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The Atom

This page is a summary of the formulas and values associated with the analysis of an atom and its component parts and may be read in conjunction with our research discussion on the atom that comprises an overview of its particles.

Whilst the levels of accuracy on this page may appear exaggerated, it is for a reason.
Between this page and our planetary theories, CalQlata is trying to establish an accurate value for Newton's gravitational constant (G).
As such, and as with our calculations for planetary motions, a great deal of effort has been made to verify the principal properties (see below and our definitions) on which all other calculations are based.
For all other properties, we have gone back to basics and recalculated from scratch.

Principal Constants & Formulas

The following constants have been used as the basis for all calculations on this page:
c = speed of light
m₁ = proton mass
m₂ = electron mass
e = Q₁ = Q₂ = 1eV = elementary charge unit
R = d = Bohr radius
μ₀ = 4.π / 1E+07 = magnetic constant
ε₀ = 1 / μ₀.c² = permittivity of free-space
εₐ = actual permittivity
permittivity inside an atom: εₐ ≡ ε₀
ε = relative permittivity inside an atom (εₐ/ε₀ (ε ≡ 1))
k = Coulomb's constant
G = Newton's gravitational constant

The following is a summary of the principal relationships governing the bonding of an electron to a nucleus:
PE = - k.e² / rn (-ve)
E = PE + KE = - Rᵧ.Z² / n² = - Z.k.e² / 2.rn (-ve)
- Z.k.e² / 2.rn = - k.e² / rn + KE
KE = k.e² / rn - Z.k.e² / 2.rn = k.e².(1 - Z/2) / rn (+ve)
δPE = - 2.Rᵧ.Z.(1/ni² - 1/nf²)

Symbols:
n = the electron number (1, 2, 3, 4, 5 etc.) counting out from the innermost shell (1s)
ni = initial electron number (shell) position
nf = final electron number (shell) position
ħ = Dirac's constant
Z = atomic number
Rᵧ = Rydberg constant
rn = the radius of electron 'n' in an unexcited state
vn = velocity of the electron
PE = potential energy
KE = kinetic energy
E = total energy

1 eV = 1.60217648753E-19 J
1 e = 1.60217648753E-19 C
# 1 eV = 1.60217648753E-19 C @ 1 V

Forces

The following is a summary of the forces holding an electron to a hydrogen proton:

According to Coulomb; the electrostatic force can be calculated thus:
Coulomb's Law: F = k.Q₁.Q₂ ÷ d².ε N
F = 8.98755178736818E+09 x 1.60217648753E-19² ÷ 5.29177209E-11²
   = 8.23872204664865E-08 N

According to Newton; the gravitational force can be calculated thus:
F = G.m₁.m₂ / R²
   = 6.67128190396304E-11 x 1.67262163783E-27 x 9.1093897E-31 ÷ 5.29177209E-11²
   = 3.62989466276708E-47 N

So [virtually] all of the attractive force holding the two masses together is electrostatic

Electron Energies

According to Coulomb; the electrostatic potential energy can be calculated thus:
Q = electric charge of the all the protons in the nucleus of the atom
Qn = electric charge of the electron
rn = orbital radius of the electron
PE = k.Q.Qn / rn J
For the electron orbiting a hydrogen atom in Shell 1s:
   = 8.98755178736818E+09 x 1.60217648753E-19² ÷ 5.29177209E-11
   = 4.3597439383723E-18 J

According to Newton; the gravitational acceleration can be calculated thus:
g = G.m₁ / r² m/s²
   = 6.67128190396304E-11 x 1.67262163783E-27 ÷ 5.29177209E-11²
   = 3.98478359397346E-17 m/s²
and the gravitational potential energy can be calculated thus:
PE = m₁.g.r
For the electron orbiting a hydrogen atom in Shell 1s:
   = 1.67262163783E-27 x 3.98478359397346E-17 x 5.29177209E-11
   = 3.52698475748778E-54 J

So [virtually] all of the potential energy holding the two masses together is electrostatic

Electron Velocity

From the above energy formulas: E = - Z.k.e² / 2.rn & PE = - k.e² / rn
therefore: E = ½.PE (for hydrogen {Z=1})
   = ½ x 4.3597439383723E-18
   = 2.17987196918615E-18 J

It takes 13.605691920662 eV (E / e) of energy to strip a 1s electron at ground state from a proton (i.e. the electrostatic potential energy in the hydrogen atom)

If E = ½.PE then ½.PE = PE+KE and KE = - ½.PE (+ve)
vₑ = √(KE/m₂) = √(½.PE/m₂) = √(½ x 4.35974393837230E-18 ÷ 9.1093897E-031)
    = 1546930.679 m/s
vₑ/c = 1546930.679 ÷ 299792458 = 0.005160005
i.e. the electron is travelling at 0.5160005% the speed of light (in a vacuum)

Properties of an Electron

The following formulas constitute the basis of the calculations that define the properties of each electron (n) attached to a nucleus dependent upon its radius (rn):

m₂.vn² / rn = Z.k.e² / 2.rn²

rn = n².ħ² / Z.k.e².m₂
KEn = ½.m₂.vn²
PEn = -Z.k.e² / rn
En = KE + PE = -Z.k.e² / 2.rn

E = PE + KE = -Z.k.e² / 2r = -Rᵧ / (Z/n)²

By manipulating the above formulas, we can define the following properties of the electron:
its orbital radius: rn = n².ħ² / Z.k.e².m₂
its orbital velocity: vn = √(Z.k.e² / rn.m₂)
its kinetic energy: KEn = ½.m₂.vn²
its potential energy: PEn = -Z.k.e² / rn
and its total energy: En = KE + PE

The discrete jump in energy required to move any electron in any atom from an initial [shell] position (ni) to a final [shell] position (nf)
or the difference in energy (δE) between any two electrons (Ei - Ef) can be calculated thus:
δE = Rᵧ.Z².(1/ni² - 1/nf²)
and an electron must receive exactly this amount of energy (δE) to make the jump.
The electron will not absorb intermediate-amounts, and therefore not make the transition.

Newton's Gravitational Constant {© 12/08/17}

It cannot have passed unnoticed that there appears to be a connection between Newton's gravitational force and Coulomb's electrostatic force:
i.e. F = G.m₁.m₂/r² {N.m²/kg²} ≡ k.Q₁.Q₂/r² {N.m²/C²}

By elliminating the radii, we get:
G.m₁.m₂ ≡ k.Q₁.Q₂
or
G/k ≡ Q₁.Q₂ / m₁.m₂

The electrical charge of both the proton and an electron at ground state are equal to 1e (eV).
Therefore G/k ≡ e² / m₁.m₂ {N.m²/kg²}

Watch this space ...

Further Reading

TBA

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