Understanding Atomic Mass Unit, Avogadro Number, Atomic Radii and Electrochemical Equivalents with 4G Model of Final Unification

1. Introduction

Following Abdus Salam et al.[1] and Roberto Onofrio [2], in our recent and previous published papers [3,4,5,6,7,8,9,10], we proposed the possible existence of three large gravitational constants assumed to be associated with the electromagnetic, strong and weak interactions. By following this idea, in analogy with Planck scale, as an immediate result, it seems possible to have three different characteristic string amplitudes corresponding to electromagnetic, strong and weak interactions. In this way, String theory [11,12] can be shaped to a model of elementary particle physics associated with 3+1 dimensions. As the subject under consideration deals with 4 different gravitational constants, our model can be called as 4G model of final unification or Microscopic Quantum Gravity. By stressing the existence of nuclear elementary charge and highlighting other applications like understanding elementary magnetic moments, understanding quantum constants and radiation constants, fitting Fermi’s weak coupling constant, understanding nuclear binding energy with 4 simple terms and one energy coefficient, understanding quarks’ electromagnetic charges and estimation of electron neutrino rest mass, in this paper, we make an attempt to estimate the unified atomic mass unit, Avogadro number, atomic electrochemical equivalents and atomic radii. It may be noted that, starting from sections 1 to 3 are in review as a ‘short letter’. Parts of sections 3 and 6 are also in review.

2. Three simple assumptions of 4G model of final unification

Our three assumptions are,

1)

There exists a characteristic weak fermion of rest energy, $M_{wf}{c}^2\cong 584.725\mathrm{GeV}$. It can be considered as the zygote of all elementary particles [7].

2)

There exists a nuclear elementary charge in such a way that, ${\left(\frac{e}{e_n}\right)}^2\cong {\alpha }_s\cong 0.1152$= Strong coupling constant [13] and $e_n\cong 2.9464e$.

3)

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

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

Unification point of view, most important relation is $\hslash c\equiv G_w{M}_{wf}^2.$Planck length can be addressed with $\sqrt{G_N\hslash /c^3}\cong \sqrt{G_w{G}_N}{M}_{wf}/c^2.$In a unified approach, Newtonian gravitational constant [14] can be addressed with $G_N\cong \frac{G_w^{21}{G}_e^{10}}{G_n^{30}}\cong 6.679851\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}.$Strange and interesting ratio associated with Avogadro like number is $\frac{G_w{G}_n}{G_N{G}_e}\cong \frac{G_n^{31}}{G_w^{20}{G}_e^{11}}\cong 6.1085\times {10}^{23}.$Clearly speaking, ratio of the product of short ranges forces and the product of long range forces somehow seems to be connected with the observed Avogadro like number. Proton-electron mass ratio can be addressed with $\frac{m_p}{m_e}\cong \frac{G_n^3}{G_w^2{G}_e}.$High energy physics point of view, we would like to emphasize the point that, currently believed strong coupling constant is inherently connected with ordinary electromagnetic charge and the proposed nuclear charge. It needs a very critical review. It can be addressed with ${\alpha }_s\cong {\left(\frac{e}{e_n}\right)}^2\cong \frac{G_e^4{G}_w^6}{G_n^{10}}\cong \frac{G_N^{1/3}{G}_e^{2/3}}{G_w}\cong \mathrm{0.1151937.}$Considering weak and electromagnetic gravitational constants, neutron life time [13] can be fitted with a very simple relation of the form,$t_n\cong \frac{G_e^2{m}_n^2}{G_w\left(m_n-m_p\right)c^3}\cong 875.0\sec .$Based on the proposed assumptions and with reference to the cosmic force of the form $\left(c^4/G_N\right)$, for the three atomic interactions, at atomic scale, interaction strength can be defined as,

For weak interaction, $\left[\left(c^4/G_w\right)/\left(c^4/G_N\right)\right]\cong G_N/G_w\cong 2.294\times {10}^{-33}.$

For nuclear or strong interaction, $\left[\left(c^4/G_n\right)/\left(c^4/G_N\right)\right]\cong G_N/G_n\cong 2.0\times {10}^{-39}.$

For electromagnetic interaction, $\left[\left(c^4/G_e\right)/\left(c^4/G_N\right)\right]\cong G_N/G_e\cong 2.811\times {10}^{-48}.$

