Understanding the Mystery of the Avogadro Number and Renaming SI Avogadro Constant as ‘Einstein-Perrin-Loschmidt-Avogadro-Newton’ Ratio

1. Introduction

1.1. The Avogadro History

The mole, an SI base unit, defines exactly $6.02214076\times {10}^{23}$ elementary entities per mole, fixed since the 2019 redefinition [1,2,3,4]. This enables precise stoichiometry: 12g ${\mathrm{o}\mathrm{f}}^{12}$C constitutes 1 mole. Yet its ~10²³ magnitude appears arbitrary-why this scale for gram-to-particle conversion? Historical estimates (Perrin 1909, Millikan 1910) converged empirically without deeper “why” [5,6,7].

The Branding Paradox: Avogadro, Loschmidt, and Perrin

The history of the “Avogadro Constant” is an illustrative case of scientific irony. While the name honours Amedeo Avogadro for his 1811 hypothesis-the conceptual leap that equal volumes of gas contain equal numbers of particles-Avogadro himself never actually calculated the value.

The Numerical Pioneer: Johann Josef Loschmidt (1865)

The first person to actually “find” the number was Johann Josef Loschmidt. Using kinetic gas theory and the relationship between a molecule’s mean free path and its liquid density, he produced the first quantitative estimate. His result, known as the Loschmidt constant (n_L ≈ 2.69x10²⁵ particles/m³ at STP), was the true breakthrough that turned a 50-year-old hypothesis into a measurable reality.

The Experimental Closer: Jean Perrin (1909)

Decades later, Jean Perrin provided a robust experimental estimation through his work on Brownian motion. Perrin was the one who officially coined the term “Avogadro’s Constant”. In doing so, he chose to honour the Italian visionary who provided the idea, effectively skipping over Loschmidt, who provided the math.

The Triple Irony of Priority

This creates a historiographic disconnect where the credit is split three ways, but unevenly:

• Avogadro gets the Name (despite no calculation).
• Perrin gets the Nobel Prize (1926, for proving the atomic nature of matter).
• Loschmidt gets Neither, remaining a niche figure primarily known for the “Loschmidt Constant”- a value often treated as a secondary gas-physics parameter rather than a primary universal constant in standard pedagogy.

1.1. Experts Opinion on Avogadro Number and Avogadro Constant

In this section we try to provide three kinds of arguments provided by field experts [8,9,10,11,12,13,14,15].

Perspective A: The Mole as a Numerical Convention

This viewpoint argues that the mole is entirely defined by a fixed numerical value-the Avogadro number.

• The Core Argument: The “amount of substance” is essentially a count of entities. Defining it through a dimensioned constant (the Avogadro constant) is seen as circular reasoning because the constant itself is derived from the unit.
• The Analogy: The mole is viewed as a scaling tool, similar to a “dozen” or “gross,” but applied specifically to elementary entities like atoms or molecules. From this perspective, the constant is a consequence of the definition, not the source of it.

Perspective B: The Mole as a Dimensional Necessity

This perspective defends the current international standards, asserting that “Amount of Substance” is a unique physical dimension (N) that cannot be reduced to a pure number.

• The Core Argument: In physics, a “number of entities” is dimensionless, whereas “amount of substance” has a specific unit. To maintain mathematical consistency in science (dimensional analysis), a bridge is required to convert counts into amounts.
• The Function of the Constant: The Avogadro constant is seen as a fundamental proportionality factor. Without it, the dimensional integrity of chemistry, such as the units for molar mass or the gas constant-would collapse into simple ratios, losing their physical meaning.

Perspective C: The Mole as a Derived Aggregate

This view attempts to resolve the “circularity” debate by proposing that the Avogadro constant has an intrinsic magnitude independent of the mole.

• The Core Argument: The constant is defined as the reciprocal of the “elementary amount” (the amount of substance contained in exactly one atom).
• The Comparison: Just as a “second” is defined by a fixed number of periods of radiation, a mole is defined by a fixed number of these “elementary amounts.” By establishing the atom itself as the base reference, this view argues that the definition is not circular, but a logical progression from the microscopic to the macroscopic.

Summary of the Debate: The debate ultimately rests on a fundamental question: Is the mole a physical property of the universe, or is it a human-made mathematical convenience for counting small things? One side prioritizes the simplicity of the “number,” while the other prioritizes the rigor of “dimensions” and “units” to ensure that scientific equations remain physically consistent across different scales.

Our Perceptive: The above points focus on the metrological definition and nomenclature of the mole, treating it as a standard for laboratory measurement and debating its status as either a numerical convention or a dimensional necessity. In contrast, the research under development explores the physical necessity and fundamental origin of the Avogadro constant itself. While the provided text examines the constant as a tool for unit consistency, the theoretical framework in progress treats the Avogadro number as a fundamental key for unlocking the relationship between gravity and quantum mechanics. This approach shifts the perspective from a debate over measurement standards to an investigation into the constant’s role in the unification of physical forces [16,17].

