Poisoning Effect of CO: How It Changes Hydrogen Electrode Reaction and How to Analyze It Using Differential Polarization Curve

Abstract

The hydrogen electrode reaction (HER) on Pt electrode in a H₂SO₄ solution when CO gas was injected/stopped was studied using polarization resistance curve. In order to elucidate and confirm the CO poisoning effect, a few curve techniques were proposed. Applying them, the kinetic parameters such as the number of electrons transferred (z) and the cathodic transfer coefficient (α_c) were determined. The HER in a 0.5 mol dm⁻³ H₂SO₄ solution saturated with H₂ was confirmed as a reversible reaction having z = 2. When the above solution was injected with CO, the reversible HER changed to an irreversible reaction having z = 1 and α_c ≈ 0.6. Once we stopped the CO injection, alteration from the irreversible to quasireversible reaction was gradually made after several cyclic polarizations. The proposed curve techniques can provide a reliable way to determine the kinetic parameters changing among reversible, irreversible, and quasireversible reactions.

1. Introduction

The hydrogen electrode reaction (HER) is one of the most fundamental electrolytic reactions in the field of electrochemistry [1,2,3,4,5,6,7,8,9,10]. This redox reaction consists of hydrogen oxidation reaction (hor) as an anodic branch reaction and hydrogen evolution reaction (her) as a cathodic branch reaction. The exchange current density of HER, which is the magnitude of the reciprocal reaction between hor and her, is widely used as a kinetic parameter to evaluate catalysts in technical fields such as fuel cells and water electrolysis. It is well known that power generation by fuel cells will be very important and become widespread in the very near future [2,3,7,8,9,10]. Since the hydrogen consumed in fuel cells is mainly produced by the steam reforming process of methane, it may contain byproducts such as CO and CO₂. Many papers reported that even a small amount of CO can poison the platinum catalyst and reduce its efficiency [11]. This poisoning phenomenon is said to be mainly caused by the firm adhesion of CO as CO_ad to the platinum electrode surface. A more detailed explanation of the CO poisoning effect from various aspects is needed to make the fuel cell fully functional. For an example, it is necessary to explain the degradation of the electrode not only from the surface inactivation caused by CO adsorption but also from the physical factors derived from it.

This paper explains the degradation of platinum catalysts from the viewpoint of polarization resistance. As far as the authors know, there are few papers that provide explanations with detailed polarization resistance curves. Exceptionally, A.C. Chialvo et al., devised their own theory of polarization resistance and applied it to HER. It was an experimental and theoretical study of CO poisoning HER [12,13,14,15,16,17,18,19,20,21,22]. However, their theory was limited to a single point at j = 0, and there was no mention or consideration from the viewpoint of the whole curve. This paper, first of all, shows the continuous change of HER from reversible to irreversible reaction caused by CO injection. Using the polarization curve and its polarization resistance curve, which is derived from the differential polarization method (DPM) [23,24,25,26], the CO poisoning effect is discussed from two main points: (1) how to analyze the curve change caused by CO injection and (2) how to determine the kinetic parameters of HER in CO-free, CO-injected, and CO-stopped solution.

2. Results

2.1. Variation of the Open Circuit Potential with Time

To detect a variation of the open circuit potential, $E_{\mathrm{ocp}}\left(t\right)$, the experiments were simulated in three environments:

Environment (I): 0.5 mol dm⁻³ H₂SO₄ solution saturated with H₂ only (abbreviation; H₂ + H₂SO₄ solution; as a reference);

Environment (II): the above H₂SO₄ solution injected with continuous CO bubbling (abbreviation; H₂ + H₂SO₄ + CO solution; as CO-contaminated solution);

Environment (III): the above H₂SO₄ solution when the CO-injection was stopped (abbreviation; H₂ + H₂SO₄ + CO-CO solution; as CO-restored solution).

An arrival time to the steady state by monitoring the $E_{\mathrm{ocp}}\left(t\right)$ was investigated. The result is shown in Figure 1.

