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Empirical Relation of the Fine-Structure Constant with the Transference Number Concept ()

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*Journal of Modern Physics*,

**9**, 2346-2353. doi: 10.4236/jmp.2018.913149.

1. Introduction

Previously, we tried to explain quantum physics using classical thermodynamics [1] [2] . However, these discussions were lacking evidential support, prompting us to search for this evidence.

Solid-oxide fuel cells (SOFCs) directly convert the chemical energy of fuel gases, such as hydrogen and methane, into electrical energy. SOFCs use a solid-oxide film as the electrolyte, and oxygen ions serve as the main charge carriers. Typically, yttria-stabilized zirconia (YSZ) is used as the electrolyte material in these cells. The open-circuit voltage (OCV) of the YSZ electrolyte is equal to the Nernst voltage (V_{th}) of 1.15 V at 1073 K. However, using samaria-doped ceria (SDC) electrolytes, the OCV is approximately 0.8 V. The low OCV was calculated using Wagner’s equation, which is based on the chemical equilibrium theory. Wagner’s equation [3] [4] is

${J}_{{O}_{2}}=-\frac{RT}{16{F}^{2}L}{\displaystyle {\int}_{\mathrm{ln}p{O}_{2}^{anode}}^{\mathrm{ln}p{O}_{2}^{cathode}}\frac{{\sigma}_{el}{\sigma}_{ion}}{{\sigma}_{el}+{\sigma}_{ion}}\text{d}\mathrm{ln}p{O}_{2}}$ (1)

where
${J}_{{O}_{2}}$ and pO_{2} are the O_{2} flux and the O_{2} partial pressure, respectively;
$p{O}_{2}^{cathode}$ and
$p{O}_{2}^{anode}$ are the O_{2} partial pressures at the cathode and anode, respectively; R, T, and F are the gas constant, the absolute temperature, and Faraday’s constant, respectively; L is the thickness of the membrane or film; and σ_{el} and σ_{ion} are the conductivities of the electrons and oxygen vacancies, respectively.

From Equations (1), Equations (2) and (3) can be deduced [5] :

$OCV={V}_{th}-{R}_{i}{I}_{i}$ (2)

where R_{i} and I_{i} are the ionic resistances of the electrolyte and the ionic current, respectively.

$OCV=\frac{RT}{4F}{\displaystyle {\int}_{\mathrm{ln}p{O}_{2}^{anode}}^{\mathrm{ln}p{O}_{2}^{cathode}}{t}_{ion}\text{d}\mathrm{ln}p{O}_{2}}$ (3)

Parameter t_{ion} is expressed as

${t}_{ion}=\frac{{\sigma}_{ion}}{{\sigma}_{el}+{\sigma}_{ion}}$ (4)

However, s_{el} is a function of the O_{2} partial pressure [6] :

${\sigma}_{el}={\sigma}_{ion}{\left(\frac{p{O}_{2}}{p{O}_{2}^{\ast}}\right)}^{-\frac{1}{4}}$ (5)

where
$p{O}_{2}^{\ast}$ corresponds to the oxygen partial pressure at which t_{ion} = 1/2. When t_{ion} is constant in the electrolytes,

$OCV={t}_{ion}\times {V}_{th}$ (6)

The low OCV was thought to be due to the low value of the ionic transference number (t_{ion}). However, experimentally, I_{i} in Equation (2) is negligible [6] [7] [8] [9] [10] . Considering the direction of the electrical field, there are serious problems in Wagner’s equation [5] [10] . Therefore, the voltage loss should be explained by other reasons.

Over the past two decades, the understanding of nonequilibrium thermodynamics has been enhanced by fluctuation and dissipation theorems such as the Jarzynski and Crooks relations [11] [12] . The autonomous Maxwell’s demon concept was proposed by Jarzynski [13] , and we independently discovered the equation for this concept [14] . In our equation, t_{ion} remains important. In addition, we determined the empirical relationship and discussed the physical meaning of this empirical relationship.

2. Equation for Autonomous Maxwell’s Demons

2.1. Main Problems in Wagner’s Equation

According to Michael Faraday, the direction of the electrical field is from the anode to the cathode. In the 1950s, Wagner studied mixed conductors with positively and negatively charged ions. However, Wagner’s equation was used for doped ceria electrolytes in which there are two negative carriers (oxygen ions and electrons). The ionic current (I_{i}) and electron drift current (I_{e_drift}) flow from the cathode to the anode. Only the electron diffusion current (I_{e_diffusion}) can flow from the anode to the cathode. A schematic drawing of the directions of I_{i}, I_{e_drift} and I_{e_diffusion} is presented in Figure 1. According to Weppner [15] , there should be a delay for I_{e_diffusion}:

$\tau =\frac{{L}^{2}}{\stackrel{\u02dc}{D}}$ (7)

where τ, L, and
$\stackrel{\u02dc}{D}$ are the equilibrium time, sample length, and chemical diffusion coefficient, respectively. According to Wang [16] ,
$\stackrel{\u02dc}{D}$ is 3.2 × 10^{−6} cm^{2}/s at 1073 K. Therefore, using 1-mm-thick SDC electrolytes, τ should be 52 min at 1073 K. However, such a delay has been never observed during the transient process, so the existence of I_{e_diffusion} can be disproved [5] [10] .

