verification-balmers-formula

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Verification of the Balmer’s Formula and measuring the
Rydberg Constant for Helium and Deuterium
Jaythan Edrick S. Salazar,* Gelli Mae P. Gloria, and Ma. Janelle G. Manuel
National Institute of Physics, University of the Philippines, Diliman, Quezon City, Philippines
*
Corresponding author: [email protected]
Abstract
The Rydberg-Ritz formula and Balmer series describe the relationship of light emitted or absorbed
by electrons of hydrogen-like atoms and their energy levels. This experiment aims to obtain the
emission spectrum of hydrogen, deuterium, and helium and evaluate their applicability to the Balmer
series by observing the constants. Using a student spectrometer and diffraction grating, the angular
displacement of emission lines from the three discharge lamps were recorded. For hydrogen and
deuterium, the calculated Rydberg constants had minimal deviations from the theoretical value. For
helium, the calculated Rydberg constant deviated from the theoretical by 74.69% and 75.94% for the
first order and second order diffraction, respectively. The deviation found in Hydrogen and Deuterium
can be caused by interference from external light sources while the deviation found in Helium can be
attributed to its non-applicability to the Balmer series and Rydberg-Ritz formula.
Keywords: Balmer, Hydrogen, Deuteurium, Helium, Rydberg, emission spectrum
1
Introduction
Deuterium is an isotope of hydrogen, with a neutron and a proton bound together in its nucleus and is
known as ”heavy hydrogen”. Both Hydrogen and Deuterium have one valence electron. Actually, singly
ionized atoms are all known as ”Hydrogen-like”, with Deuterium as an example. Helium has 2 protons
and 2 neutrons in its nucleus, and has 2 valence electrons. As a noble gas, it is difficult to get it to share
its electrons, even though a singly ionized He+ is Hydrogen-like. [1]
The Rydberg formula is used to calculate the wavelength of energy emitted or absorbed when an
electron in a hydrogen-like atom transitions from one quantum state to the other.It is given by
1
1
1
2
= RZ
− 2
λ
n22
ni
(1)
where R is the Rydberg constant, Z is the atomic number of the hydrogen-like atom, n2 is the final
quantum state, and ni is the excited state levels of the atom.
The Balmer series is a set of spectral emission lines for Hydrogen that are a result of electron transitions
from higher levels (ni =3,4,5,6) to low energy levels (n2 = 2) [Figure 1]. The formulation is a special case
of the Rydberg formula.
1
1
1
=R 2 − 2
(2)
λ
n2
ni
The experiment was conducted to obtain the emission spectra of deuterium, helium, and hydrogen
discharge source. From these spectra, the applicability of the spectrum of each gas to the Balmer series
of Hydrogen and the Balmer’s equation were compared and evaluated by observing their experimental
Rydberg constant.
2
Methodology
2.1 Materials
A student spectrometer was the main apparatus used in the experiment, and a diffraction grating (300
lines/mm) was used. A magnifying glass was used to read the measurements on the Vernier scale of
the spectrometer. 2 lamps, a desk lamp for illumination while reading measurements, and a discharge
lamp as the main light source of the experiment, were used. Different discharge lamps were used, namely
Hydrogen, Deuterium, and Helium [4].
1
(a)
(b)
Figure 1: (a)Different spectral series of a hydrogen atom [2] and(b) Emission spectra of different gases[3]
2.2 Procedure
After calibrating the student spectrometer using the first red line and making sure that the diffraction
grating is perpendicular to the optical axis (absolute difference should be less than 10’), the angular
displacements of each visible bright line in the first order and second order for all discharge lamps were
measured and recorded. In the calculation, the direct image angle was subtracted. The wavelength from
the angular displacements was calculated for every 1st and 2nd order color [4].
3
Results and Discussion
Using a diffraction grating of grating constant equal to 300 lines/mm, the student spectrometer was first
aligned. Using the Hydrogen discharge lamp, the direct image and the visible first order red were used
as a basis for the alignment. The angular displacements recorded is shown by Table 1.
Table 1: Angular displacement of a hydrogen emission line separated using grating
Trial
1
Direct Image
θ0,cw
θ0,ccw
176.5o 356.5o
Color line
θcw
θccw
165o
8o
∆θcw
∆θccw
11o 30’
11o 30’
Absolute
Difference
0’
The determination of the experimental Rydberg constants of the different gases were done by locating
the first- and second-order visible spectral lines. Tables 2, 3, and 4 show the different measurement for
Hydroen, Deuterium, and Helium, respectively.