1.2. The Silicon Sphere Experiment – The kg Tension

Chemistry favours gram-moles (12 g ${\mathrm{o}\mathrm{f}}^{12}$C/mol), while SI metrology uses kg-moles (12 kg). The nuclear physics community recognizes $N_A\approx 6.022\times {10}^{26}$ atoms/kg as natural. This paper derives both scales from first principles, proposing clearer terminology.

The silicon sphere experiment [18,19,20,21], while ingeniously precise, ultimately ties the Avogadro constant to atomic measurements in a potentially circular manner: a perfect 28Si crystal sphere of known volume is weighed against the SI kilogram to yield approximately 6.022×10²³ entities per mole, then defined as exact. Defining this particle count by anchoring to a kilogram-scale crystal and retrofitting atomic weights introduces a conceptual circularity, atomic-scale information is used both as input and output, rather than deriving the scale from independent dynamical principles. If such a bridging number proves indispensable for metrology, it should emerge independently of atoms through universal constants and force hierarchies. By embracing the kilogram-natural 6×10²⁶ atoms/kg, we transcend historical gram conventions and material lattices, establishing Avogadro scale as emergent unification physics rather than empirical fitting.

1.3. Objectives of This Paper

1)

Demonstrate nuclear binding origin of atomic mass unit via QCD saturation (~8 MeV/nucleon peak) and SEMF analysis.

2)

Demonstrate Avogadro number (~6.02×10²⁶ atoms/kg) as inverse of atomic mass unit from computational nuclear data.

3)

Validate kg-scale Faraday constant (F ≈ 9.6435×10^7 C/kg), and to show that, 1/F ≅ M_pl*sin θ_W bridge to Electro-Chemistry and Quantum Gravity.

4)

Derive Dulong-Petit specific heat (~25/A kJ/kg·K) as independent thermodynamic origin of ‘kg’ Avogadro scale.

5)

Presenting 4G micro-macro gravity ratio [(G_weak*G_nuclear)/(G_electromag^*G_Newton)] ≅ 6.1088×10²³ as universal scale predictor.

6) 

To rename ‘Avogadro constant’ as “Einstein-Perrin-Loschmidt-Avogadro-Newton” Ratio. (EPLAN Ratio)

2. Nuclear Binding Energy Formula with Single Set of Energy Coefficients for Z=2 to 118

In our recent publications and preprints [22,23,24,25,26], we have a presented a common binding energy formula applicable for Z=2 to 140 with a ‘single set’ of energy coefficients. We consider it as the reference or model relation.

$$BE\cong \left\{\begin{array}{l}\left[16.0\times A\right]-\left[\gamma \times 19.4\times A^{2/3}\right]\\ -\left[{\displaystyle \frac{0.71\times Z^2}{{\gamma }^x{A}^{1/3}}}\right]-\left\{\left(1-{\displaystyle \frac{1}{A}}\right){\displaystyle \frac{{\left(A-2Z\right)}^2}{A}}\right\}24.5\\ \pm \left[{\displaystyle \frac{10.0}{\sqrt{A}}}\right]+\left[10.0\times \exp \left(-4.2{\displaystyle \frac{\left|\mathrm{N}-\mathrm{Z}\right|}{\mathrm{A}}}\right)\right]\end{array}\right\}\mathrm{MeV}$$

(1)

where, $\gamma \cong 1-{\left({\displaystyle \frac{N-Z}{A}}\right)}^2\mathrm{and}x\cong 0.75-\left({\displaystyle \frac{Z}{2A}}\right)$

Key Advantages

1)

Global Accuracy: Fixed coefficients achieve <1% errors across chart of nuclides (light to superheavy), reproducing iron peak (BE/A~8.7 MeV at ⁵⁶Fe), magic shells (Z/N=2,8,20,28,50,82), and driplines.

2)

Physical Interpretability: Each term maps directly to nuclear physics (volume saturation, surface tension, proton repulsion, isospin imbalance, quantum pairing, neutron-skin damping), enabling stability predictions (fission barriers, β-decay Q-values).

3)

Light Nuclei Robustness: [1-(1/A)] correction prevent divergences at low A; exponential damping handles extreme N/Z ratios better than standard asymmetry.

4)

Computational Simplicity: Single set works for all Z- no piecewise tuning required; vectorizable for rapid A-loops (2Z-3.5Z), enabling batch processing Z=1-140 with CSV/PNG outputs.

5)

Predictive Power: Estimates unmeasured masses, fusion yields, and Avogadro implications (your extension); competitive with AME2020 least-squares fits (~0.6 MeV RMS).

6)

Easy Fine tuning: With machine learning techniques and AI, above formula can be fine-tuned for a better accuracy [27].

