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The Classical Atom {© 10/01/2018}

This paper, which was released by Keith Dixon-Roche (one of CalQlata's Contributors) on the 10th of January 2018 explains why classical theory does not apply to electrons in an atom.

Note: All the input data in these calculations has been provided by CalQlata's Constants page.
All calculations are the sole copyright priority of Keith Dixon-Roche © 2018
Keith Dixon-Roche is also responsible for all the other web pages on this site related to atomic theory
A 'pdf' version of this paper can be found at: The Classical Atom - The Paper

Conclusions

This paper 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.

Premise

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 though 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

Calculations

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)

Findings

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 (parabola) 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

Whilst the [classical theory] calculations in their current form may appear 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

ShellKinetic EnergyShell RadiusShell AreaShell Volume
FormulasR∞.h.c.(Z/n)²aₒ.n²/Z
Rn-₁ + aₒ.(2.n - 1)
4 π.R²4/3 π.R³
UnitsJm
nRᵧRAV
12.17987197684936E-185.291772106700E-113.51894216858592E-206.207146670938E-31
25.44967994212340E-192.116708842680E-105.63030746973747E-193.972573869400E-29
32.42207997427707E-194.762594896030E-102.85034315655460E-184.525009923114E-28
41.36241998553085E-198.466835370720E-109.00849195157996E-182.542447276416E-27
58.71948790739744E-201.322943026675E-092.19933885536620E-179.698666673340E-27
Table 1: Shell Size

ShellSemi-Major AxisEccentricityHalf ParameterShell NumberShell Capacity
FormulasR / (1-e) ⁽¹⁾(-R+√[R²-4.a.{R-a}] ) / 2.a
where:
a = ³√[G.mp / (2π/t)²]
a.(1-e) ²R / 3.d
√[R / R₁]
√[t / n.t₁]
2.n³
Unitsm
naeCNp
12.6462441580938E-11-0.9997293486751.43222513204E-1412
21.0584976632375E-10-0.9997293486755.72890052825E-1428
32.3814074108147E-10-0.9999076488984.39830888227E-14318
44.2339906529500E-10-0.9997293486752.29156021121E-13432
56.6136997935644E-10-1.0003070413970550
Table 2: Shell Shape
1) focus distance ƒ = R in this Table

ShellWave LengthFrequencyLinear MomentumLinear VelocityPeriodAngular VelocityAngular Momentum
Formulas2π.R / nv/λmₑ.v√[2.Rᵧ/mₑ]
v = v₁/n
2π.R / v
t₁.n³
2π / t2/5.mₑ.R²
UnitsmHzkg.m/sm/ssᶜ/skg.m²
nλƒpvtωJ
13.325E-106.58E+151.993E-242.188E+061.52E-164.134E+161.02E-51
26.65E-101.645E+159.964E-251.094E+061.216E-155.168E+151.633E-50
39.975E-107.311E+146.643E-257.292E+054.104E-151.531E+158.265E-50
41.33E-094.112E+144.982E-255.469E+059.727E-156.46E+142.612E-49
51.663E-092.632E+143.986E-254.375E+051.9E-143.307E+146.377E-49
Table 3: Electron Velocities

ShellElectrostaticMagneticGravitational (acceleration)
Formulask.Qp / R²μ₀.Qp / 2πRG.mp / R²
Unitsm/s²
nEBg
15.142206313558E+116.055349532159E-163.986163589659E-17
23.213878945974E+101.513837383040E-162.491352243537E-18
36.348402856245E+096.728166146844E-174.921189616863E-19
42.008674341234E+093.784593457600E-171.557095152211E-19
58.227530101693E+082.422139812864E-176.377861743455E-20
Table 4: Proton Fields

ShellElectrostaticMagneticGravitationalCentrifugal
Formulask.Qₑ.Qp / R²Qp.(E + B.v)G.mₑ.mp / R²mₑ.v² / R
UnitsNNNN
nFₑFmFgFc
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

ShellKineticElectrostaticMagneticGravitationalCentrifugalTotal
Formulas½m.v² + ½J.ω²Fₑ.RFm.RFg.RFc.RKE+PEₑ
UnitsJJJJJJ
nKEPEₑPEmPEgPEcE
13.05182E-18-4.35974E-18-4.35974E-18-1.92152E-57-4.35974E-18-1.30792E-18
27.62955E-19-1.08994E-18-1.08994E-18-4.80381E-58-1.08994E-18-3.26981E-19
33.39091E-19-4.84416E-19-4.84416E-19-2.13503E-58-4.84416E-19-1.45325E-19
41.90739E-19-2.72484E-19-2.72484E-19-1.20095E-58-2.72484E-19-8.17452E-20
51.22073E-19-1.7439E-19-1.7439E-19-7.68609E-59-1.7439E-19-5.23169E-20
Table 6: Energies

ShellElectron Separation DistanceNucleus Repulsion EnergyElectron Repulsion Force
FormulasV/Ak.Z².Qₑ.Qp / 2.Rk.Qₑ.Qp / d²
UnitsmJN
ndESFS
11.763924035567E-112.179871976849E-187.41484984465015E-07
23.527848071133E-114.904711947911E-181.85371246116254E-07
35.291772106700E-112.930716768875E-178.23872204961127E-08
47.055696142267E-111.145795207831E-164.63428115290634E-08
58.819620177833E-113.244521450343E-162.96593993786006E-08
Table 7a: Electron Separation

ShellSeparation Between ShellsElectron SeparationShell NumberAtomic Number
FormulasRn - Rn-1ES / FS√[PEₑ₁ / PEₑ]√[ℓ / ℓ₁]
Unitsmm
nδnZ
15.29177210670E-112.93987339261111E-1211
21.58753163201E-102.64588605335E-1123
32.64588605335E-103.55724680505945E-10311
43.70424047469E-102.47243352318594E-09429
54.76259489603E-101.09392688939059E-08561
Table 7b Electron Separation

Shell
FormulasFₑ / FgFS / PEₑKE / RᵧPEₑ / RᵧKE / PEₑ
Units
n
14.40742111792335E-40-5.87974678522222E-121.4-2-0.7
24.40742111792335E-40-5.87974678522222E-121.4-2-0.7
34.40742111792335E-40-5.87974678522222E-121.4-2-0.7
44.40742111792335E-40-5.87974678522222E-121.4-2-0.7
54.40742111792335E-40-5.87974678522222E-121.4-2-0.7
Table 8: Constants

Claims

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)

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