Based on these values, at atomic scale, it seems logical to say that, based on $\left(c^4/G_N\right)$ –

a)

Interaction range is inversely proportional to interaction strength.

b)

Effective magnitude of the operating gravitational constant is, $G_x\cong \frac{G_N}{\mathrm{Interaction}\mathrm{strength}}$

3. Atomic, nuclear and sub-nuclear applications

Clearly speaking gravity associated with weak interaction seems to play a crucial role in exploring the mystery of quantum phenomena. Fermi’s weak coupling constant can be addressed with,

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

Important relation pertaining to final unification can be expressed as, $\hslash c\cong G_w{M}_{wf}^2.$With reference to the assumed gravity associated with electron, characteristic radius of electron can be addressed with $R_{ele}\cong \frac{2G_e{m}_e}{c^2}\cong 4.813\times {10}^{-10}\mathrm{m}\cong 0.48\mathrm{nm}.$ We are working on understanding the applications of this length at nano-scale. The most puzzling root mean square radius of proton can be addressed with $R_p\cong \sqrt{\left(\frac{4\pi {\epsilon }_0{\hslash }^2}{e_n^2{m}_p}\right)\left(\frac{\hslash }{m_{p}c}\right)}\cong \sqrt{\frac{{\alpha }_s}{\alpha }}\left(\frac{\hslash }{m_{p}c}\right)\cong 0.835\mathrm{fm}.$Quantum of magnetic flux can be addressed [15] with $\frac{h}{e}\cong \frac{e_n}{e}\sqrt{\frac{{\mu }_0}{4\pi }\left(G_e{m}_e^2\right)}.$Most fundamental Planck’s radiation constant can be addressed with $h\cong \sqrt{\left(\frac{e_n^2}{4\pi {\epsilon }_{0}c}\right)\left(\frac{G_e{m}_e^2}{c}\right)}\cong \frac{e_n}{e}\sqrt{\left(\frac{e^2}{4\pi {\epsilon }_{0}c}\right)\left(\frac{G_e{m}_e^2}{c}\right)}.$ Reduced Planck’s constant can be addressed with $\hslash \cong \left(\frac{e}{e_n}\right)\frac{G_n{m}_p^2}{c}.$ Proton magnetic moment can be expressed as ${\mu }_p\cong \frac{e{G}_n{m}_p}{2c}.$ Bohr magneton can be expressed as ${\mu }_e\cong \frac{e\hslash }{2m_e}\cong \sqrt{\left(\frac{e{G}_n{m}_p}{2c}\right)\left(\frac{e{G}_e{m}_e}{2c}\right)}\cong \left[\left(\frac{e}{m_e}\right)÷\left(\frac{e_n}{m_p}\right)\right]{\mu }_p.$Clearly speaking, ratio of Bohr magneton to proton magnetic moment is close to the ratio of specific charge ratio of electron having a charge$e$ and proton having a charge$e_n\cong 2.9464e.$Electron’s anomalous magnetic moment can be addressed with a factor $\left[1+\left(\frac{\alpha }{2\pi }\right)\right]{\mu }_{Bohr}\cong \left[1+\left(\frac{e^2}{4\pi {\epsilon }_{0}hc}\right)\right]{\mu }_{Bohr}.$ Fine structure ratio can be addressed with $\alpha \cong \left(\frac{e^2}{4\pi {\epsilon }_0\hslash c}\right)\cong \frac{e{e}_n}{4\pi {\epsilon }_0{G}_n{m}_p^2}.$Based on these expressions, it seems possible to understand the discreteness of elementary angular momentum with the discrete nature of elementary particle rest masses.