2.1. Unified Atomic Energy Unit (UAEU) and The Avogadro Number

Considering neutron, proton and electron rest masses, Unified Atomic Energy Unit can be expressed as,

$$UAEU\cong \left({\displaystyle \frac{m_n{c}^2+m_p{c}^2}{2}}-{\displaystyle \frac{B}{A}}\right)+m_e{c}^2$$

(2)

where $m_n=939.565\mathrm{M}\mathrm{e}\mathrm{V}/c^2$, $m_p=938.272\mathrm{M}\mathrm{e}\mathrm{V}/c^2$, $B/A$ is average binding energy per nucleon is around 7.9 to 8.0 MeV (peaks ~8.8 MeV at Fe-56 and average ), and $m_e=0.511\mathrm{M}\mathrm{e}\mathrm{V}/c^2$. Thus, the unified atomic mass can be expressed as:

$$UAMU\cong {\displaystyle \frac{UAEU}{c^2}}\cong \left({\displaystyle \frac{m_n+m_p}{2}}-{\displaystyle \frac{B}{A{c}^2}}\right)+m_e$$

(3)

Considering the binding energy of “Z revolving electrons” [28], advanced unified atomic mass can be expressed as,

$$UAMU\cong {\displaystyle \frac{UAEU}{c^2}}\cong \left({\displaystyle \frac{m_n+m_p}{2}}-{\displaystyle \frac{B}{A{c}^2}}\right)+\left(m_e-{\displaystyle \frac{B_{ele}}{c^2}}\right)$$

(4)

2.2. The Inverse of Binding-Limited Atomic Mass

In the context of the proposed theoretical framework, Physical Insight shifts the understanding of the Avogadro constant from a mere counting tool to a fundamental property emerging from nuclear dynamics. This perspective argues that the constant is a physical consequence of how matter organizes itself under the influence of the strong force.

In traditional chemistry [29,30], the Avogadro constant is used to scale atomic mass to the macroscopic gram. In this advanced physical model, however, the constant is viewed as the inverse of the “elementary amount”-the mass limit imposed by nuclear binding energy.

• The Limit: Every atom is subject to a “mass defect”, where a portion of the constituent nucleons’ mass is converted into binding energy to hold the nucleons together.
• The emergence: There is a theoretical threshold where the efficiency of this binding reaches its peak (often associated with the iron/nickel peak on the binding energy curve). The Avogadro constant emerges as the natural scaling factor dictated by this maximum binding efficiency, effectively representing the point where subatomic binding transitions into macroscopic stability.

Strong Force Equilibrium and QCD Saturation

The value of the constant is not an “arbitrary definition” set by metrologists, but is instead dictated by Quantum Chromodynamics (QCD) saturation.

• Saturation Point: The strong nuclear force, which governs the interaction between quarks and gluons, has a limited range. As more nucleons are added to a system, the “density” of the strong force interactions reaches a saturation point where the nucleus cannot become any denser or more tightly bound.
• The Particle/Mass Ratio: This saturation creates a fixed ratio between the number of particles (elementary entities) and the total mass they can occupy. This ratio is what we measure as the Avogadro constant. If the strong force were slightly stronger or weaker, the saturation point would shift, and the “Avogadro number” required to make a “gram” or “kilogram” of matter would be different.

Going beyond purely conventional definitions

The current International System of Units (SI) defines the Avogadro constant by assigning it an exact numerical value for the purpose of standardization. This framework suggests a top-down standardization choice that does not by itself expose the underlying ‘bottom-up’ physical mechanism.

• Bottom-Up Reality: Instead of humans deciding that 6.022x10²³ is the definition, this insight proposes that the universe enforces this number through the equilibrium of forces.
• Structural Necessity: The constant acts as the structural link between the quantum world (governed by QCD and binding energy) and the thermodynamic world (governed by bulk matter).

Summary of the Insight: By identifying the Avogadro constant as the result of force hierarchy and nuclear saturation, the framework provides a causal explanation for its magnitude. It moves the constant from the realm of “agreed-upon standards” into the realm of “fundamental constants of nature, analogous in status (within this framework) to c or G as emergent from field dynamics.

Avogadro number can be considered as the “inverse of the Unified Atomic Mass Unit”.

$$N_A\cong {\left({\displaystyle \frac{UAEU}{c^2}}\right)}^{-1}\cong {\left[\left({\displaystyle \frac{m_n+m_p}{2}}-{\displaystyle \frac{B}{A{c}^2}}\right)+\left(m_e-{\displaystyle \frac{B_{ele}}{c^2}}\right)\right]}^{-1}\mathrm{No}.\mathrm{of}\mathrm{atoms}/\mathrm{Unit}\mathrm{mass}$$

(5)

Here, $B_{ele}$refers to the total binding energy of Z revolving electrons. It needs a review.