The $E_{\mathrm{ocp}}$ showing $\approx 1.0 \mathrm{V}$ sharply descended to 0.0 V when the H₂ bubbling was started. The steady state showing $E_{\mathrm{ocp}}=0.0 \mathrm{V}$ was obtained in the period of $0.5 \mathrm{ks}\lesssim t\lesssim 2 \mathrm{ks}$. The $E_{\mathrm{ocp}}\left(t\gtrsim 2 \mathrm{ks}\right)$ increased to $\approx$0.34 V at start of CO injection and decreased to $\approx$0.25 V at the gradual stop of CO injection. In this experiment, it was found that the steady state condition is obtained at $t\gtrsim 4 \mathrm{ks}$ for the environment (I), $3 \mathrm{ks}\lesssim t\lesssim 5 \mathrm{ks}$ for the environment (II), and $t>5.4 \mathrm{ks}$ for the environment (III).

2.2. $E_{\exp }\left(j\right)$ and $h_{\exp }\left(j\right)$

The polarization curves, $E_{\exp }\left(j\right)$, in the above three solutions were measured at the immersion periods of 1.7 ks for environment (I), 4.9 ks for environment (II), and 5.4 ks for environment (III). Their polarization resistance curves, $h_{\exp }\left(j\right) \left(=\mathrm{d} E_{\exp }\left(j\right)/\mathrm{d} j\right)$, were calculated using the finite difference method in a software.

2.2.1. $E_{\exp }\left(j\right)$ and $h_{\exp }\left(j\right)$ in the Environment (I)

Figure 2 shows the voltammogram (CV with seven cycles) curves in the H₂ + H₂SO₄ solution.

The starting point is (A), and the finishing point is (E). The potential-reverse operation is carried out at points of (B) and (D). We can see that $E_{\exp }\left(j\right)$ follow same route and formed no hysteresis loop. The dark red curve of (B)–(C)–(D) was selected for analysis as a representative. Figure 3 shows that its $h_{\exp }\left(j\right)$ is drawn in a deep blue bold line together with the experimental $h_{\exp }\left(j\right)$ (sky blue fine).

The important readings in Figure 2 and Figure 3 are summarized in

Table 1. Experimental readings of $E\left(0\right)$ in Figure 2 and $h_{\exp }\left(0\right)$, $w_{\mathrm{c}} j_{{ \mathrm{H}}^{+},\mathrm{L}}$, $w_{\mathrm{a}} j_{{ \mathrm{H}}_2,\mathrm{L}}$, and $l/\kappa$ in Figure 3.

Item	Reading	Remarks
$E\left(0\right)/ \mathrm{V} \mathrm{vs}. \mathrm{SHE}$	0	(C) and (E) in Figure 2
$h_{\exp }\left(0\right)/{\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$	$2.2\times {10}^{-2}$	(C) and (E) in Figure 3, $\approx \left[h_{\exp }\left(\lesssim {10}^{-3}\right)\right]$
$\left|w_{\mathrm{c}} j_{{ \mathrm{H}}^{+},\mathrm{L}}\right|/{\mathrm{mA} \mathrm{cm}}^{-2}$	${10}^2<$	we cannot observe it in Figure 3
$w_{\mathrm{a}} j_{{ \mathrm{H}}_2,\mathrm{L}}/{\mathrm{mA} \mathrm{cm}}^{-2}$	$\approx 0.7$	vertical line having (D) in Figure 2 and Figure 3
${\left(l/\kappa \right)}_{\mathrm{c}}/{\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$	$\approx {10}^{-3}$	asymptotic horizontal line; $\lesssim$(B) in Figure 3

. Symbols used in this paper are shown and explained in Appendix A.

A question will occur to readers that it is impossible to read the value of $h_{\exp }\left(0\right)$ in logarithm expression. The above answer is that the $h_{\exp }\left(0\right)\approx h_{\exp }\left(\lesssim {10}^{-3}\right)$ is acceptable because the $h_{\exp }\left(\lesssim {10}^{-3}\right)$ is almost horizontal line. We can see that the j at (D) is $\approx 0.7$ mA cm⁻², which is related to the limiting anodic current density of H₂. We cannot observe the j relating to the limiting cathodic current density of H⁺ due to the over-scaled value.