2.2. Autonomous Maxwell’s Demons Explanation

We discovered the following empirical equation using SDC electrolytes [14] :

$OCV={V}_{th}-\frac{{E}_{a}}{2e}$ (8)

where e is elemental charge. E_{a} is the ionic activation energy, which is 0.7 eV for SDC electrolytes. Therefore, the OCV in Equation (1) is 0.80 V (=1.15 V − 0.7 eV/2e). This equation is explained in Figures 2-4. The Boltzmann distribution of oxygen ions in the electrolyte at 1073 K is displayed in Figure 2. The ions with energies exceeding E_{a} become carriers (hopping ions). Figure 3 presents an incorrect carrier distribution. The Boltzmann distribution cannot be separated using passive filters because of the phenomenon known as “Maxwell’s demon”, and an accurate distribution is provided in Figure 4. The loss of Gibbs energy is illustrated in Figure 3. Equation (8) is correct, when t_{ion} is zero. When t_{ion} is not zero, the equation for autonomous Maxwell’s demon [14] [17] is

$OCV={V}_{th}-\left(1-{t}_{ion}\right)\times \frac{{E}_{a}}{2e}$ (9)

The direction of I_{e_drift} is the same as that of I_{i}.

Figure 1. Schematic drawing indicating the directions of I_{i}, I_{e_drift} and I_{e_diffusion} for the open-circuit case.

Ions with energies exceeding the ionic activation energy are converted into charge carriers (i.e., hopping ions).

Figure 2. Boltzmann distribution at 1073 K.

This distribution is forbidden according to Maxwell’s demon.

Figure 3. Forbidden distribution of hopping ions.

The shape of the distribution in this figure should be the same as the shape of the distribution in Figure 4.

Figure 4. Correct distribution of hopping ions.

3. Empirical Relations of the Fine-Structure Constant with the Transference Number Concept

The fine-structure constant (α) is

$\alpha =\frac{{e}^{2}}{4\text{\pi}{\epsilon}_{0}\hslash c}$ (10)

where π, ħ, c and ε_{0} are the mathematical constant pi, the reduced Planck constant, the speed of light in a vacuum and the electric constant or permittivity of free space, respectively.

${R}_{K}=\frac{h}{{e}^{2}}$ (11)

Here, R_{K} is the von Klitzing constant.

${Z}_{0}=\frac{1}{{\epsilon}_{0}c}$ (12)

Here, Z_{0} is the characteristic impedance. Therefore,

$\alpha =\frac{{Z}_{0}}{2{R}_{k}}$ (13)

When the interaction coefficient is 1/137, the transference number should be 136/137. The parameter t_{ion} is expressed as

${t}_{ion}=\frac{{\sigma}_{ion}}{{\sigma}_{el}+{\sigma}_{ion}}=\frac{{R}_{el}}{{R}_{el}+{R}_{ion}}$ (14)

where R_{e} and R_{ion} are the resistance values for electrons and ions, respectively. Here, σ_{ion} can be defined even when the ions are blocked to move. In Equation (13), we assumed that the main carriers are electrons that must move with two unknown carriers belonged to the environment. Then, the transference number unknown carriers is

${t}_{unknown}=\frac{\frac{{R}_{el}}{2}}{\frac{{R}_{el}}{2}+{R}_{unknown}}=\frac{{R}_{el}}{2\left(\frac{{R}_{el}}{2}+{R}_{unknown}\right)}$ (15)

where R_{unknown} is the resistance of unknown particles belonged to the environment. Equation (15) is similar to Equation (13), and α^{-}^{1} is 137.035. Therefore,

$\frac{\frac{{R}_{el}}{2}}{{R}_{unknon}}=136.035$ (16)

Next, we consider the mobility (μ):

$\sigma =ne\mu $ (17)

where n is the number of carriers.

$2{n}_{el}={n}_{unknown}$ (18)

Here, n_{el} and n_{unknown} are the number of electrons and the number of unknown particles, respectively.

$\mu =\frac{e}{{m}^{*}\tau}$ (19)

Here, m^{*} is the carrier effective mass, and τ is the average scattering time. When τ is constant,

$\frac{{m}_{unknown}}{{m}_{el}}=136.035$ (20)

where m_{el} and m_{unknown} are the mass of electrons and the mass of unknown particles, respectively, and m_{el} is 0.511 MeV. Therefore, we must search for the mass with an energy value of 69.50 MeV (=0.511 × 136). The rest mass of a negatively charged pion has an energy of 139.57 MeV. Then, consider the following equation:

${m}_{{\pi}^{-}}=2{m}_{quark}+{m}_{el}$ (21)

where m_{p}_{-} and m_{quark} are the mass of the negatively charged pion and the mass of quarks, respectively. From Equation (21), m_{quark} is 69.53 MeV, which is similar to 69.50 MeV. Therefore, our empirical equation is

$\frac{{m}_{quark}}{{m}_{el}}=\frac{1}{\alpha}-1$ (22)

4. Discussion

We proposed a model in which there should be one free electron and two quarks belonged to the environment. Electrons receive the 1/137 energy of photons in the presence of an electrical field. Two quarks receive the 136/137 energy of photons. However, movement of the two quarks with the usual energy is blocked for unknown reasons. Thus, the 136/137 energy of photons should diffuse to the environment, meaning that the transference number of the space for electrons is 136/137, instead of 1, in the presence of an electrical field. We proposed that the quantity of 257,934 ohms (from the calculation of 258,123 − (377/2)) should be measured.

When two quarks can move with higher energy, the interaction coefficient of quarks should be 136/137 and the transference number of quarks should be 1/137. This is the explanation for the strong interaction. The diffusion response time of the mixed electronic and quark conductors depend exponentially on the distance. So, Yukawa potential can be explained.

5. Conclusion

Using the transference number concept, we proposed an empirical relationship in which the fine-structure constant is related to the mass ratio of electrons and quarks. This empirical equation is determined to be correct with a 99.96% (69.50/69.53) accuracy. Furthermore, we proposed that the quantity of 257,934 ohms should be measured.

Conflicts of Interest

The authors declare no conflicts of interest.

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