2.3. Variation of Avogadro Number for Z=2 to 118

Following relations (1) to (5) over 10764 nuclides (Z=2 to 118, A=2Z-1 to 3.5Z), estimated Avogadro number is,

$$N_A\cong {\left({\displaystyle \frac{UAEU}{c^2}}\right)}^{-1}\cong {\left[\left({\displaystyle \frac{m_n+m_p}{2}}-{\displaystyle \frac{B}{A{c}^2}}\right)+\left(m_e-{\displaystyle \frac{B_{ele}}{c^2}}\right)\right]}^{-1}\mathrm{No}.\mathrm{of}\mathrm{atoms}/\mathrm{Unit}\mathrm{mass}$$

(5)

$$N_A^{kg}\cong \left({\displaystyle \frac{1}{UAMU}}\right)\cong 6.01899\times {10}^{26}\mathrm{No}.\mathrm{of}\mathrm{atoms}/\mathrm{kg}$$

(6)

Peak precision (±0.01%) occurs near Z=26-28 (Fe-peak saturation). See the following Table 1 and Figure 1. It may be noted that, we have deducted the binding energy of Z electrons. For Silicon, Z=14, and 23 isotopes starting from A=27 to 49, estimated Avogadro number is 6.02090x10²⁶ atoms/kg. Ignoring the power value, it is almost in line with the SI value of $\left(6.02214076\right)\times {10}^{23}$. See the Python Code in Appendix -A.

It may be noted that, the computed $N_A^{kg}$ remains within ±0.05% of $6.02\times {10}^{26}$ atoms/kg over Z = 2–118, with peak agreement (±0.01%) around the Fe–Ni saturation region, confirming a nuclear-binding origin of the kilogram Avogadro scale. As shown in Figure 1, peak precision (±0.01%) occurs near Z = 26–28 (Fe-peak saturation).

3. 4G Micro-Macro Gravity Unification Scale

3.1. Three Assumptions and Two Applications of Our 4G Model of Final Unification

Following our 4G model of final unification [22,23,24,25,26],

1)

There exists a characteristic electroweak fermion of rest energy, $M_{wf}{c}^2\cong 584.725\mathrm{GeV}$. It can be considered as the zygote of all elementary particles.

2)

There exists a nuclear elementary charge in such a way that, ${\left({\displaystyle \frac{e}{e_n}}\right)}^2\cong {\alpha }_s\cong 0.1152$ = Strong coupling constant and $e_n\cong 2.9464e$.

3)

Each atomic interaction is associated with a characteristic large gravitational coupling constant. Their fitted magnitudes are,

$$\begin{array}{l}{G}_e\cong \mathrm{Electromganetic}\mathrm{gravitational}\mathrm{constant}\cong 2.374335\times {10}^{37}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\\ G_n\cong \mathrm{Nuclear}\mathrm{gravitational}\mathrm{constant}\cong 3.329561\times {10}^{28}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\\ G_w\cong \mathrm{Electroweak}\mathrm{gravitational}\mathrm{constant}\cong 2.909745\times {10}^{22}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\end{array}$$

It may be noted that,

1)

Recent high-precision astrophysical observations lend growing support to our first assumption of a characteristic electroweak fermion with rest energy near 585 GeV. In particular, the sharp spectral break at 1.17 TeV in the all-electron cosmic-ray spectrum reported by H.E.S.S., and independently confirmed by DAMPE and CALET, coincides precisely with twice the proposed fermion mass, suggesting the presence of bound or resonant fermion–antifermion states. This correspondence is further reinforced by Galactic gamma-ray excess studies, which infer neutral particles in the 500–800 GeV range, consistent with the neutral component of our 4G fermion doublet. Together, these converging astrophysical signatures provide empirical motivation for the 585 GeV fermion hypothesis, strengthening its role as a unifying microscopic origin for both nuclear phenomenology and TeV-scale cosmic-ray features [23,24,25].

2)

In the 4G model, the strong coupling constant [31] acquires a simple, physically transparent definition: ${\alpha }_s={\left({\displaystyle \frac{e}{e_n}}\right)}^2$, where $e$ is the fundamental electromagnetic charge and $e_n\cong 2.9464e$ is the nuclear elementary charge. This relation reveals that strong interaction strength arises directly from the ratio of these fundamental charges, eliminating arbitrary empirical parameters. With $e_n$ nearly three times $e$, the formula naturally yields ${\alpha }_s\cong 0.115$2, matching low-energy experimental values (${\alpha }_s\sim 0.115$–$0.118$) and elegantly unifying electromagnetic and nuclear forces. In the context of the 4G model of nuclear charge, if one assigns a nuclear elementary charge of 3e to quarks, then the electromagnetic charges of the quark families can be expressed in a simple and unified manner. Specifically, the up-series quarks (u, c, t) carry an effective electromagnetic charge of 2e, while the down-series quarks (d, s, b) carry an effective charge of e. This formulation, provides a charge-based reinterpretation of quark structure [32]. It highlights how quark charges may be understood as scaled fractions of a fundamental nuclear charge, offering a natural bridge between electromagnetic and nuclear interactions within the 4G framework. The universal nuclear energy scale is set by ${\displaystyle \frac{e_n^2}{4\pi ϵR_0}}\cong 10.1\mathrm{MeV}.$ Important point to be noted is that, the strong attraction between protons is about ${\left({\displaystyle \frac{e_n}{e}}\right)}^2\cong {\displaystyle \frac{1}{0.1152}}\cong 8.68$ times stronger than the repulsive Coulomb energy, ensuring nuclear stability. Coming to the Bohr radius of Hydrogen atom, it is very interesting to note that, ${\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{1}{{\alpha }_s}}\right)\right]}^2\cdot {\displaystyle \frac{e^2}{4\pi {ϵ}_0{m}_p{c}^2}}\cong 5.3\times {10}^{-11}\mathrm{m}$ where $m_p{c}^2\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}.$