With reference to currently believed Semi Empirical Mass Formula (SEMF), we call our formula as ‘Strong and Electroweak Mass Formula’ (SEWMF).It constitutes 4 simple terms and single binding energy potential or coefficient addressed with$\frac{e_n^2}{8\pi {\epsilon }_0\left(0.62\mathrm{fm}\right)}\cong 10.1\mathrm{MeV}$where $\frac{G_n{m}_p}{c^2}\cong 0.62\mathrm{fm}\mathrm{and}\frac{\alpha }{2\pi }\cong \frac{e^2}{4\pi {\epsilon }_{0}hc}\cong \sqrt{\frac{13.6\mathrm{eV}}{10.1\mathrm{MeV}}}.$This is a very interesting observation. Fortunately this energy coefficient is very close to the average rest energy of 3 Up quarks and 3 Down quarks. Simplified formula can be expressed as, $BE\cong \left(A-A_{free}-A_{rad}-A_{asy}\right)10.1\mathrm{MeV}$

where $\left\{\left.\begin{array}{l}{A}_{free}\cong \left(2-\frac{N}{Z}\right)+\left[0.0016\left(\frac{Z^2+A^2}{2}\right)\right]\\ A_{rad}\cong A^{1/3},A_{asym}\cong \frac{{\left(A_s-A\right)}^2}{A_s}\end{array}\right\}\right.$and$\left\{\begin{array}{l}{A}_s\cong 2Z+0.0016{\left(2Z\right)}^2\cong 2Z+0.0064Z^2\\ \frac{m_p}{M_{wf}}\cong \frac{938.272\mathrm{MeV}}{584725\mathrm{MeV}}\cong \left(\frac{\sqrt{{\left(m_{\pi }{c}^2\right)}^0{\left(m_{\pi }{c}^2\right)}^{\pm }}}{\sqrt{{\left(m_z{c}^2\right)}^0{\left(m_w{c}^2\right)}^{\pm }}}\right)\\ \cong \left(\frac{\sqrt{134.98\text{×}139.57}\mathrm{MeV}}{\sqrt{80379.0\text{×}91187.6}\mathrm{MeV}}\right)\cong 0.0016\end{array}\right\}$

First term is a volume term, second term is associated with free nucleons pertaining to electroweak interaction, third term is a radial term and fourth one is an asymmetry term about a light house like stable mass number associated with weak interaction coefficient 0.0016. Here it is very important to note that, 0.0016 is the ratio of proton rest energy of 938.272 MeV and the assumed weak fermion of rest energy 584.725 GeV. It is noticed that, the ratio of mean mass of pions to mean mass of weak bosons is accurately matching with 0.0016. Thus, we have clearly established the combined and independent roles of strong and weak interactions in basic nuclear physics [16]. With this kind of approach, cold nuclear fusion like complicated concepts can be understood independent of Coulombic repulsions [17,18].We are working on finding alternative expressions for estimating the number of free nucleons. Approximate new expressions are [19],$A_{free}\approx 0.0016\left[\left(Z^2+N^2+{\left(\frac{Z^2}{N}\right)}^2\right)-{\left(\frac{A-2Z}{2}\right)}^2\right],$$A_{free}\approx \left[0.0016A\left(\sqrt{NZ}+\sqrt{A}+\sqrt{A_s-2Z}\right)\right]$and $A_{free}\approx \left[0.004156\left(N-1\right)\left(Z-1\right)\right]$where $\sqrt{10.1\mathrm{MeV}/584.725\mathrm{GeV}}\cong \mathrm{0.004156.}$Based on the proposed nuclear charge, nuclear stability can also be understood with$A_s\cong \left\{{\left(Z+2.9464\right)}^{1.2}\right\}-1$$\mathrm{and}{\left(\frac{e_n}{e}\right)}^{1/6}\cong \mathrm{1.2.}$This can be confirmed with Green’s beta stability relation [20,21], $Z\cong \frac{A_s}{2}-\frac{0.2A_s^2}{A_s+200}.$Proceeding further, Myer and Swiatecki’s mass formula energy coefficients can be fixed with a reference potential energy of $\frac{e_n^2}{4\pi {\epsilon }_0}{\left(0.62\mathrm{fm}\right)}^{-1}\cong 20.2\mathrm{MeV}.$Volume and surface energy coefficients can be addressed with$\left(1-2{\alpha }_s\right)20.2\mathrm{MeV}$ and $\left(1-{\alpha }_s\right)20.2\mathrm{MeV}$ respectively. Interesting point to be noted is that all other minor energy coefficients are interlinked with expressions like$\frac{1}{2}\left({\alpha }_s\right)20.2\mathrm{MeV},$$\left({\alpha }_s\right)20.2\mathrm{MeV},$$\left(2{\alpha }_s\right)20.2\mathrm{MeV}$and $\left(3{\alpha }_s\right)20.2\mathrm{MeV}.$Isospin coefficients of volume and surface energy terms can be fitted with $2\mp {\left[\frac{\left(1+{\alpha }_s\right)}{\left(1-{\alpha }_s\right)}\right]}^2{\alpha }_s.$We have developed the above relations with reference to the following advanced mass formula [22].

