Mr Casey Ray McMahon, B.Sci (Hons), B.MechEng (Hons)
Version: 6th October, 2013
Page: 1 of 4
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Hydrogen spectral series limit equations
Abstract: Here, I present equations to predict the convergence points or limit points for
all the different spectral series of Hydrogen. Hence, one can determine where a particular
spectral series will converge without knowing anything about the spectral series in
question.
Theory:
From “Wikipedia (2013) Hydrogen spectral series”, we are presented with the Rydberg
equation which predicts the spectral lines for hydrogen. This equation is presented as
equation 1 below.
……….equation (1
Where: n’ = the series number, n = integer values greater than n’
For example, for the Lyman series, we set n’ = 1, and vary n as follows:
=λ (nm)
n
2
122
3
103
4
97.3
5
95.0
6
93.8
(convergence point)
91.2
For the Balmer series, we set n’ = 2, and vary n as follows:
n
3
4
5
6
7
(convergence point)
=λ (nm)
656
486
434
410
397
365
For the Paschen series, we set n’ = 3, and vary n as follows:
=λ (nm)
n
4
5
6
7
1870
1280
1090
1005
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8
9
10
11
(convergence point)
Mr Casey Ray McMahon, B.Sci (Hons), B.MechEng (Hons)
Version: 6th October, 2013
Page: 2 of 4
954
923
902
887
820
For the Brackett series, we set n’ = 4, and vary n as follows:
=λ (nm)
n
5
6
7
8
9
(convergence point)
4050
2620
2160
1940
1820
1460
For the Pfund series, we set n’ = 5, and vary n as follows:
=λ (nm)
n
6
7
8
9
10
(convergence point)
7460
4650
3740
3300
3040
2280
For the Humphreys series, we set n’ = 6, and vary n as follows:
=λ (nm)
n
7
8
9
10
11
(convergence point)
12400
7500
5910
5130
4670
3280
The infinity values in each of the equations above represent the convergence points of the
respective series. Ie: as n increases, the wavelength approaches the wavelength
corresponding to the n infinity value.
From this data, using the wavelength of the convergence point for the Lyman series (91.2
nm), the following holds true:
Mr Casey Ray McMahon, B.Sci (Hons), B.MechEng (Hons)
Version: 6th October, 2013
Page: 3 of 4
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Table 1: factor table:
Base wavelength
x factor
91.2nm
91.2nm
91.2nm
91.2nm
91.2nm
91.2nm
1
4
9
16
25
36
= Convergence limit (nm)
for:
Lyman series (n’=1)
Balmer (n’=2)
Paschen (n’=3)
Brackett (n’=4)
Pfund (n’=5)
Humphreys (n’=6)
For example: 91.2 nm x 25 = 2280nm, the convergence point for the Pfund series.
Table 1 can be re-written as table 2.
Table 2: factor table:
Base wavelength
x factor
91.2nm
91.2nm
91.2nm
91.2nm
91.2nm
91.2nm
12
22
32
42
52
62
= Convergence limit (nm)
for:
Lyman series (n’=1)
Balmer (n’=2)
Paschen (n’=3)
Brackett (n’=4)
Pfund (n’=5)
Humphreys (n’=6)
From table 2, we see that:
91.2nm x (n’)2 = convergence wavelength (nm) for the corresponding n’ series
……….equation (2
To express this in terms of frequency, considering that c = fλ, we have:
……….equation (3
Thus, equations 2 and 3 allow us to directly determine the convergence wavelength and
frequency points for any spectral series for hydrogen.
The paper McMahon, C.R. (2013) “The McMahon equations” discusses spectral lines in
great detail, as well as depicts exactly what they are, something that is not currently
known in conventional science. Refer to this paper to see exactly what spectral lines are,
and why they exist.
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Mr Casey Ray McMahon, B.Sci (Hons), B.MechEng (Hons)
Version: 6th October, 2013
Page: 4 of 4
References:
McMahon, C.R. (2013) “The McMahon equations” The general science Journal.
Wikipedia (2013) Hydrogen spectral series. Link:
http://en.wikipedia.org/wiki/Hydrogen_spectral_series
Link last accessed 6th October, 2013.