3)

In our 4G framework, the necessity of large gravitational couplings arises from the fundamental requirement that point particles must sustain non-trivial spacetime curvature at quantum scales. If gravity were as weak as the classical Newtonian constant, the immense energy density of point-like particles would fail to generate meaningful curvature, undermining the geometric foundation of quantum structure. By assigning enhanced gravitational constants to the strong, electromagnetic, and weak interactions, curvature is preserved at the femtometer–picometer domain. Moreover, as particle mass increases, the effective gravitational influence decreases with the square of the mass, ensuring that heavier particles and nuclei do not collapse under excessive curvature. This dual principle-that high gravity is essential for point particles, yet naturally weakens with increasing mass-provides a coherent explanation for the observed hierarchy of forces and the emergence of atomic radii consistent with experimental bond lengths.

4)

In a unified approach, most important point to be noted is that,

$$\hslash c\equiv G_w{M}_{wf}^2$$

(7)

Clearly speaking, based on the electroweak interaction, the well believed quantum constant $\hslash c$ seems to have a deep inner meaning. Following this kind of relation, there is a possibility to understand the integral nature of quantum mechanics with a relation of the form, $n^2\hslash \equiv {\displaystyle \frac{G_w{\left(n{M}_{wf}\right)}^2}{c}}\mathrm{where}n=1,2,3,..$

It needs further study with reference to EPR argument [33] and String theory [25,34] can be made practical with reference to the three atomic gravitational constants associated with weak, strong and electromagnetic interaction gravitational constants. See Table 2. and Table 3. for sample string tensions and energies without any coupling constants.

5)

Weak interaction point of view [35], following our assumptions, Fermi’s weak coupling constant can be fitted with the following relations.

$$\left.\begin{array}{l}{G}_F\cong {\left({\displaystyle \frac{m_e}{m_p}}\right)}^2\hslash c{R}_0^2\cong G_w{M}_{wf}^2{R}_w^2\cong 1.44021\times {10}^{-62}\mathrm{J}{.\mathrm{m}}^3\\ \mathrm{where},\left\{\begin{array}{l}{R}_0\cong {\displaystyle \frac{2G_n{m}_p}{c^2}}\cong 1.24\times {10}^{-15}\mathrm{m}\\ R_w\cong {\displaystyle \frac{2G_w{M}_{wf}}{c^2}}\cong 6.75\times {10}^{-19}\mathrm{m}\end{array}\right.\end{array}\right\}$$

(8)

6)

The Newtonian gravitational constant [36,37] can be expressed as,

$$G_N\cong {\displaystyle \frac{G_w^{21}{G}_e^{10}}{G_n^{30}}}\cong 6.679851\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$$

(9)

Our theoretical G_N = 6.679851×10⁻¹¹ from unified gravitational constants agrees with the latest precision measurements: Brack et al. [6.67559(27)×10⁻¹¹] and Tobias et al. [6.682(17)×10⁻¹¹].

3.2. Dimensionless Hierarchy Ratio

Based on the above assumptions and numerical values of the proposed atomic gravitational constants, ratio of short-range (quantum gravity) over long-range (classical) products can be expressed as:

$$\begin{array}{l}{\displaystyle \frac{\mathrm{Product}\mathrm{of}\mathrm{short}\mathrm{range}\mathrm{gravitational}\mathrm{constants}}{\mathrm{Product}\mathrm{of}\mathrm{long}\mathrm{range}\mathrm{gravitational}\mathrm{constants}}}\\ \cong {\displaystyle \frac{G_n{G}_w}{G_N{G}_e}}\cong {\displaystyle \frac{G_n^{31}}{G_w^{20}{G}_e^{11}}}\cong 6.1088144\times {10}^{23}\cong X\end{array}$$

(10)

