and work out a value for Rydberg’s
constant.
As a group, I think Part B of the experiment worked out very well. When comparing the
group’s results to the established value of Rydberg’s constant we found it to be within our
uncertainty.
I would suggest this is due to these factors:
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The effectiveness of the experimental procedure – from a practical perspective this
experiment was relatively simple. All that was required from students was that they
place different lamps in one end and look down the other. The only difficult part was
the reading of the Vernier scales. But after taking a few results this difficulty reduces
dramatically.
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The spectrometers used by every member of the group were very similar which
meant that the comparison of the results was accurate. Also, each member of the
group used the same spectrometer for each part of their experiment. This reduced
errors in setting up the experiment.
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Controlling uncertainties :
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When recording telescope positions, both Vernier scales were read
and an average position was calculated. Also, a magnifying glass
was used when reading the scales as they are very small. This
allowed for a reduction in reading error.
The light entering the telescope was made sure to be well focused so
lining up crosshairs was easier.
The graph was made as accurate as possible by taking many results,
meaning many points. This allowed for the best fit line to be drawn as
accurately as possible.
While the experiment worked out very well I think that there could have been ways to improve
its effectiveness:
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A more accurate scale – the Vernier scales used were incredibly tiny, it was quite
difficult to see exactly which line matched up with which. If the scale was physically
larger it would make it easier to read, and more accurate.
More emission spectra examined – if more lamps with different emission spectra
were used then more points could have been recorded on the graph therefore making
it easier for the best fit line to be drawn.
A different method of creating the best fit line – as mentioned earlier, I believe the
drawing of the best fit line was the greatest source of error. If there was some method
by which it could be created more accurately this would greatly reduce error in this
experiment.