The Classical Atom

{Keith Dixon-Roche © 10/01/2018}

A summary of the mathematical theory of the atom and its component parts according to classical theory.

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 classical atomic theory; elliptical orbits?


This study appears to show that the elliptical orbit predicted by classical theory does not apply to electrons in an atom and therefore disqualifies it for such calculations.


Devise a simple mathematical procedure using classical theory conjointly with Newton's and Coulomb's laws for force attraction/repulsion to describe the behaviour of electrons in an atom for all atomic and shell numbers.

The successful procedure must show it is possible to begin the calculation procedure with any shell and atomic number working through all motion, force and energy characteristics and return to the same shell and atomic numbers.

Constants & Formulas

Radius of electron shell 1: 5.2917721067E-11 m
Isaac Newton's gravitational constant: G = 6.67359232004332E-11 m³/kg/s²
Coulomb's constant: k = 8.98755184732667E+09 N.m²/C²
Elementary charge unit: Qₑ = -Qp = -1.60217648753000E-19 C
Ultimate density: ρ = 7.12660796350450E+16 kg/m³


The levels of accuracy on this page have been set to assist CalQlata in its effort to ensure that the calculations are not approximations.
To this end, Calqlata has confirmed all constants used in these calculations via original formulas and indisputable data where possible.

Every part of every atom (from Z = 1 to Z = 92) has been fully analysed during the generation of the above paper.

All calculation formulas are provided in the Tables of Results (below)


The calculations in this paper are based upon classical theory developed by Johannes Robert Rydberg, Charles-Augustin de Coulomb and Isaac Newton.
They reveal a working model of the atom that is currently regarded as an accurate description of the behaviour of the sub-atomic particles in any atom.

The results listed in Tables of Results (below) provide the properties and characteristics of the electrons in the first five shells according to these theories. In reaching these results, all the associated properties for the first 92 elements in the periodic Table were calculated and the results found to be consistent.

However, there are a number of issues with the calculation results:

1) They assume that an electron's (and a proton's) electrostatic angular moment apply at the same centre as its mass, which is unlikely

2) The resultant eccentricity of 1.0 (straight-line) is incompatible with the electron orbital shape

3) If the eccentricities are indeed ellipses, they can only apply to electrons at their perigee

4) If the calculated eccentricities are correct, according to elliptical theory (b = a.√[1 - e²]) the apogee radii reduce with shell number ultimately becoming smaller than the perigee radius of shell 1 (e.g. Shell 6: 2.58899E-11)

5) It is clear from Coulomb's law (and Dalton's law) that electrons will distance themselves equally throughout an atomic structure, which can only be achieved with circular orbits

6) Shell radii calculated according to this theory appear not to comply with generally accepted atomic radii for most elements

Even if the [classical theory] calculations in their current form could be considered valid for an electron at its perigee, it is necessary to apply Newton's laws of motion for the rest of the orbit. But Newton's laws of motion show the electron orbit to be circular as is to be expected from numerous electron-microscope photographs.

It is therefore the conclusion of this paper that, with the exception of electron and proton separation (Tables 7a & 7b) Newton's 'laws of motion' must apply to electron behaviour in an atom (further work).

Tables of Results

Shell Kinetic Energy Shell Radius Shell Area Shell Volume
Formulas R∞.h.c.(Z/n)² aₒ.n²/Z
Rn-₁ + aₒ.(2.n - 1)
4 π.R² 4/3 π.R³
Units J m
n Rᵧ R A V
1 2.179871976849E-18 5.291772106700E-11 3.518942168586E-20 6.207146670938E-31
2 5.449679942123E-19 2.116708842680E-10 5.630307469737E-19 3.972573869400E-29
3 2.422079974277E-19 4.762594896030E-10 2.850343156555E-18 4.525009923114E-28
4 1.362419985531E-19 8.466835370720E-10 9.008491951580E-18 2.542447276416E-27
5 8.719487907397E-20 1.322943026675E-09 2.199338855366E-17 9.698666673340E-27
Table 1: Shell Size

Shell Semi-Major Axis Eccentricity Half Parameter Shell Number Shell Capacity
Formulas R / (1-e) ⁽¹⁾ (-R+√[R²-4.a.{R-a}] ) / 2.a
a = ³√[ / (2π/t)²]
a.(1-e) ² R / 3.d
√[R / R₁]
√[t / n.t₁]
Units m
n a e C N p
1 2.6462441580938E-11 -0.999729348675 1.43222513204E-14 1 2
2 1.0584976632375E-10 -0.999729348675 5.72890052825E-14 2 8
3 2.3814074108147E-10 -0.999907648898 4.39830888227E-14 3 18
4 4.2339906529500E-10 -0.999729348675 2.29156021121E-13 4 32
5 6.6136997935644E-10 -1.000307041397 0 5 50
Table 2: Shell Shape
1) focus distance ƒ = R in this Table