$$\left.BE\cong \left\{\left[1+\left(\frac{4k_v}{A^2}\right)\left|T_z\right|\left(\left|T_z\right|+1\right)\right]a_v*A\right\}+\left\{\left[1+\left(\frac{4k_s}{A^2}\right)\left|T_z\right|\left(\left|T_z\right|+1\right)\right]a_s*A^{\frac{2}{3}}\right\}+\left\{a_c*\left(\frac{Z^2}{A^{1/3}}\right)\right\}+\left\{f_p*\frac{Z^2}{A}\right\}+E_p\right\}$$

where, $T_z\cong 3\mathrm{rd}\mathrm{component}\mathrm{of}\mathrm{isospin}=\frac{1}{2}\left(Z-N\right)$

$\left\{\left.\begin{array}{l}{a}_v=-15.4963\mathrm{MeV},a_s=17.7937\mathrm{MeV}\\ k_v=-1.8232,k_s=-2.2593\\ a_c=0.7093\mathrm{MeV},f_p=-1.2739\mathrm{MeV}\\ d_n=4.6919\mathrm{MeV},d_p=4.7230\mathrm{MeV}\\ d_{np}=-6.4920\mathrm{MeV}\end{array}\right\}\right.$and$\left.\left\{\begin{array}{l}\mathrm{for}\left(Z,N\right)\mathrm{Odd},E_p\cong \frac{d_n}{N^{1/3}}+\frac{d_p}{Z^{1/3}}+\frac{d_{np}}{A^{2/3}}\\ \mathrm{for}\left(\mathrm{Odd}Z,\mathrm{Even}N\right),E_p\cong \frac{d_p}{Z^{1/3}}\\ \mathrm{for}\left(\mathrm{Even}Z,\mathrm{Odd}N\right),E_p\cong \frac{d_n}{N^{1/3}}\\ \mathrm{for}\left(\mathrm{Even}Z,\mathrm{Even}N\right),E_p\cong 0\end{array}\right.\right\}$

Medium and heavy atomic nuclides’ charge radii [23,24] can be addressed with $R\left({}_{Z,N}\right)\cong \left[Z^{1/3}+{\left(\sqrt{NZ}\right)}^{1/3}\right]0.62\mathrm{fm}.$Considering our proposed $e_n\cong 2.9464e$ as a characteristic quark charge, it seems possible to understand the integral nature [25,26,27] of quarks’ electromagnetic charge. A detailed paper is in review on magnetic and radiation constants, quarks’ electromagnetic charges and other applications [28].

a)

Up, Charm and Top quark’s electromagnetic charge is $+\frac{2}{3}{e}_n\cong +2e.$

b)

Down, Strange and Bottom quark’s electromagnetic charge is $-\frac{1}{3}{e}_n\cong -e.$

c)

Quarks having an electromagnetic charge of $\pm 2e$ and $\mp e$ can be called as integral charge quarks.

d)

There exists no repulsion between any two particles having same kind of nuclear charge.

For example, proton can be assumed to have one up quark pair and one anti down quark. Neutron can be assumed to have two anti down quarks and one anti up quark. Positive and negative pions can be assumed to have one Up quark and one down quark or one anti up quark and anti down quark. Neutral pions can be assumed to have one Up quark pair or one down quark pair. Thus, decay of neutrons, protons and pions can be understood very easily. For the case of neutron decay, one anti up quark transforms to one down quark and releases one electron. For the case of proton decay, one up quark transforms to one anti down quark and releases one positron. In case of charged pions, up quark plays a vital role in the decay process. For positive pion, up quark transforms positive electron or positive muon and generates a neutral pion. For negative pion, anti up quark transforms to electron or negative muon and generates a neutral pion. With reference to Up quark pair, neutral pion can decay into 2 positive particles and two negative particles. With reference to down quark pair, neutral pion can decay into one positive particle and one negative particle. In this way, neutron, proton and pions decay can be understood very easily. In all the cases, up quark of charge $\left(\pm 2e\right)$ seems to play a crucial role in final decay. In the given Table 1, it may be noted that, in the second column, quark charges expressed in ‘( )’ indicate their own internal decay in the third column. This proposal can be applied to all 6 quarks without any difficulty.