2.2.2. $E_{\exp }\left(j\right)$ and $h_{\exp }\left(j\right)$ in the Environment (II)

Similarly, the $E_{\exp }\left(j\right)$ and its $h_{\exp }\left(j\right)$ in the H₂ + H₂SO₄ + CO solution are shown in Figure 4 and Figure 5, respectively.

Except for the first route of $\left(\mathrm{F}\right)\to \left(\mathrm{G}\right)$ (downward, red line), we can see that all curves (pink fine) repeat similar hysteresis loop of $\left(\mathrm{G}\right)\to \left(\mathrm{H}\right)\to \left(\mathrm{I}\right)\to \left(\mathrm{J}\right)\to$ $\left(\mathrm{K}\right)\left(\mathrm{or} \left(\mathrm{G}\right)\right)\to \left(\mathrm{L}\right) \left(\mathrm{or} \left(\mathrm{H}\right)\right)\to \left(\mathrm{I}\right)$. The representative tracks of $\left(\mathrm{G}\right)\to \left(\mathrm{H}\right)\to \left(\mathrm{I}\right)$ (upward, green line) and $\left(\mathrm{I}\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{K}\right)$ (downward, blue line) are shown with their bold colors. Characteristic readings appeared on Figure 4 and Figure 5 are summarized in

Table 2. Experimental readings in Figure 4 and Figure 5 are shown. Explanations of items and symbols are shown in Section 3.9 and Appendix A.

Item	Reading	Remarks
$E\left(0\right)/ \mathrm{V} \mathrm{vs}. \mathrm{SHE}$	0.24	(H) or (L) in Figure 4
	0.02	(J) in Figure 4
$h_{\exp }\left(0\right)/{\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$	$\approx 43$	(H) or (L) in Figure 5
	$\approx 190$	(J) in Figure 5
$\left|w_{\mathrm{c}1} j_{{ \mathrm{H}}^{+},\mathrm{L}}\right|/{\mathrm{mA} \mathrm{cm}}^{-2}$	${10}^2<$	we cannot observe it in Figure 5
$\left|w_{\mathrm{c}2} j_{{ \mathrm{CO}}_2,\mathrm{L}}\right|/{\mathrm{mA} \mathrm{cm}}^{-2}$	$1.3\times {10}^{-3}$	red vertical line in Figure 5
$w_{\mathrm{a}5} j_{{ \mathrm{H}}_2,\mathrm{L}}$$+w_{\mathrm{a}2} j_{{ \mathrm{CO}}_{\mathrm{ad}},\mathrm{L}}/{\mathrm{mA} \mathrm{cm}}^{-2}$	$1.3\times {10}^{-3}$	green vertical line in Figure 5
$\left(w_{\mathrm{a}2}+w_{\mathrm{a}5}\right) j_{{ \mathrm{H}}_2,\mathrm{L}}+w_{\mathrm{a}4} j_{{ \mathrm{CO}}_{\mathrm{ad}},\mathrm{L}}/{\mathrm{mA} \mathrm{cm}}^{-2}$	$3\times {10}^{-3}$	green vertical line in Figure 5
${\left(l/\kappa \right)}_{\mathrm{c}}/{\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$	$\approx {10}^{-3}$	asymptotic line; (G) or (K) in Figure 5

.

2.2.3. $E_{\exp }\left(j\right)$ and $h_{\exp }\left(j\right)$ in the Environment (III)

The results in the H₂ + H₂SO₄ + CO-CO solution are shown in Figure 6 for $E_{\exp }\left(j\right)$ and Figure 7 for $h_{\exp }\left(j\right)$.