Shell Wave Length Frequency Linear Momentum Linear Velocity Period Angular Velocity Angular Momentum
Formulas 2π.R / n v/λ mₑ.v √[2.Rᵧ/mₑ]
v = v₁/n
2π.R / v
2π / t 2/5.mₑ.R²
Units m Hz kg.m/s m/s s ᶜ/s kg.m²
n λ ƒ p v t ω J
1 3.325E-10 6.58E+15 1.993E-24 2.188E+06 1.52E-16 4.134E+16 1.02E-51
2 6.65E-10 1.645E+15 9.964E-25 1.094E+06 1.216E-15 5.168E+15 1.633E-50
3 9.975E-10 7.311E+14 6.643E-25 7.292E+05 4.104E-15 1.531E+15 8.265E-50
4 1.33E-09 4.112E+14 4.982E-25 5.469E+05 9.727E-15 6.46E+14 2.612E-49
5 1.663E-09 2.632E+14 3.986E-25 4.375E+05 1.9E-14 3.307E+14 6.377E-49
Table 3: Electron Velocities

Shell Electrostatic Magnetic Gravitational (acceleration)
Formulas k.Qp / R² μ₀.Qp / 2πR / R²
Units m/s²
n E B g
1 5.142206313558E+11 6.055349532159E-16 3.986163589659E-17
2 3.213878945974E+10 1.513837383040E-16 2.491352243537E-18
3 6.348402856245E+09 6.728166146844E-17 4.921189616863E-19
4 2.008674341234E+09 3.784593457600E-17 1.557095152211E-19
5 8.227530101693E+08 2.422139812864E-17 6.377861743455E-20
Table 4: Proton Fields

Shell Electrostatic Magnetic Gravitational Centrifugal
Formulas k.Qₑ.Qp / R² Qp.(E + B.v) G.mₑ.mp / R² mₑ.v² / R
Units N N N N
n Fₑ Fm Fg Fc
1 -8.238722049611E-08 -1.318195527938E-10 -3.631151754616E-47 -8.238722049611E-08
2 -5.149201281007E-09 -5.149201281007E-09 -2.269469846635E-48 -5.149201281007E-09
3 -1.017126178964E-09 -1.017126178964E-09 -4.48290340076E-49 -1.017126178964E-09
4 -3.218250800629E-10 -3.218250800629E-10 -1.418418654147E-49 -3.218250800629E-10
5 -1.318195527938E-10 -1.318195527938E-10 -5.809842807385E-50 -1.318195527938E-10
Table 5: Potential (attraction) Forces

Shell Kinetic Electrostatic Magnetic Gravitational Centrifugal Total
Formulas ½m.v² + ½J.ω² Fₑ.R Fm.R Fg.R Fc.R KE+PEₑ
Units J J J J J J
n KE PEₑ PEm PEg PEc E
1 3.05182E-18 -4.35974E-18 -4.35974E-18 -1.92152E-57 -4.35974E-18 -1.30792E-18
2 7.62955E-19 -1.08994E-18 -1.08994E-18 -4.80381E-58 -1.08994E-18 -3.26981E-19
3 3.39091E-19 -4.84416E-19 -4.84416E-19 -2.13503E-58 -4.84416E-19 -1.45325E-19
4 1.90739E-19 -2.72484E-19 -2.72484E-19 -1.20095E-58 -2.72484E-19 -8.17452E-20
5 1.22073E-19 -1.7439E-19 -1.7439E-19 -7.68609E-59 -1.7439E-19 -5.23169E-20
Table 6: Energies

Shell Electron Separation Distance Nucleus Repulsion Energy Electron Repulsion Force
Formulas V/A k.Z².Qₑ.Qp / 2.R k.Qₑ.Qp / d²
Units m J N
n d ES FS
1 1.763924035567E-11 2.179871976849E-18 7.41484984465015E-07
2 3.527848071133E-11 4.904711947911E-18 1.85371246116254E-07
3 5.291772106700E-11 2.930716768875E-17 8.23872204961127E-08
4 7.055696142267E-11 1.145795207831E-16 4.63428115290634E-08
5 8.819620177833E-11 3.244521450343E-16 2.96593993786006E-08
Table 7a: Electron Separation

Shell Separation Between Shells Electron Separation Shell Number Atomic Number
Formulas Rn - Rn-1 ES / FS √[PEₑ₁ / PEₑ] √[ℓ / ℓ₁]
Units m m
n δ n Z
1 5.29177210670E-11 2.93987339261111E-12 1 1
2 1.58753163201E-10 2.64588605335E-11 2 3
3 2.64588605335E-10 3.55724680505945E-10 3 11
4 3.70424047469E-10 2.47243352318594E-09 4 29
5 4.76259489603E-10 1.09392688939059E-08 5 61
Table 7b Electron Separation

Formulas Fₑ / Fg FS / PEₑ KE / Rᵧ PEₑ / Rᵧ KE / PEₑ
1 4.40742111792335E-40 -5.87974678522222E-12 1.4 -2 -0.7
2 4.40742111792335E-40 -5.87974678522222E-12 1.4 -2 -0.7
3 4.40742111792335E-40 -5.87974678522222E-12 1.4 -2 -0.7
4 4.40742111792335E-40 -5.87974678522222E-12 1.4 -2 -0.7
5 4.40742111792335E-40 -5.87974678522222E-12 1.4 -2 -0.7
Table 8: Constants


Claim 1: The classical theory of electron behaviour in atoms is not correct

Claim 2: Newton's 'laws of motion' should be used to evaluate the behaviour of electrons in atoms


Further Reading

Laws of Motion
Planetary Spin
Rydberg Atom
Planck Atom
Newton Atom

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