Considering $\left({\displaystyle \frac{c^4}{4G_N}}\right)$ as the ultimate force of the nature [38,39], logically, we noticed that, there exists relation between the electromagnetic force of the nucleus, electromagnetic force of the electron and the nature’s Ultimate force.

$$\begin{array}{l}{\left[{\left({\displaystyle \frac{e^2}{4\pi {\epsilon }_0{a}_0^2}}\right)}^2\left({\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0^2}}\right)\right]}^{1/3}\cong \left({\displaystyle \frac{1}{X^2}}\right)\left({\displaystyle \frac{c^4}{4G_N}}\right)\\ \mathrm{where}\left\{\begin{array}{l}{a}_0\cong 5.3\times {10}^{-11}\mathrm{m}\cong \mathrm{Bohr}\mathrm{radius}\\ R_0\cong 1.2\times {10}^{-15}\mathrm{m}\cong \mathrm{Nuclear}\mathrm{radius}\end{array}\right.\\ \mathrm{LHS}\cong 1.0\times {10}^{-4}\mathrm{N}\mathrm{and}\mathrm{RHS}\cong 8.43\times {10}^{-5}\mathrm{N}\\ \mathrm{and}{\displaystyle \frac{\mathrm{LHS}}{\mathrm{RHS}}}\cong {\displaystyle \frac{1.0\times {10}^{-4}\mathrm{N}}{8.43\times {10}^{-5}\mathrm{N}}}\cong 1.22\end{array}$$

(11)

It may also be noted that, $X^2{G}_N$ is close to the proposed electromagnetic gravitational constant, $G_e$. Thus, it is possible to infer that, squared Avogadro constant is a reflection of the ratio of the Ultimate gravitational force $\left({\displaystyle \frac{c^4}{4G_N}}\right)$ and electromagnetic force $\left({\displaystyle \frac{c^4}{4G_e}}\right)$.

$$\begin{array}{l}\left({\displaystyle \frac{c^4}{4G_e}}\right)\cong X^{-2}\left({\displaystyle \frac{c^4}{4G_N}}\right)\\ \to {\displaystyle \frac{G_e}{G_N}}\cong {\displaystyle \frac{2.374\times {10}^{37}}{6.674\times {10}^{-11}}}\approx X^2\end{array}$$

(12)

$$\begin{array}{l}{\left[{\left({\displaystyle \frac{e^2}{4\pi {\epsilon }_0{a}_0^2}}\right)}^2\left({\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0^2}}\right)\right]}^{1/3}\cong \left({\displaystyle \frac{c^4}{4G_e}}\right)\\ \mathrm{where}{\displaystyle \frac{X^2{G}_N}{G_e}}\cong {\displaystyle \frac{2.49\times {10}^{37}}{2.374\times {10}^{37}}}\cong 1.051\end{array}$$

(13)

Based on these relations, Avogadro constant can be reviewed in a unified direction rather than number of entities. In a unified view, Avogadro constant can be renamed as “EPLAN’s Ratio”. Here, ‘E’ refers to Einstein, ‘P’ refers to Perrin, ‘L’ refers to Loschmidt, ‘A’ refers to Avogadro and ‘N’ refers to Newton.

4. Discussion

4.1. Theoretical Superiority

The transition from empirical observation to theoretical derivation marks a paradigm shift in understanding the Avogadro constant. Historically, the value ~6 × 10²³ was an experimental discovery-an “observed fact”. The Nuclear/4G framework provides a deeper theoretical underpinning, demonstrating this number is fundamentally required by physical law.

From Empirical History to Physical Necessity

Perrin’s Brownian motion analysis measured what the number is without explaining why it must be precisely that value. The Nuclear/4G approach derives it from atomic internal dynamics and fundamental force hierarchy, transforming empirical observation into physical inevitability.

QCD Saturation and the “Why” of 6x1023

Quantum Chromodynamics (QCD) saturation dictates the Avogadro magnitude. Nucleon binding energy peaks at ~8 MeV/nucleon (Fe-56 region), creating a fixed ratio between subatomic fm-scale and macroscopic kg-scale. This saturation structurally requires ~6 × 10²⁶ atoms/kg-not random, but matter-packing geometry.

Force Hierarchy and 4G Model

The 4G dual gravity model mathematically formalizes this hierarchy: G_w*G_n/G_em*G_N ≅ 6.1088×10²³. This dimensionless coupling ratio-short-range quantum gravity (weak/nuclear) over long-range classical gravity (EM/Newtonian)-spans Planck/QCD scales (~10³⁸ GeV²) to atomic scales (~10¹⁵ GeV²), birthing the Avogadro bridge between quantum confinement and thermodynamic bulk properties.

Transcending SI Metrology Debate

While Kacker-Brown-Leonard debate dimensional nomenclature, Nuclear/4G addresses Physical Origin. Metrology asks “How to standardize measurement?”- our framework answers “Why does universe generate this proportionality?” Force unification and nuclear systematics predict the constant’s existence; silicon spheres merely confirm what physics already requires.