4. Understanding the rest mass of electron neutrino

Based on the above applications, we noticed that, there exists an electron neutrino [29,30] or gravitino like a neutral fermion of rest mass, $m_{xf}\cong \left(m_e^6/m_p^5\right)\cong 4.365\times {10}^{-47}\mathrm{kg}.$Similar to the proposed, $m_{xf}\cong \left(m_e^6/m_p^5\right),$ current recommended value of electron neutrino mass can be fitted with, $\frac{m_e^3}{m_p^2}\cong 2.7\times {10}^{-37}\mathrm{kg}\cong 0.15\mathrm{eV}/c^2.$Mass ratio of proposed and believed electron neutrino is${\left(m_e/m_p\right)}^3$and needs a review. Considering $M_{wf}{c}^2\cong 584.725$ GeV as the weak fermion, electron and electron neutrino rest masses can be addressed with $m_e\cong \left(\frac{G_w}{G_n}\right)M_{wf}$ and $m_{xf}\cong \left(\frac{\sqrt{G_w{G}_N}}{G_n}\right)M_{wf}$respectively where $G_w,$$G_n$and $G_N$represent electroweak, nuclear and Newtonian gravitational constants respectively. In a verifiable approach with other relations, our adopted procedure is mainly associated with interpreting the large numbers as ${\left(m_p/m_e\right)}^{10}\cong {\left(m_e/m_{xf}\right)}^2$and${\left(m_p/m_e\right)}^{12}\cong {\left(m_p/m_{xf}\right)}^2.$ With reference to strong coupling constant, we noticed that, $\ln {\left(m_e/m_{xf}\right)}^2\cong \ln \left(G_w/G_N\right)\cong 1/{\alpha }_s^2\cong {\left(0.1152\right)}^{-2}.$With reference to the proposed electron neutrino rest mass, Planck length can be addressed with a very interesting relation of the form,$\sqrt{G_N\hslash /c^3}\cong G_n{m}_{xf}/c^2.$

Considering nuclear unit charge, unit volume, Fermi’s weak coupling constant and based on proton, electron and proposed neutrino rest masses, neutron life time[13] can be fitted with, $t_n\cong \left(\frac{m_p{m}_e}{m_{xf}\left(m_n-m_p\right)}\right)\left(\frac{4\pi }{3}{R}_0^3\right)\left(\frac{\hslash }{G_F}\right)\cong 884.0\sec .$ By considering the relations, $G_F\cong {\left(\frac{m_e}{m_p}\right)}^2\hslash c{R}_0^2$and $R_0\cong \frac{2G_n{m}_p}{c^2},$ neutron life can be fitted with, $t_n\cong \frac{8\pi G_n{m}_p^4}{3m_{xf}{m}_e\left(m_n-m_p\right)c^3}\cong \frac{4\pi m_p^3}{3m_{xf}{m}_e\left(m_n-m_p\right)}\left(\frac{R_0}{c}\right).$ In this way, with our 4G model of final unification electron neutrino rest mass can be inferred and can be confirmed by fitting it with the neutron life time. Accuracy point of view, neutron life time seems to depend on the characteristic nuclear radius and nuclear volume. With further research, rest masses of electron neutrino and gravitino can be understood in a unified approach.