The complex shape of $E_{\exp }\left(j\right)$ in Figure 5 has disappeared, and the similar shape of Figure 1 appears. Close observation on $E_{\exp }\left(j\right)$ leads to an interesting fact that there is open loop in upward and close loop in downward. The $h_{\exp }\left(j\right)$ also shows complex shape, but there are two remarkable points: (1) the crowded route of $\left(\mathrm{P}\right)\to \left(\mathrm{Q}\right)\to \left(\mathrm{R}\right)$ in Figure 7 is almost similar to the route of $\left(\mathrm{B}\right)\to \left(\mathrm{C}\right)\to \left(\mathrm{D}\right)$ in Figure 3, and (2) the first line of $\left(\mathrm{M}\right)\to \left(\mathrm{N}\right)$, on which the Tafel slope was satisfied, was disappeared and gathered into $\left(\mathrm{Q}\right)\to \left(\mathrm{R}\right)$ after several CV. Characteristic readings appeared in Figure 4 and Figure 5 are summarized in

Table 3. Experimental readings in Figure 2 and Figure 3.

Item	Reading	Remarks
$E\left(0\right)/ \mathrm{V} \mathrm{vs}. \mathrm{SHE}$	$\approx 0$	(Q) in Figure 6
$h_{\exp }\left(0\right)/{\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$	$0.03~0.06$	(Q) in Figure 7
$\left|w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}\right|/{\mathrm{mA} \mathrm{cm}}^{-2}$	${10}^2<$	we cannot observe it in Figure 3
$w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}/{\mathrm{mA} \mathrm{cm}}^{-2}$	$0.6~0.9$	vertical line having (P) in Figure 7
${\left(l/\kappa \right)}_{\mathrm{c}}/{\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$	$\approx {10}^{-3}$	asymptotic horizontal line; (B) in Figure 7

.

3. Discussion

3.1. Single Electrode Reaction and Its Classification

The single electrode reaction is basic redox reaction in electrochemical field. In this paper, it is expressed as:

$$\mathrm{Red}\rightleftarrows {\mathrm{Ox}}^{\mathrm{z}+}+{\mathrm{ze}}^{-}.$$

(1)

The Nernst equation for the above is expressed as:

$$E_{\mathrm{eq}}=E^{\varnothing }+\frac{R T}{z F}\ln \frac{{\left[{\mathrm{Ox}}^{\mathrm{z}+}\right]}_{\mathrm{bulk}}}{{\left[\mathrm{Red}\right]}_{\mathrm{bulk}}}$$

(2)

When electrons are pumped up or into the system, the equilibrium state is lost. Consequently, the equilibrium potential, $E_{\mathrm{eq}}$ changes to a new $E_{\mathrm{eq}}$. The difference between applied potential, E and the $E_{\mathrm{eq}}$ is the overpotential, $\eta$:

$$\eta =E-E_{\mathrm{eq}}$$

(3)

Applying $\eta$ to the system, a net current, j is observed. The curve plotting between $j$ and $\eta$ is the polarization curve, $j\left(\eta \right)$. When charge transfer and diffusion transfer processes simultaneously occur, the $j\left(\eta \right)$ is expressed to [27]:

$$j\left(\eta \right)=\frac{\exp \left(f_{\mathrm{a}}\eta \right)-\exp \left(-f_{\mathrm{c}}\eta \right)}{1/j_0+\exp \left(f_{\mathrm{a}}\eta \right)/j_{\mathrm{Red},\mathrm{L}}+\exp \left(-f_{\mathrm{c}}\eta \right)/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}=j_{\mathrm{a}}\left(\eta \right)+j_{\mathrm{c}}\left(\eta \right),$$

(4)

where the $j_{\mathrm{a}}\left(\eta \right)$ and $j_{\mathrm{c}}\left(\eta \right)$ are the anodic and cathodic branch current density, respectively:

$$j_{\mathrm{a}}\left(\eta \right)=\frac{\exp \left(f_{\mathrm{a}}\eta \right)}{1/j_0+\exp \left(f_{\mathrm{a}}\eta \right)/j_{\mathrm{Red},\mathrm{L}}+\exp \left(-f_{\mathrm{c}}\eta \right)/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}} \left(>0\right),$$

