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Balmer and Rydberg Equations for Hydrogen Spectra Revisited
Raji Heyrovska
Institute of Biophysics, Academy of Sciences of the Czech Republic, 135
Kralovopolska, 612 65 Brno, Czech Republic. Email: [email protected]
Abstract
Balmer equation for the atomic spectral lines was generalized by Rydberg. Here it is shown that 1) while Bohr's theory explains the Rydberg constant in terms of the ground state energy of the hydrogen atom, quantizing the angular momentum does not explain the Rydberg equation, 2) on reformulating Rydberg's equation, the principal quantum numbers are found to correspond to integral numbers of de
Broglie waves and 3) the ground state energy of hydrogen is electromagnetic like that of photons and the frequency of the emitted or absorbed light is the difference in the frequencies of the electromagnetic energy levels.
Subject terms: Chemical physics, Atomic physics, Physical chemistry, Astrophysics
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An introduction to the development of the theory of atomic spectra can be found in
[1, 2]. This article is divided into several sections and starts with section 1 on
Balmer's equation for the observed spectral lines of hydrogen.
1. Balmer's equation
Balmer's equation [1, 2] for the observed emitted (or absorbed) wavelengths (λ) of the spectral lines of hydrogen is given by,
2 2 2 λ = h B,n1[n2 /(n2 - n 1 )] (1)
2 where n1 and n2 (> n1) are integers with n1 having a constant value, hB,n1 = n1 hB is a constant for each series and hB is a constant. For the Lyman, Balmer, Paschen,
Bracket and Pfund series, n1 = 1, 2, 3, 4 and 5 respectively. Note: In equation (1),
Balmer used different symbols. The symbols used here conform with those below.
For each series, λ varies between two limits. When n2 >> n1, λ = λmin,n1 = hB,n1 =
2 n1 hB is the minimum wavelength, and it has the values, hB, 4hB, 9hB , 16hB and 25hB for the Lyman, Balmer, Paschen, Bracket and Pfund series, respectively. Balmer's equation (1) can be written in a new form as,
2 2 2 2 λ = λmin,n1 [n 2 /(n 2 - n 1 )] = λmin,n1 /sin θ (2)
2 2 2 2 where sin θ = (n 2 - n 1 )/ n 2 in a right angled triangle with n2 as the hypotenuse and
2 n1 as the side which makes an angle θ with the hypotenuse. λ = λmin,n1 for sin θ = 1.
For the higher wavelength limit, n2 = n 1 + 1 and λ = λmax,n1 is given by
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2 λmax,n1 = λmin,n1 [(n1 + 1) /(2n1 +1)] (3)
and is equal to (4/3) λmin,n1 , (9/5) λmin,n1 , (16/7) λmin,n1 , (25/9) λmin,n1 and (36/11) λmin,n1 , for the Lyman, Balmer, Paschen, Bracket and Pfund series, respectively.
The wavelength data from [3] for the various spectral lines of hydrogen are given in Table 1. Fig. 1 shows the linear graphs of λ versus sin -2θ. The slopes give the
2 limiting values, λmin,n1 = hB,n1 = n1 hB. These values are also included in Table 1 and
2 shown as the intercepts in Fig. 1. For the Pfund series (n1 = 5), λmin,n1 = n1 hB, where hB = λmin,n1 is the value for the Lyman series (n1 = 1).
2. Rydberg equation
Balmer's equation (1) was generalized by Rydberg as shown [1, 2],
2 2 1/ λ = RH[(1/n 1 ) - (1/n 2 )] (4)
where 1/ λ is the wave number of the emitted (or absorbed) light and RH = 1/hB =
2 2 n1 /hB,n1 = n1 /λmin,n1 is the Rydberg's constant. For each series, 1/ λ varies linearly
2 2 with (1/n 2) , with RH (= 1/hB) as the slope and RH/n 1 (= 1/hB,n1 = 1/ λmin,n1 ) as the intercept. The interpretation of the Rydberg equation (4) amounts to that of the
2 2 Rydberg constant, RH and of the term in the square brackets, [(1/n 1 ) - (1/n 2 )].
3. Rydberg constant (RH) and the energy (EH) of hydrogen
Bohr [4] made a major advance in the interpretation of the Rydberg constant [1, 2].
On multiplying both sides of equation (4) by hc, where c is the speed of light in 4 vacuum and h is the Planck's constant, the energy, hν, of emitted (or absorbed) light is given by,
2 2 hc/ λ = hν = EH[(1/n 1) - (1/n 2) ] = EH,n1 - EH,n2 (5)
where ν = c/ λ is the frequency of the emitted (or absorbed) light, EH = hcRH is the
2 2 ground state energy of the hydrogen atom, E H,n1 = EH/n 1 and EH,n2 = E H/n 1 .
EH = eI H , where e is the unit of charge and I H is the measured ionization potential and it is given by the equivalent terms, [1, 2, 4],
2 EH = (1/2) HvH = (1/2)p ωωH (6)
where H is the reduced mass of the hydrogen atom, vH = ωHaH is the linear velocity,
ωH is the angular velocity, aH is the Bohr radius (distance between the electron and proton), p ω is angular momentum. These constants are given by the following equations,
pω = HvHaH = h/2 π (7a)