5. Understanding and estimating atomic radii

With reference to 4G model of final unification, we emphasize the point that, there exists 3 different large gravitational constants for weak, strong and electromagnetic interactions. In this paper, by considering the effective behavior of strong and electromagnetic interactions, similar to Rutherford's nuclear radii formula [23], we propose a very simple formula for understanding atomic radii. It can be expressed as, $R_A\cong A^{1/3}\left(\frac{2\sqrt{G_n{G}_e}{M}_u}{c^2}\right)\cong A^{1/3}\times 32.86\mathrm{pm}$where $A$represents atomic mass number, $G_n,G_e$represent strong and electromagnetic gravitational constants respectively and $M_u$represents the unified atomic mass unit [14]. Starting from A= (1 to 340), obtained rage of atomic radii is (32.86 to 229.35) pm. It is in line with the measured range of various atomic radii [31,32,33,34,35,36] and needs a fine tuning. In this context, for 3 to 7 periods, we have developed an approximate formula, $R_A\cong \left[4-\left(A/Z\right)\right]{\left(Z_{fp}/Z\right)}^2{A}^{1/3}\times 32.86\mathrm{pm}.$Here,$Z_{fp}$ represents first element of any period and $Z$ represents any element in that period. For Helium and the 2^nd period, $R_A\cong \left[4-\left(A/Z\right)\right]\left(Z_{fp}/Z\right)A^{1/3}\times 32.86\mathrm{pm}.$We sincerely appeal the field experts to see the possibility of implementing this idea with possible changes for a better understanding at fundamental level. See the following Figure 1.

6. Understanding Unified atomic mass unit, Avogadro number and Electrochemical equivalents

We have noticed a very simple and accurate relation for understanding and estimating the Unified atomic mass unit$M_u$. In terms of energy, it can be considered as ‘average rest energy of nucleons minus average binding energy per nucleon of all atomic nuclides plus electron rest energy’. By considering the ‘binding energy per electron’ of all atomic electrons there is a scope for correction to the estimated$M_u$. It needs a review. It may be noted that, ‘average binding energy per nucleon’ of all atomic nuclides can be estimated in three ways. One simple way is, to consider the average of experimental ‘binding energy per nucleon’ values of all observed 2500 atomic nuclides. Second possible way is, starting from A=4 to 294, selecting all 291 mass numbers having maximum binding energy, finding their binding energy per nucleon and taking the average of 291 values. Alternatively, based on our 4G model of final unification, starting from A= 5 to 340, considering isobars, based on our three assumptions and considering maximum binding energy associated with any stable mass number, we have noticed a very simple relation [37]. It can be expressed as,

$$\begin{array}{l}\left.\begin{array}{l}{\left(BE\right)}_{A_s}\cong \left(A_s-\left(\left(\frac{1}{2}\right)+0.000935A_s^2\right)-A_s^{1/3}\right)10.1\mathrm{MeV}\\ \frac{{\left(BE\right)}_{A_s}}{A_s}\cong A_s^{-1}\left(A_s-\left(\left(\frac{1}{2}\right)+0.000935A_s^2\right)-A_s^{1/3}\right)10.1\mathrm{MeV}\end{array}\right\}\\ \mathrm{where}\sqrt{\frac{G_w}{G_n}}\cong \sqrt{\frac{m_e{c}^2}{M_{wf}{c}^2}}\cong \sqrt{\frac{0.511\mathrm{MeV}}{584725\mathrm{MeV}}}\cong 0.000935\end{array}$$

With reference to stability against beta decay, proposed coefficient 0.000935 can be addressed with a relation of the form, $\sqrt{\left(\frac{m_p{c}^2}{M_{wf}{c}^2}\right)\left(\frac{m_e{c}^2}{m_p{c}^2}\right)}\cong \sqrt{\left(0.00160464\right)\left(0.0054462\right)}\cong \mathrm{0.000935.}$It needs a review. See Figure 2 and Figure 3 for the estimated binding energies of $A_s=4\mathrm{to}294$. It may be noted that, with reference to experimental binding energy of isobars having maximum binding energy [38], average error in estimated binding energy seems to be 0.93 MeV for 291 atomic nuclides and corresponding standard deviation is 3.97 MeV. It needs a review with respect to the following relations [20,21],$Z\cong \frac{A_s}{1+\sqrt{0.0064A_s+1}}\cong \frac{A_s}{2}-\frac{0.2A_s^2}{A_s+200}$ and the proposed asymmetry term, $A_{asy}\cong \frac{{\left(A_s-A\right)}^2}{A_s}.$ Table 2, Figure 2 and Figure 3 clearly demonstrate the existence of our proposed weak fermion of rest energy 584.725 GeV.

