Physics 133

Atomic Spectra Lab

Austin Provansal

Lab Partners
Nicolas Blanc
Edward Conley

Abstract
The goal of this lab was to calculate a value for the Rydberg constant. We attempted this by
using a two lamps, with helium and hydrogen respectively, a spectrometer, and a grating to disperse the
light coming from the lamps. Our method was to use the helium lamp as a standard with known
wavelengths in order to determine a value for the spacing of our diffraction grating (3163.127768 nm).
Then, we would use this value to take the various wavelength calculations for the hydrogen lamp,
which would finally give us our Rydberg constant ( R H =14247751.74±4.220626006 ).
 Introduction
In this lab, we set out to calculate the value of the Rydberg constant. To measure this we used a
spectrometer to both view the dispersed light given off by a helium lamp and hydrogen lamp and
measure where along the spectrum where the characteristic dark lines that appear along the spectrum.
Where these dark lines appear along the spectrum are unique to an excited element. By measuring the
angles where these lines appear (fig 1), we can determine the Rydberg constant with the below
equations.
1 1 1
=R H ( 2 − 2 ) Eq 1.
λ n2 n1
R H is our Rydberg constant. The values n 1 and n 2 are the principle quantum numbers of the
initial and final energy states of the element that are predetermined based on the color that is given off.
The value of our wavelengths were found with Eq 2
mλ
dsin θ= Eq 2.
cos Δ

Where m is the order of the line, and λ is the wavelength where it is found. The values for d,
θ and Δ given by Eq 3, Eq 4 and found are derived from measured values.

1
θ= (θ 1−θ 2) Eq 3.
2

1
Δ= (θ 1−θ2 ) Eq 4.
2

The two diagrams below show the geometry that give us the value for our important angles and
how we measured them.
 Fig 1: Geometry for how the spectral lines are measured
using a spectrometer
This was done using both helium and hydrogen lamps and a standard spectrometer.
 Apparatus
Spectrometer: (Fig 2)The device that allows us to view the dispersed light while simultaneously
measuring the angle where the spectral lines appear. The right side is the where the light
viewed, the center holds the diffraction grating, and the left side is placed directly in
front of the helium or hydrogen lamp. Below the diffraction grating in the center, is a
vernier scale which measures the difference in degrees and minutes between the center
left view point and the center of the grating.

Fig 2: Similar Spectrometer to the one we used. Image Courtesy of

http://www.abbomedindia.com/spectrometer.htm
Procedure
The first step in our experiment was to determine the spacing of our diffraction grating using a
helium lamp. In order to measure and θ2 from Eq 3 using the vernier scale we first measure the
“zero” angle, α 0 , that the scale reads when the eye piece is pointing directly at the grating, and then
the angle that it reads when the eye piece is lined up with the desired spectral line, α1 or α 2 . Then,
θ1 and θ 2 are determined below.
θ1=∣(α 1−α 0)∣ Eq. 5

θ2 =∣(α2 +α0 )∣ Eq. 6

With these values, the known values of wavelengths and the orders, we can use Eq. 2 to find our
value of d, which is the spacing of our diffraction grating.
To determine the error on d, we took a weighted sum, which is given by Eq. 7.
σd=
∑ ω d summing over d Eq 7.
∑ω
Where ω is the weight determined by Eq 2.
Next, to find the value of the Rydberg we switched to a hydrogen lamp and continued to
measure the angle of each line and using the above equations to determine our wavelength, λ ,
instead of d. After calculating this, we can use this value along with the quantum numbers of hydrogen
to find our Rydberg constant with Eq 1. All of the errors are found using the same method above.
 Data and Analysis

Spectral Lines for

Helium
Literature Spectral
LINE Line Observed ? Color Intensity
Lines (nm)
1 438.793 No violet weak
2 443.755 Yes violet weak
3 447.148 Yes violet-blue strong
4 471.314 Yes blue medium
5 492.193 Yes teal medium
6 501.567 Yes yellow strong
7 504.774 No yellow weak
8 587.562 Yes red strong
9 667.815 Yes red medium

Table 1: Summary of Helium Lines

Fig 3: Example Helium Spectra. Image Courtesy o f

http://hyperphysics.phy-
astr.gsu.edu/hbase/hyde.html#c4

According to Table 1, our findings matched the example we found very closely. The only two
lines that we did not observe, are the more difficult ones to observe in the image above. However, these
two missing lines take away two measurements that would have otherwise given us a larger pool of
angles. If we could have found these lines, it would have helped us get a more accurate measurement.
All angular measurements are in radians.

