# Rydberg's Atom

A summary of the mathematical theory of the atom according to Johannes Rydberg.

Sources: Planck's Atom; Newton's Atom; Laws of Motion; Physical Constants
Related Books: Philosophiæ Naturalis Principia Mathematica Rev. IV; The Atom; The Mathematical Laws of Natural Science
Related Calculators: Atomic Elements; Orbital Motion; Atoms; Physics

## Introduction

The purpose of this study is to answer the following question:
1) Can the atom be explained using Rydberg's atomic theory?

## Conclusion

This study appears to show that Rydberg's atom cannot work because;
a) none of the properties of any atom can be predicted using it; and, b) its orbital eccentricities are equal to 1; a straight line.

## Constants & Formulas

The constants used on this page can be found in our Constants page.
n = the electron shell number (1, 2, 3, 4, 5 etc.) counting out from the innermost shell (1s)
Z = atomic number
c = speed of light in a vacuum
v = velocity of electron
λ = the wavelength of an electron
ƒ = the frequency of an electron
t = the orbittal period of an electron
e = elementary charge unit
h = Planck's constant
ħ = Dirac's constant
h = Newton's motion constant
p = momentum of the electron
Rᵧ = Rydberg energy
R = Rydberg constant
R = the orbittal radius of an electron
PE = potential energy
KE = kinetic energy
E = total energy
m₁ = proton mass
m₂ & mₑ = electron mass

## Formulas and Properties

The following tables contain the formulas and properties of a ground-state electron in a given shell (n) orbitting a single proton (Z=1)
To calculate the properties of a ground-state electron orbitting more than one proton, you must change 'Z' in the respective formulas to the correct number of protons where appropriate.

Shell KE = Rᵧ.(Z/n)²
= mₑ.R.(2.π/t)²
= mₑ.h² / R³
PE = -2.KE
= -h.ƒ
= -mₑ.v²
E = KE+PE
= -KE
(J) (J) (J)
1 2.17987197684936E-18 -4.35974395369872E-18 -2.17987197684936E-18
2 5.44967994212340E-19 -1.08993598842468E-18 -5.44967994212340E-19
3 2.42207997427707E-19 -4.84415994855413E-19 -2.42207997427707E-19
4 1.36241998553085E-19 -2.72483997106170E-19 -1.36241998553085E-19
5 8.71948790739744E-20 -1.74389758147949E-19 -8.71948790739744E-20
6 6.05519993569267E-20 -1.21103998713853E-19 -6.05519993569267E-20
7 4.44871832010073E-20 -8.89743664020147E-20 -4.44871832010073E-20
Kinetic, Potential and Total Energies in an Atom with one Proton and One Electron

Shell v = 2.KE / mₑ
= 2.π.R / t
= √[k.Q₁.Q₂ / mₑ.R]
R = aₒ.n² / Z t = v.R
= n.h / 2.Rᵧ
= n³ / 2.Z².c.R
= n³ . [π.aₒ]¹˙⁵ . [16.ε₀.mₑ]² / e
= n.λ / v
(m/s) (m) (s)
1 2187690.35053551 5.2917721067E-11 1.51983047973957E-16
2 1093845.17526775 2.11670884268E-10 1.21586438379166E-15
3 729230.11684517 4.76259489603E-10 4.10354229529685E-15
4 546922.587633877 8.46683537072E-10 9.72691507033327E-15
5 437538.070107102 1.322943026675E-09 1.89978809967447E-14
6 364615.058422585 1.905037958412E-09 3.28283383623748E-14
7 312527.192933644 2.592968332283E-09 5.21301854550674E-14

Shell h = R.v p = mₑ.v
(m²/s) (kg.m/s)
1 1.15767587750606E-04 1.99285239459576E-24
2 2.31535175501211E-04 9.96426197297878E-25
3 3.47302763251817E-04 6.64284131531919E-25
4 4.63070351002422E-04 4.98213098648939E-25
5 5.78837938753028E-04 3.98570478919151E-25
6 6.94605526503633E-04 3.32142065765959E-25
7 8.10373114254239E-04 2.84693199227965E-25
Newton's Motion Constants and Momenta

Shell λ = 2πR / n
= p / h
ƒ = v / λ
(m) (Hz)
1 3.32491847497602E-10 6.57968117714912E+15
2 6.64983694995204E-10 1.64492029428728E+15
3 9.97475542492806E-10 7.31075686349903E+14
4 1.32996738999041E-09 4.11230073571820E+14
5 1.66245923748801E-09 2.63187247085965E+14
6 1.99495108498561E-09 1.82768921587476E+14
7 2.32744293248321E-09 1.34279207696921E+14
Electron Wavelengths and Frequencies

Shell Fg = G.m₁.m₂ / R²
= G.m₁.m₂ / R³.(2.π/t)²
Fₑ = k.Q₁.Q₂ / R².ε φ = Fg/Fₑ
= G.m₁ / R.(2.π.R/t)²
= G.m₁ / R.v²
= G.m₁.R / h²
(N) (N)
1 8.23872204961127E-08 3.63115175461573E-47 4.40742111792333E-40
2 5.14920128100705E-09 2.26946984663483E-48 4.40742111792333E-40
3 1.01712617896435E-09 4.48290340076016E-49 4.40742111792333E-40
4 3.21825080062940E-10 1.41841865414677E-49 4.40742111792333E-40
5 1.31819552793780E-10 5.80984280738517E-50 4.40742111792333E-40
6 6.35703861852722E-11 2.8018146254751E-50 4.40742111792333E-40
7 3.43137111603968E-11 1.51234975202654E-50 4.40742111792333E-40
Gravitational and Electrostatic Electron Holding Forces and their ratio (φ)

Shell KEn-1/KEn - 1 = [n/(n-1)]² - 1 KE₁/KEn - 1 = n² - 1
1
2 3 3
3 1.25 8
4 0.777777778 15
5 0.5625 24
6 0.44 35
7 0.361111111 48
Kinetic Energy Jump Factors Between Shell Numbers (n)
n=1 to n: KEn = KE₁ / n²
n-1 to n: KEn = KEn-1 . [(n-1)/n]²

## Coupling Force

There are two principal forces holding an electron to its nucleus; Gravitational and Electrostatic
The gravitational force was defined by Newton's formula:
Fg = G.m₁.m₂ ÷ R² N
The electrostatic force was defined by Coulomb's formula:
Fₑ = k.Q₁.Q₂ ÷ R².ε N
where; ε = εₐ/εₒ = 1 inside an atom
The ratio of these forces is defined Thus:
φ = Fg/Fₑ = 4.40742111792333E-40 (see above Table)
which is a constant

Applying φ to a ground-state electron and a proton we get:
ψ = φ.m₁/m₂ = 2.40035855253320E-43
which is the force coupling factor between both masses
This is not a constant as it varies with the masses concerned.