Planck's Atom

An atomic model that can be constructed from Max Planck's three constants (time, length & mass).

Sources: Newton's Atom; The True 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
CalQlata comment: This is the only study we (at CalQlata) know on this subject that manifestly works.

Introduction

The purpose of this study is to answer the following questions:
1) Can a theoretical atom be created from Planck's three constants; time, length and mass?
2) What would be the purpose of such a model?

Conclusion

The answer to the above questions is yes.
A Planck atom can be created if his constants are extended to include Plank Energy and Planck Force.
This theoretical atom, in which the electron and the proton possess identical properties and dimensions, can be used to establish Isaac Newton's gravitational constant 'G' and to finalise the Newtonian atomic model.

Constants & Formulas

The following Table, which should be read in conjunction with our Rydberg Atom page, contains modified constants used in the calculations for the Planck atom values:

Sym (units) Formula Planck Atom Values Rydberg Atom Values
k (N.m²/C²) = 1 / 4π.ε₀ 8.98755184733E+09 8.98755184733E+09
c (m/s) 299792459 299792459
ε₀ (s²/m²) = 1 / μ₀.c² 8.85418775855E-12 8.85418775855E-12
μ₀ = 4π / 1E+07 1.256637061436E-06 1.256637061436E-06
NA 6.02214129E+23 6.02214129E+23
φ 4.4074211179E-40 4.4074211179E-40
G (m³ / kg.s²) = c.aₒ/m₁ (or m₂) 9.25892493947763E-58 6.67359232004334E-11
F (C) = e.NA 96485.331794 5.3800516756E+25
ρᵤ (kg/m³) = m/V 3.97381844498046E+37 7.12660796350450E+16
Vᴾ : Vₑ & Vₚ (m³) Vᴾ=√Σ & √[Vₑ.Vₚ] : Vₑ=mₑ/ρᵤ
Vₚ=mₚ/ρᵤ
5.47722557505167E-46 1.27822236702922E-47
2.34700946985653E-44
m₁ (kg) = (ħ.c/G)⁰˙⁵ 2.176550002E-08 1.6726216E-27
m₂ (kg) = (ħ.c/G)⁰˙⁵ 2.176550002E-08 9.1093897E-31
h (J.s) = (π.m₂.aₒ.e² / ε₀)⁰˙⁵ 5.0232407302E-15 6.6260717447E-34
ħᵨ (J.s) = h / 2π 7.9947359256E-16 1.0545720714E-34
e (C) = Q 89.337852045 1.60217649E-19
Rᵧ (J) = R∞.h.c.(Z/n)² 8.76103166894E+49 2.17987197685E-18
Rᵧ (eV) = e / Rᵧ 9.8066289578E+47 13.60569197
R (/m) = m₂.e⁴ / 8.ε₀².h³.c 5.817689712E+55 1.0973726956E+07
PE (J) = -k.e² / aₒ = ½.Rᵧ -1.7522063338E+50 -4.3597439537E-18
aₒ (m) = λ / (2π)² 4.0938052242E-37 5.2917721067E-11
Q (C) = Q 89.337852045 1.602176488E-19
Q (J) = (G.m₁² / k.φ)⁰˙⁵ 89.337852045 1.60217649E-19

The following formulas are provided to assist with the calculation method used to identify G

Planck's original formulas:
Planck's time; tᵨ = (ħ.G / c⁵)⁰˙⁵
Planck's length; λᵨ = (ħ.G / c³)⁰˙⁵
Planck's mass; mᵨ = (ħ.c / G)⁰˙⁵

CalQlata's formulas:
Planck's energy; Eᵨ = (ħ.c⁵ / G)⁰˙⁵
Planck's force; Fᵨ = c⁴ / G

Along with the above formulas ...
ħ = h / 2π
v = 2πR/t
λ = h / m.v
... we can establish the following for a Planck atom:
(ħ.G/c³)⁰˙⁵ = h ÷ (ħ.c/G)⁰˙⁵ ÷ 2πR/t
(ħ.G/c³)⁰˙⁵ = h ÷ 2πR x t ÷ (ħ.c/G)⁰˙⁵
(ħ.G/c³)⁰˙⁵ = ħ/R x (ħ.G/c⁵)⁰˙⁵ ÷ (ħ.c/G)⁰˙⁵
ħ.G/c³ = ħ²/R² x ħ.G/c⁵ ÷ ħ.c/G
ħ.G/c³ = R².ħ² x ħ.G/c⁵ x G/ħ.c
G/c³ = G².ħ / R².c⁶
R² = G.ħ / c³
R = (G.ħ/c³)⁰˙⁵ = λ {λ = 2.π.R/n for non-Planck values}
i.e. in Planck's atom, the radius of separation between its nucleus and its orbiting mass is equal to its wavelength, and its shell number is equal to 2.π
moreover, if R = λ in Planck's atom;
G = λ².c³ / ħ
from which; G = 6.67359232004332E-11 using Rydberg Atom Values (see above Table) verifying the above formula, however, regarding its units:
m² x m³/s³ ÷ J.s = m² x m³/s³ ÷ kg.m².s/s² = m⁵/m² x s²/s⁴ ÷ kg = m³ ÷ kg.s²
are missing; kg/kg
i.e. kg/kg x m³ ÷ kg.s² = kg.m/s² x m²/kg² = N.m²/kg²
In order to create the correct units we need to apply the mass ratio m₁/m₂, which in the Planck atom equals 1.0 (both the force-centre and the orbiting mass are the same)

