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
The purpose of this study is to answer the following question:
1) Can the atom be explained using Rydberg's atomic theory?
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.
The constants used on this page can be found in our Constants page.
The symbols used on this page are identified as follows:
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
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 |
| Orbital Velocities, Radii and Periods | |||
| 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 |
|||
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.
You will find further reading on this subject in reference publications(55, 60, 61 & 62)