Table 2: Balmer series for the 1st and 2nd order diffraction of Hydrogen
ni
3 (Red)
4 (Blue-green)
5 (Blue-Violet)
∆θ
11o 30’
8o 15’
7o 23’
First Order
−1
λexp (nm) λexp
(mm−1 )
645.5
1549
478.3
2091
428.4
2335
(a)
(ni )−2
0.1111
0.0625
0.0400
∆θ
23o
16o 44’
15o
Second Order
−1
λexp (nm) λ−1
)
exp (mm
651.2
1536
479.9
2084
431.4
2318
(ni )−2
0.1111
0.0625
0.0400
(b)
Figure 2: (a) First and (b) Second-order spectra of Hydrogen
From the data shown in Table 2, the Rydberg constant for Hydrogen was determined as the slope
2
from a graph of λ−1 vs n2i of the first and second-order diffraction.
Based on the slope of the graphs from Figure 2, the experimental R for the first and second-order were
calculated to be 1.1061 x 107 m− 1 (relative deviation: 0.85 %) and 1.1050 x 107 m− 1 (relative deviation
of 0.75 %), respectively
Hydrogen has a proton number, Z, equal to 1. The experimental n2 for both orders were also calculated
and were found to be both equal to 2, with no deviation from the theoretical. Only three out of four
emission lines were obtained for Hydrogen.
Moreover, only three emission lines were seen from the two orders of diffraction of Hydrogen.
Table 3: First- and second-order-diffraction series of Deuterium
ni
3 (Red)
4 (Blue-green)
5 (Blue-Violet)
∆θ
11o 13’
8o 19’
7o 30’
First Order
−1
λexp (nm) λexp
(mm−1 )
648.4
1542
482.1
2074
435.1
2298
(ni )−2
0.1111
0.0625
0.0400
(a)
∆θ
22o 54’
16o 44’
14o 50’
Second Order
−1
λexp (nm) λ−1
)
exp (mm
648.5
1542
479.9
2084
426.7
2344
(ni )−2
0.1111
0.0625
0.0400
(b)
Figure 3: (a) First and (b) Second-order spectra of Deuterium
The experimental R for the first and second-order were calculated to be 1.0682 x 107 m− 1, with a
relative deviation of 2.61 %, and 1.1255 x 107 m− 1 (relative deviation of 2.62 %), respectively.
Since Deuterium is an isotope of Hydrogen, its proton number, Z, is also 1. The experimental n2
for both orders were also calculated. 1.98, with a relative deviation of 1.00 %, was obtained from the
first-order emission spectra while n2 =2 with no deviation from the theoretical was obtained for the
second-order spectral lines. Only three out of four emission lines were obtained for Deuterium.
Table 4: First- and second-order-diffraction series of Helium
ni
3 (Red)
4 (Blue-green)
5 (Blue-Violet)
∆θ
11o 30’
8o 30’
7o 30’
First Order
−1
(mm−1 )
λexp (nm) λexp
664.6
1505
492.7
2030
435.1
2298
(a)
(ni )−2
0.1111
0.0625
0.0400
∆θ
23o 24’
17o 20’
15o 22’
Second Order
−1
)
λexp (nm) λ−1
exp (mm
661.9
1511
496.6
2014
441.7
2264
(ni )−2
0.1111
0.0625
0.0400
(b)
Figure 4: (a) First and (b) Second-order spectra of Helium
The experimental R for the first and second-order were calculated to be 2.7755 x 106 m− 1 (relative
deviation: 74.69%) and 2.639 x 106 m− 1 (relative deviation of 75.94 %), respectively.
The experimental n2 was also calculated for helium. For this case, Z=2. The values obtained were
2.01 with relative deviation of 0.50% and 1.98 with relative deviation of 1.00% for the first and second
order, respectively.
3
The same trend was also made in the emission spectra for Helium where only 3 emitted wavelengths
were observed.
The deviation found in Hydrogen and Deuterium can be caused by interference from external light
sources. In addition, only three out of four emission lines were observed.
The deviation found in Helium can be attributed to its non-applicability to the Balmer series and
Rydberg-Ritz formula.
4
Conclusion
For hydrogen and deuterium, the calculated Rydberg constants had minimal deviations from the theoretical value. For helium, the calculated Rydberg constant deviated from the theoretical by 74.69% and
75.94% for the first order and second order diffraction, respectively.
From this, we can verify that the Balmer series is not applicable for non-Hydrogen like atoms like
Helium.
References
[1]
[2]
[3]
[4]
R. Harris, Modern Physics, 2nd Edition (University of California, Davis, 2008).
Langsam (Retrieved from http : //eilat.sci.brooklyn.cuny.edu/cis15 /OldHW s/HW 2dC .htm, nd).
M. Bolte (Retrieved from http : //astronomy.nmsu.edu/geas/lectures/lecture19/slide02.html, nd).
Physics 104.1 Laboratory Manual: Rydberg Constant (National Institute of Physics, University of the
Philippines Diliman, 2018).
4
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