Line theta one theta two

1 N/A N/A
2 N/A N/A
3 0.142244334 0.1332267996
4 0.1480620982 0.1413716694
5 0.1509709803 0.1483529864
6 0.1786053601 0.1733693724
7 N/A N/A
8 0.2004219758 0.1989675347
9 0.2135119452 0.2103121749
Line Theta Delta
1 N/A N/A
2 N/A N/A
3 0.1377355668 0.004508767234
4 0.1447168838 0.0033452144
5 0.1496619834 0.001308996939
6 0.1759873662 0.002617993878
7 N/A N/A
8 0.1996947552 0.0007272205217
9 0.21191206 0.001599885148
Table 2: Angle Measurements of each Helium Line.

After our calculations we determined the value for our grating was:
d =3163.127768±9.372380694 nm

Our grating is labeled to have a spacing of 300 lines/mm or 3300 nm/line which is on the same
order of magnitude as our calculated value is satisfactory to our level of accuracy.
Moving on to the hydrogen lamp,

Theta 1 Theta 2 Theta Delta

0.1323541349 0.1250819297 0.1287180323 0.003636102608
0.1468985454 0.1416625576 0.1442805515 0.002617993878
0.1995493111 0.1911135531 0.1953314321 0
Table 3: Angle Measurements for First Order Hydrogen Spectrum

note: the “0” value above should read “0.004217879026 “.

Theta 1 Theta 2 Theta Delta
0.2687807048 0.2617993878 0.2652900463 0.003490658504
0.2932153143 0.2929244261 0.2930698702 0.0001454441043
0.3982259577 0.402880169 0.4005530633 -0.002327105669

Table 4: Angle Measurement for Second Order Hydrogen Spectrum

For the hydrogen lamp, we were able to confidently measure the second order lines, giving us a
larger sample size and therefore a more accurate final calculation.

Line Lambda (nm) Error On Lambda

1 406.0282394 1.825008205
2 454.7961059 1.821111218
3 613.9368118 1.805236992
Table 5: Wavelength and Errors for First Order Hydrogen Spectrum

Error on
Line Lambda (nm)
Lambda
1 414.6689661 0.8879270953
2 456.9020687 0.8808836359
3 616.6955936 0.8472846392
Table 6: Wavelength and Errors for Second Order Hydrogen Spectrum
Comparing our values to other accepted values below (http://hyperphysics.phy-
astr.gsu.edu/hbase/hyde.html#c4). These values are higher than our measured values, however, the
differences between them are much closer, making them reliable enough to keep.

Line 1 = 434.037 nm
Line 2 = 486.133 nm
Line 3 = 656.272 nm
 Line RH (1/m)
m=1 1 17732756.74
2 11726866.75
3 7756343.438
m=2 1 17363247.77
2 11672815.03
3 7721645.511
Table 7: Rydberg Calculations for First and Second Orders of Hydrogen

Performing another weighted sum of the Rydberg constant, we obtained:

Rydberg Value = 14247751.74 m−1
Total Error on Rydberg = 4220626.006 m−1

The accepted value for the Rydberg constant is 10967758 m−1 . Which, after error is included
is close to our calculated value. For the level of accuracy of our experiment both the calculated value
and the error seem valid.

Conclusion
In this lab, we attempted to calculate the Rydberg constant. This was to be done experimentally
though the measurement of the diffraction grating used and the spectral lines of helium and hydrogen
lamps. These values were found with acceptable levels of accuracy and and errors were most likely
caused by our inability to observe some lines in the initial helium measurement, potentially propagating
error throughout the rest of our data. Our final value for the Rydberg value was also accurate enough
for our level of accuracy, matching within the order of magnitude of its accepted value.