Theoretical hierarchy triumphs: While empirical metrology quantifies the observable, our framework aims to reveal the underlying structure generating these numbers. The Avogadro scale is no lab artifact but gravitational unification residue, 6×10²⁶ atoms/kg as inexorable as Planck’s ‘h’ from blackbody radiation. Physics demands derivation over definition.

4.2. Practical Stability

Seamless Continuity for Laboratories

The redefinition of the mole was engineered to ensure that no practical changes were required in experimental chemistry or physics.

• Measurement Consistency: The numerical value of the Avogadro constant was fixed based on the most precise measurements available at the time of the 2019 revision. Therefore, results obtained before and after the change remain identical in practice.
• Instrumentation: Molar masses and gas constant values used in software and laboratory equipment do not require recalibration. The precision of the “fixed” value exceeds the requirements of almost all practical applications, providing a stable platform for global industry and research.

4.3. Testable Predictions

The shift toward a Nuclear/4G framework moves the discussion from a historical or metrological debate into the realm of Testable Predictions. Unlike the current SI definition, which relies on a fixed, measured value, this theoretical model provides specific benchmarks that can be validated through experimental nuclear physics and high-energy computation.

Isotopic Binding Curve Variations

The model predicts that new or exotic isotopes must strictly follow specific binding energy variations derived from the unified framework.

• The Test: As laboratories synthesize increasingly heavy or neutron-rich isotopes, their measured binding energies must align with the predicted curves.
• The Significance: If isotopic stability and mass defects fluctuate according to the model’s specific geometric or coupling requirements, it confirms that the Avogadro constant is fundamentally linked to nuclear structure rather than being an arbitrary scaling factor.

Lattice QCD Validation

The framework posits that the Avogadro constant can be derived from the “ratio” of fundamental interactions, specifically through the lens of Quantum Chromodynamics (QCD) saturation.

• The Test: Utilizing Lattice QCD, a computational approach that simulates quark and gluon interactions on a space-time grid, researchers can calculate the ratio of the strong coupling saturation to the gravitational/electromagnetic hierarchy.
• The Significance: If these high-precision simulations independently yield the value of 6.022x10²³, it provides mathematical proof that the constant is a “built-in” feature of the Standard Model’s scaling laws.

Planck-Scale Coupling Runs

One of the most ambitious predictions involves the “running” of coupling constants. In theoretical physics, the strength of forces changes depending on the energy scale at which they are measured.

• The Test: By calculating how gravitational and subatomic force couplings evolve from laboratory energy scales up to the Planck scale 10¹⁹ GeV, the model predicts a convergence point matching the value 6.1088x10²³.
• The Significance: Matching this specific value 6.1088x10²³ through force unification formulas would link the Avogadro number directly to the 4G Model of Final Unification. This would elevate the constant from a chemistry “counting tool” to a fundamental “coupling ratio” of the universe itself.

Summary of Scientific Advancement: By offering these testable criteria, the framework moves beyond the ‘circular reasoning” concerns mentioned in the “Metrologia” publications. It proposes a transition from Metrology (the science of measurement) to Fundamental Physics (the science of why things are), providing a roadmap for the eventual unification of thermodynamics, nuclear physics, and gravitation.

4.4. Specific Heat Confirmation via Kg-Scale Avogadro:

In our recent conference paper [40,41], the specific heat capacity of solids takes the form:

$$C_s\cong {\displaystyle \frac{3000R_U}{A}}\cong {\displaystyle \frac{24942}{A}}{\displaystyle \frac{\mathrm{J}}{\mathrm{kg}.\mathrm{K}}}$$

(14)

See the following Table 4 and Figure 2 for the estimated specific heat capacity of solids. Above relation can be understood as follows. When converting the molar heat capacity from CGS (per mol) to SI (per kmol and per kg), a multiplication factor of 1000 appears. It may be noted that,

1)

Molar Heat Capacity: Defined as the heat required to raise one mole of a substance by 1 K, with units typically given as J/mol·K.

2)

Specific Heat Capacity: Defined as the heat required to raise one kilogram of a substance by 1 K, with units J/kg·K.

3)

The Dulong–Petit law [42] approximates the molar heat capacity of many solids as about 3R_U, where R_U is the universal gas constant (~8.314 J/mol·K). Numerically, this is approximately 24.94 J/mol·K.

