Rydberg's Atom

{© 21/10/2017}

This page contains a summary of the mathematical theory of the atom and its component parts according to Johannes Rydberg.

The levels of accuracy on this page have been set to assist CalQlata in its effort to establish an accurate value for Newton's gravitational constant (G).
To this end, a great deal of effort has been made by Calqlata to confirm all constants used in these calculations using original formulas and indisputable data where possible.

Constants & Formulas

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

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
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

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.

Further Reading

You will find further reading on this subject in reference publications(55, 60, 61 & 62)

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