Properties and Formulas

The following Tables contain the formulas and properties of a Planck electron in the specified shells (n) orbiting a single Planck proton (Z=1) using the same formulas for a Rydberg Atom and the above constants.

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 8.76103166894E+49 -1.75220633379E+50 -8.76103166894E+49
2.21919524656E+48 -4.43839049313E+48 -2.21919524656E+48
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 8.972393222074E+28 4.09380522418E-37 2.866808910219E-65
1.428000732657E+28 1.616169522311E-35 7.111125620784E-63

Shell h = R.v p = mₑ.v
(m²/s) (kg.m/s)
1 3.67312302459346E-08 1.95288624831699E+21
2.30789126195887E-07 3.10811499715836E+20
Newton's Motion Constants and Momenta

Shell λ = 2πR / n
= p / h
ƒ = v / λ
(m) (Hz)
1 2.57221368350306E-36 3.48819900913307E+64
1.61616952231127E-35 8.83571130963482E+62
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 1.88643835639E+47 4.28014093938E+86 4.40742111792E-40
1.21038391821E+44 2.7462406832E+83 4.40742111792E-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
38.4784176043574 38
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

Planck's Atomic Particle

The Planck atom is artificial, equivalent to the Newton atom but based upon Planck's time, length and mass, all the particles of which are identical in size and substance to each other.

Newton's and Planck's atomic particles share a commonality in the product of their volumes:
Vᵨ.Vₑ = 3E-91 (exact) for both Newton's and Planck's atomic particles where: Vᵨ is the volume of the force-centre and Vₑ is the volume of the satellite

The radius (Rᴾ) of a Planck particle can therefore be calculated thus;
Rᴾ = ⁶√[9 x 3E-91 ÷ 16π²] = 5.07563837996471E-16 m
and Vᴾ = Vᵨ = Vₑ = ⁴/₃π.Rᴾ³ = 5.47722557505167E-46 m³

As Planck's mass; mᴾ = 2.1765500017459E-08 kg
Planck's density; ρᴾ = mᴾ/Vᴾ = 3.97381844498046E+37 kg/m³

Planck-Newton Commonality
Fᴾ = c⁴/G = 299792459⁴ / 6.67359232004334E-11 = 1.21038391820525E+44 N
Fᴺ = G.mᵨ.mₑ/aₒ² = 3.63115175E-47 N
Fᴺ/Fᴾ = 3E-91 (exact)
Therefore; Fᴺ/Fᴾ ≡ Vᴺᵨ.Vᴺₑ = Vᴾᵨ.Vᴾₑ = 3E-91 (exact)

G

Planck's formulas (see above) are used to calculate his electron properties first with 'ħ' for the standard atom and again with 'ħᵨ' (see above), the results from which are summarised below:

Property Values using Planck's constant 'ħᵨ': A Values using Planck's constant 'ħ':B Ratio A/B
t (s) 1.48432887846076E-34 5.39096122598358E-44 2753365895.68949
λ (m) 4.44990604438463E-26 1.61616952231127E-35 2753365895.68949
m (kg) 59.9283854507007 2.17655000174590E-08 2753365895.68949
E (J) 5.38609471364750E+18 1.95618559889903E+09 2753365895.68949
F (N) 1.21038391820525E+44 1.21038391820525E+44 1.0

Note:
E = 2.KE.φ; which means that Planck's energy must be gravitational (Newtonian)
t = λ.(m/E)⁰˙⁵; so it too must be gravitational (Newtonian)

The above ratios reveal that G = 6.67359232004332E-11 N.m²/kg² must be correct because:
2753365895.68949 exactly equals (ħᵨ/ħ)⁰˙⁵ and shows that Planck's constants vary with mass;
a value of exactly 1.0 also shows that Planck's gravitational force and Newton's gravitational force may be calculated using conventional Newtonian theory.