Spectroscopic measurement of Rydberg’s constant for Hydrogen
H. Potter, and E. Kager
(Completed 11 September 2005)
Rydberg’s Constant for Hydrogen was determined to be about 11,214,870m-1,
resulting in only about a 2.25% discrepancy with the presently accepted value of
RH1 using a spectrometer and a diffraction grating.
I. Introduction
Long before Bohr developed his orbital model of the atom and derived the general
equation
1/λ = RH(1/(nf2)-1/(ni2))
(1)
relating the wavelength of the light emitted by an excited atom to its orbital transition,
Balmer had already determined that a similar equation,
1/λ = k((1/4)-1/(m2)),
(2)
where m is an integer and k is a constant, accurately described the spectroscopic lines
when visible light was emitted. Balmer was able to come up with his formula empirically
due to the abundance of data available for visible emissions.
The scattering from a set of Bragg planes establishes the Bragg condition for
constructive interference, referred to in the context of this experiment as the grating
equation, that
nλ = 2dsin(θ),
(3)
where n is the order of the observed diffraction, λ is the wavelength of the incident light,
d is the spacing of the diffraction grating, and θ is the angle through which the light is
diffracted.
II. Experiment
The ease of measurement of diffractions in the visible spectrum due to the ability
of the human eye to observe them was utilized in this experiment. The visible spectra
emitted by excited sodium and hydrogen atoms were observed using a spectrometer in
order to determine the diffraction angles of each atom’s various emissions in both the
first and second order through a diffraction grating.
For the sodium, the presently accepted values for the wavelengths of the sodium
emissions2 were taken as known quantities. This enabled the calculation of the spacing in
the diffraction grating using the grating equation (3). The mean of the calculated values
for the spacing of the diffraction grating was taken to be the spacing of the diffraction
grating for the remainder of the experiment.
For the hydrogen, the wavelengths of the various spectral lines were calculated
for each measurement using the grating equation (3). Four spectral lines were observed.
The diffraction angles for the red, turquoise, and light purple spectral lines were observed
and recorded in both the first and second order as they bent both to the left and right side
of the spectrometer. The diffraction angles for the dark purple line, however, could only
be recorded in the first order as it bent to the left and right side of the spectrometer.
Using Bohr’s orbital transition model for atomic emissions each observed value
of λ could reasonably be paired with its orbital transition. The orbital transitions for each
spectral line were always to the second orbital because if they were not they would not
have been observable in the visible spectrum. The orbital from which this transition was
made was different for each spectral line, increasing incrementally from 3 to 6 as the
color changed from red to dark purple. This was justified because a more energetic
emission requires a larger orbital transition, and light has more energy at shorter
wavelengths. This pairing led to14 data points of the form ((1/(nf2)-1/(ni2)), 1/λ).
Comparing this plot to equation (1), it is easy to see that, since equation (1) is linear with
regards to the quantities being plotted, a linear regression on these data points should
yield a line with slope RH. The value of RH thus obtained was taken to be the
experimentally calculated value of RH.
III. Results
Sodium Data:
n = Order of Line
θ
n
1
2
1
2
20.9
46.0
21.1
45.9
Accepted λ (m) 2
5.896E-07
5.896E-07
5.890E-07
5.890E-07
Mean:
θ = Measured Angle
Grating Equation:
2dsin(θ) = nλ
d (m)
Uncertainty in d
8.26E-07
1.35E-09
8.20E-07
9.94E-10
8.18E-07
1.33E-09
8.20E-07
9.96E-10
8.21E-07
1.17E-09
Table 1: Data obtained for spectral lines of sodium that was used to calculate the spacing
of the diffraction grating. The uncertainty in d was calculated using error propagation
analysis by assuming the uncertainty in all quantities but the angle measurement were
negligible.