Even though our estimation is on lower side for A=4 and some other atomic nuclides, considering A=5 to 340, average of ‘average binding energy per nucleon’ is around 7.93 MeV. Thus, currently believed unified atomic mass unit [14] can be addressed with,

$$\begin{array}{l}{M}_u{c}^2\cong \left(\frac{m_n{c}^2+m_n{c}^2}{2}\right)-{〈\frac{B{E}_{Max}}{A}〉}_{Ave}+m_e{c}^2\cong 931.51\mathrm{MeV}\cong 1.6605494\times {10}^{-27}\mathrm{kg}.c^2\\ \mathrm{where},\left\{\begin{array}{l}{m}_n{c}^2=\mathrm{Rest}\mathrm{energy}\mathrm{of}\mathrm{neutron},m_p{c}^2=\mathrm{Rest}\mathrm{energy}\mathrm{of}\mathrm{proton}\\ m_e{c}^2=\mathrm{Rest}\mathrm{energy}\mathrm{of}\mathrm{electron}\end{array}\right.\end{array}$$

Avogadro number [14,39] can be addressed with ‘inverse of the unified atomic mass unit’ as,

$$N_A\cong {\left(M_u\right)}^{-1}\cong 6.0221034\times {10}^{26}\mathrm{atoms}/\mathrm{kg}\cong 6.0221034\times {10}^{23}\mathrm{atoms}/\mathrm{gram}$$

Thus, number of atoms per kg can be called as Kg-mole and number of atoms per gram can be called as gram-mole.

Atomic electrochemical equivalents (ECE) [40] can be expressed as,

$$EC{E}_{\left(A,v\right)}\cong \frac{A{M}_u}{ve}\mathrm{kg}/\mathrm{C}\cong \left(\frac{A}{v}\right)\left(\frac{M_u}{e}\right)\mathrm{kg}/\mathrm{C}$$

where $A,v$represent Atomic mass number and valency number respectively. It may be noted that, $\left(\frac{M_u}{e}\right)$can be called as ‘Faraday mass’.

$$M_F\cong \frac{M_u}{e}\cong 1.0364334\times {10}^{-8}\mathrm{kg}/C\cong 1.0364334\times {10}^{-5}\mathrm{gram}/\mathrm{C}.$$

In a very simple view, ECE can be defined as the ratio of mass $\left(A{M}_u\right)$of the ion to the charge $\left(ve\right)$ carried by the ion. See Table 3 for the estimated electrochemical equivalents. Readers are encouraged to refer and cross check at ‘https://environmentalchemistry.com/yogi/periodic/’.

Inverse of Faraday mass can be called as ‘Faraday charge’,

$$F_C\cong \frac{e}{M_u}\cong 96484731.7\mathrm{C}/\mathrm{kg}\cong 96484.732\mathrm{C}/\mathrm{gram}$$

With reference to Faraday mass, Planck mass $M_p\cong \sqrt{\hslash c/G_N}$and weak coupling angle [13,14]${\mathrm{Sin}}^2{\theta }_W$, we noticed a very strange relation,

$$\frac{G_N{M}_F^2}{\hslash c}\cong {\mathrm{Sin}}^2{\theta }_W\mathrm{and}\frac{M_F^2}{M_p^2}\cong {\mathrm{Sin}}^2{\theta }_W$$

$$\frac{M_F}{M_p}\cong \mathrm{Sin}{\theta }_W\mathrm{and}{M}_F\cong \mathrm{Sin}{\theta }_W{M}_p$$

This is a very simple relation and seems to connect gravity, weak interaction, electromagnetic interaction and quantum mechanics. It may be an accidental coincidence also. With reference to the recommended value of the Newtonian gravitational constant, obtained value of the weak coupling angle is${\mathrm{Sin}}^2{\theta }_W\cong \mathrm{0.2267732.}$Even though its recommended value is 0.23122, based on the rest masses of neutral and charged weak bosons, its defined value is 0.22339. To a very good approximation, Newtonian gravitational constant can be expressed as,

$$G_N\cong {\mathrm{Sin}}^2{\theta }_W\left(\frac{\hslash c}{M_F^2}\right)\cong \left(6.574728\mathrm{to}6.8051776\right)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$$

7. Conclusion

By following our 4G model of final unification, there is a scope for understanding and estimating the values of Unified atomic mass unit, Avogadro number and Electrochemical equivalents. We humbly appeal the science community to recommend our views for further research.