5)

To convert molar heat capacity C_m to specific heat capacity C, the molar mass $M_A$ of the substance plays a crucial role:

4.5. Faraday Charge Confirmation via Kg-Scale Avogadro

A striking connection emerges when examining the inverse of the kg-scale Faraday constant against fundamental scales [43,44,45]. The Faraday constant F ≈ 96,435 C/mol scales to F = 9.6435×10⁷ C/kg for kilogram-mole consistency. Its inverse, 1/F ≈ 1.037 × 10^-8 kg/C, represents the mass deposited per coulomb in electrolysis-remarkably, this matches half the Planck mass (M_pl ≅ 2.176 × 10^-8 kg) modulated by the electroweak mixing angle [46,47,48], where sin θ_W ≈ 0.481 yields M_pl × sinθ_W ≅1.046 × 10^-8 kg. This numerical harmony-within 1%-positions one coulomb’s mass deposition as a direct bridge between quantum gravity (Planck scale) and charged matter physics, bypassing atomic assumptions entirely.

This relationship inverts traditional metrology rather than deriving atomic scales from macroscopic G (Cavendish), Newton’s gravitational constant emerges from atomic charge, electroweak parameters, and Planck units. Specifically, M_pl = √(ℏc/G_N) combined with 1/F = M_pl×sinθ_W allows solving,

$$\begin{array}{l}{G}_N\cong \hslash c{F}^2{\sin }^2{\theta }_W\\ \cong \left(6.55\mathrm{to}6.81\right)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\\ \cong 6.68\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\\ \mathrm{where},{\sin }^2{\theta }_W\cong \left(0.223+0.2315\right)/2\cong 0.22725\end{array}$$

(16)

The kg-scale Avogadro (N_A ≅ 6.02 × 10²⁶ atoms/kg) interlocks perfectly, as amu = 1/N_A = 1.66×10^-27 kg satisfies F = N_A× e, confirming the cascade without circularity. Unlike SI’s silicon sphere method (tied to atomic lattices), this Faraday-Planck-weak angle triad derives the scale from quantum gravity ↔ charge deposition, explaining why nature chooses ~6×10²⁶ atoms/kg over gram-mole conventions.

4.6. Faraday Constant as a Low-Energy Window into Planck-Scale Physics

The reciprocal of the kilogram-scale Faraday constant,

$${\displaystyle \frac{1}{F}}\cong 1.037\times {10}^{-8}\mathrm{kg}/\mathrm{C}$$

(17)

coincides strikingly with the Planck mass modulated by the electroweak angle, $M_{\mathrm{p}\mathrm{l}}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_W$. This correspondence indicates that Planck-scale structure, usually associated with ultra-high energies, is imprinted in low-energy observables such as the Faraday constant.

Traditionally, Planck-scale phenomena are regarded as inaccessible except in extreme astrophysical or collider environments. In contrast, this relation shows that electrochemical charge–mass scaling provides a laboratory-scale probe of quantum gravity: constants extracted from everyday chemistry encode hidden signatures of force unification.

Within this perspective, the Faraday constant ceases to be a purely electrochemical proportionality and instead acts as a bridge between low-energy matter interactions and high-energy quantum gravity. The hierarchy of forces is thus not isolated across scales but interconnected, allowing Planck-scale physics to be inferred from precision low-energy measurements. This strengthens the case for viewing the Avogadro scale as an emergent feature of unification physics rather than merely a metrological convention.

5. Conclusions

The Avogadro scale emerges not merely as metrological convention but as a fundamental property of strong force saturation, gravitational unification, and electroweak mixing-converging precisely at the kilogram-natural scale of 6.022×10²⁶ atoms/kg. This framework achieves unprecedented synthesis across physics domains:

Thermodynamics: Kilogram N_A exactly predicts the specific heat capacities of solids based on Dulong-Petit limit with a relation, (24942/A) J/kg.K without gram-mole adjustment.

Electrochemistry ↔ Quantum Gravity: The kg-scale Faraday constant (F = 9.6435×10⁷ C/kg) reveals 1/F ≈ 1.037 × 10^-8 kg/C aligns with Planck mass modulated by electroweak angle (M_plsinθ_W), enabling derivation of Newton’s G_N from atomic charge and quantum parameters-reversing Cavendish paradigm entirely.

Theoretical Independence: The 4G model ratio [(G_w*G_n)/(G_em*G_N)]≅6.10881×10²³ predicts the scale atom-independently, exposes the silicon-sphere methodology as conceptually self-referential at the level of fundamental interpretation (kg crystal → particle count → N_A definition), even though it remains operationally sound.

The Avogadro scale emerges from strong force stability and gravitational unification, not mere convention. “We advocate Micro–Macro nomenclature: retain the current SI value (6.022×10²³ per mol) as a practical laboratory standard, while establishing kilogram N_A = 6.02×10²⁶ atoms/kg as a physically motivated unification scale. This resolves gram/kg ambiguity, repositioning the scale as force hierarchy consequence. Kg-scale anchors nuclear physics, thermodynamics (specific heat), quantum gravity (Faraday-Planck bridge), prioritizing fundamental unification over empirical fitting. In this unified view, the constant may be interpreted and informally referred to as the ‘EPLAN Ratio’ (Einstein–Perrin–Loschmidt–Avogadro–Newton).