2
Hydrogen Data:
Left = Light Shield
Right = No Light Shield
Color
Dark Purple
Dark Purple
Light Purple
Light Purple
Light Purple
Light Purple
Turquoise
Turquoise
Turquoise
Turquoise
Red
Red
Red
Red
Side
Left
Right
Left
Right
Left
Right
Left
Right
Left
Right
Left
Right
Left
Right
n
1
1
1
1
2
2
1
1
2
2
1
1
2
2
Balmer's Formula:
1/λ = RH(1/(nf2) - 1/(ni2))
Use Mean d From Above
λ = wavelength
θ
14.3
14.3
15.2
15.0
31.7
31.5
17.0
17.0
36.4
35.9
23.5
23.2
53.0
52.5
"x"
"y"
-1
n f n i 1/(n f 2 ) - 1/(n i 2 ) 1/λ (m ) Uncertainty in λ Uncertainty in "y"
λ (m)
4.06E-07 2 6
0.2222 2465474
3.35E-09
2.04E+04
4.06E-07 2 6
0.2222 2465474
3.35E-09
2.04E+04
4.31E-07 2 5
0.2100 2322635
3.38E-09
1.82E+04
4.25E-07 2 5
0.2100 2352878
3.37E-09
1.87E+04
4.31E-07 2 5
0.2100 2317802
1.83E-09
9.85E+03
4.29E-07 2 5
0.2100 2330991
1.83E-09
9.95E+03
4.80E-07 2 4
0.1875 2082861
3.42E-09
1.49E+04
4.80E-07 2 4
0.1875 2082861
3.42E-09
1.49E+04
4.87E-07 2 4
0.1875 2052411
1.85E-09
7.78E+03
4.81E-07 2 4
0.1875 2077075
1.85E-09
7.96E+03
6.55E-07 2 3
0.1389 1527200
3.56E-09
8.30E+03
6.47E-07 2 3
0.1389 1545836
3.55E-09
8.49E+03
6.56E-07 2 3
0.1389 1525025
1.79E-09
4.17E+03
6.51E-07 2 3
0.1389 1535179
1.80E-09
4.24E+03
Table 2: Data obtained for spectral lines of hydrogen that was used to calculate λ values
and to create a linear plot of Balmer’s Formula (1) with the experimental data. The
uncertainties in both λ and in “y” were calculated using error propagation.
IV. Analysis and Discussion
Linear Plot
1/Lambda m
-1
2.5E+06
2.3E+06
y = 11,214,870x - 23,269
R2 = 0.999
2.1E+06
1.9E+06
1.7E+06
1.5E+06
0.13
0.18
0.23
1/4 - 1/(n*n)
Figure 1: Linear plot of the hydrogen data according to Balmer’s Formula (1). The slope
of the line obtained via linear regression is the experimentally obtained value of RH.
Using a least-squares linear regression on the 14 data points, the linear fit shown
in Figure 1 was obtained. The slope of the best-fit line is 11,214,870m-1, compared with
the accepted value1 for RH of 10,967,760m-1, an absolute discrepancy of 247,110m-1, but
a percentage discrepancy of only 2.25%. The linear fit was also fantastic, with r2 = .9990
and correlation coefficient r = .9995.
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The linear fit, however, has a nonzero additive constant of -23,269m-1.
Fortunately, in analogy with the absolute discrepancy, this is small relative to the
enormity of the slope, being only about 0.2% as large.
The uncertainty in the measurement of diffraction angles was only .1˚; however,
since the measurement of the wavelength of the light in each case was dependent upon
both the diffraction grating spacing and the diffraction angle, and the measurement of the
diffraction grating spacing was also dependent upon the angle of diffraction, the
uncertainty in the wavelength of the light in each case was doubly dependent upon the
uncertainty in the angle. Such error propagation analysis enabled error bars to be
included on the linear plot. This allowed a maximum and minimum value of RH to be
estimated by calculating a maximum and minimum reasonable slope of the linear fit.
Taking points (5/36, 1500000) and (2/9, 2500000), and (5/36, 1600000) and (2/9,
2400000) leads to RHmax = 12,000,000m-1 and RHmin = 9,600,000m-1.
V. Conclusion
The calculated value for RH was within the specified experimental uncertainty of
the accepted value1; however, several avenues are available for improving the precision
of this experiment. One source of uncertainty was the measurement of each angle when
reading the Vernier scale on the spectrometer. The telescope of the interferometer tended
to interfere with a precisely vertical reading of the Vernier scale, and the eye strain
resulting from finding and centering the spectral lines in the eyepiece compounded the
difficulty. A finely calibrated digital display would alleviate these problems. A method
for focusing the spectrometer automatically and with less uncertainty would also make
the determination of the center of each of the spectral lines in the eyepiece more precise.
Conducting the experiment in a pitch black dark room would also ease the process of
finding and centering spectral lines, possibly enabling the detection of the second order
dark purple line, though data collection would be impeded in such a situation.
1
Tipler, Paul A. and Ralph A. Llewellyn, Modern Physics, 3rd ed. (W. H. Freeman and
Company, New York 2003), p.164.
2
CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and
Physical Data, edited by Robert C. Weast (CRC Press, Cleveland, OH, 1975), p.E-213.
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