Finding the Rydberg Constant From the Hα Wavelength From
The Hydrogen Spectrum
G. S. Sagoo
University College London
10th December 2008
Abstract: The Rydberg Constant was obtained by diffracting light emitted by a hydrogen
lamp and obtaining the wavelength of the Hα line. The obtained value of
1.100 × 10 7 m −1 ± 0.000480 × 10 7 m −1 is just outside the range on the accepted value of
1.097 × 107 m −1 [1]. This suggests that there were some errors in the experimental
procedures.
Introduction
The aim of the experiment was to find the red component of the hydrogen spectrum from
the energy level transition of an electron from quantum number 3 to quantum number 2;
leading to the determination of the Rydberg Constant.
Electrons in atoms are bound to certain energy
states called “quantum” numbers (n). The electrons
are usually free to move to and from each energy
state providing there is enough energy for the
electron to be promoted to a higher quantum
number. Upon the electron receiving enough
packets of energy it is promoted to a higher energy
level. Upon losing energy the electron drops to a
lower quantum number, emitting a photon. In the
experiment, for the emission of the red line, the
electron in hydrogen is promoted to n=3 from n=2
energy states, which is part of the Balmer series as
shown in Figure 1 on the left.
Figure 1: Energy level diagram
for Hydrogen showing the
Balmer series [2]
The Rydberg constant, R, relates the reciprocal of
the wavelength (λ) of the photon emitted when an
electron drops an energy level, so for the quantum number n=3 to n=2 transition:
1⎞
⎛ 1
= R⎜ 2 − 2 ⎟
λ
3 ⎠
⎝2
1
(1)
By obtaining the wavelength for the n=3 to n=2 transition R can be determined. When
light emitted from the hydrogen it is not monochromatic, thus the light needs to be “split”
so that the red component wavelength can be found. This can be done by using a
diffraction grating. Diffraction gratings work by letting light through the gaps, which are
a multiple of the wavelength of the light passing through it as shown by equation (2).
1
sin θ p = pλ
N
(2)
Where N= number of gratings per metre, θ p = diffraction angle and p is a wavelength
multiplier.
Method
Figure 2 shows a schematic diagram of the apparatus used in the setup. The diffraction
grating used had 300 lines mm-1, which is about the wavelength of the Hα line. To
calibrate the apparatus, the telescope was focused on a distant object to ensure that it was
focusing properly and then crosshair centered in relation to the light emitted by the
hydrogen lamp through the slit. The angle from the vernier scale was recorded. It doesn’t
matter what the angle is in relation to because the diffraction angle will be the difference
in this angle and angle through which a Hα line is seen. The telescope was then rotated
through 90° and a mirror was placed instead of the grating. The grating turntable was
then rotated until light was viewable through the telescope and the crosshair centered on
beam of light. This is so that when the telescope is parallel the beam of light emitted by
the lamp, the grating will be orthogonal to the light. The angle of the turntable was
recorded and the grating turntable was rotated back 45° and the telescope back 90°, the
equipment is now calibrated. The telescope was then rotated so that a red beam of light
was visible, this is the 1st order of diffraction and the difference between the starting
angle and the rotation angle was recorded. This was repeated until 3 diffraction orders
were recorded in the positive and negative direction.
Figure 2: Schematic diagram showing setup of apparatus [2]
Results and Analysis
From equations (1) and (2) it can be seen that:
pN
⎛ 1 1⎞
= R⎜ 2 − 2 ⎟
sin θ p
⎝2 3 ⎠
(3)
Figure 3 shows sin θp vs order of diffraction from the results collected. By performing
linear fits on the data using MATLAB, the gradient and y-intercept was found with their
associated errors, which are 0.1963± 0.0000186 and -0.001276± 0.00003847
respectively. Hence:
λ=
m
n
where m= gradient
(4)
Figure 3: Graph showing Sin θp vs order of diffraction from results collected
Since the recorded values for θp had errors associated with them, there is an uncertainty in
the calculated wavelength. This is related by:
∂ ( sin θ p )
(5)
Δθ p = cos θ p Δθ p
∂θ
However the calculated error in wavelength was done differently. The individual error on
the wavelength was done by:
Δ sin θ p =
Δλ =
1
cos θ p Δθ
pN
(5)
Then the highest value of the individual errors was used as the overall error because it
was felt that it contributed most towards the overall error. The same techniques were used
to calculate the error on the Rydberg constant:
ΔR =
36
Δλ
5λ 2
From equation (3) and (6) the experimental value obtained was:
R = 1.100 × 107 m −1 ± 0.000480 × 107 m −1
(6)
(7)
The accepted value for the Rydberg constant is 1.097 × 107 m −1 [1]. The value measured
is just outside the error set, and our value differs by ≈ 0.003 × 107 m −1 or ≈ 0.3%
suggesting that errors were minimized and that uncertainties were taken into
consideration.
However looking Figure 3, the y-intercept was not zero, as expected. This could have
been because of errors not taken into consideration such us:
•
•
•
The vernier scale could not be calibrated properly thus giving systematic errors. This
may be the cause of the y-intercept not being zero because sin θp was not zero,
ultimately giving error in θp.
The grating may not be what the manufacturer states of 300mm-1. This will mean the
wrong wavelengths will be recorded therefore causing an inaccurate measurement of
the Rydberg constant. This error could be taken into consideration by asking the
manufacturer of the grating for a mean percentage error of the number of lines per
grating.
The light beam through the telescope may not be parallel, ultimately creating an
inability to focus on a parallel beam of light, thus giving incorrect angles. This could
be solved by firstly manually checking each telescope, but this is time consuming.
Conclusion
In conclusion, the Rydberg constant measured by spectroscopy was found to be
R = 1.100( ±0.000480) × 107 m −1 which differs from the accepted value of 1.097 × 107 m −1
by 0.003 × 107 m −1 ; indicating that errors were minimized but some systematic errors did
remain such as the vernier scale not being calibrated properly, the blazed grating not
being accurate, and the light beam not being parallel through the telescope.
References
1. Kay and Laby. Tables of Physical and Chemical Constants.
2. UCL, Department Physics of Astronomy. Atomic Physics and Quantum Phenomena.