(5)

$$j_{\mathrm{c}}\left(\eta \right)=\frac{-\exp \left(-f_{\mathrm{c}}\eta \right)}{1/j_0+\exp \left(f_{\mathrm{a}}\eta \right)/j_{\mathrm{Red},\mathrm{L}}+\exp \left(-f_{\mathrm{c}}\eta \right)/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}} \left(<0\right)$$

(6)

The $j_{\mathrm{a}}\left(0\right)$, total exchange current density is expressed as the reciprocal of the summation of the reciprocals of $j_0$, $j_{\mathrm{Red},\mathrm{L}}$, and $-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$:

$$j_{\mathrm{a}}\left(0\right)=-j_{\mathrm{c}}\left(0\right)=\frac{1}{1/j_0+1/j_{\mathrm{Red},\mathrm{L}}+1/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}$$

(7)

Therefore, the $j_{\mathrm{a}}\left(0\right)$ is a harmonic mean among $j_0$, $j_{\mathrm{Red},\mathrm{L}}$, and $-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$. Its value has a tendency to be close to the minimum value among them. The $j_{\mathrm{Red},\mathrm{L}}$ and $j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$ are shortened into $j_{\mathrm{d}}$ in this paper:

$$j_{\mathrm{d}}=\frac{1}{1/j_{\mathrm{Red},\mathrm{L}}+1/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}$$

(8)

Then, Equation (7) is simplified as below:

$$j_{\mathrm{a}}\left(0\right)=\frac{1}{1/j_0+1/j_{\mathrm{d}}}$$

(9)

In order to simplify the complicated Equation (4), it is convenient to divide $j\left(\eta \right)$ into three approximations by considering relationship between $j_0$ and $j_{\mathrm{d}}$.

(A)

Reversible reaction; $j_0\gg j_{\mathrm{d}} \left(\mathrm{or} j_{\mathrm{a}}\left(0\right)\approx j_{\mathrm{d}}\right)$;

(B)

Irreversible reaction; $j_0\ll j_{\mathrm{d}} \left(\mathrm{or} j_{\mathrm{a}}\left(0\right)\approx j_0\right)$;

(C)

Quasireversible reaction; $j_0\approx j_{\mathrm{d}} \left(\mathrm{or} j_{\mathrm{a}}\left(0\right)\approx j_{\mathrm{d}}/2\approx j_0/2\right)$.

The above classification may be archaic, but it plays an important role in the $h\left(j\right)$ expression.

Note that, at the equilibrium state ($j=0 \mathrm{or} \eta =0$), the following relationship holds:

$$j_{\mathrm{a}}\left(\eta \right)+j_{\mathrm{c}}\left(\eta \right)=0,$$

(10)

$$\eta =E-E_{\mathrm{eq}}=\left(E-E^{\varnothing }\right)+\left(E^{\varnothing }-E_{\mathrm{eq}}\right)={\eta }^{\varnothing }-\frac{R T}{ z F}\ln \frac{{\left[{\mathrm{Ox}}^{\mathrm{z}+}\right]}_{\mathrm{bulk}}}{{\left[\mathrm{Red}\right]}_{\mathrm{bulk}}}$$

(11)

Substituting them into Equation (4), we can obtain same Equation (2). The same result shows that the Nernst equation is a special case of $j\left(\eta \right)$.

3.2. Single Electrode Reaction and Its Polarization Resistance

Polarization curve has been frequently employed as a simple and basic electrochemical method to elucidate reaction mechanism. Usually, the polarization curve has been expressed using two functions: $j\left(\eta \right)$ and $E\left(j\right)$. The former, $j\left(\eta \right)$, is possible for direct plotting using measuring data. It has a decided merit that current can be superposed. The latter, $E\left(j\right)$, is also commonly used, which is obtained by exchanging between the horizontal and vertical axes in $j\left(\eta \right)$. Having same vertical axis with potential scale, there is a superiority that $E\left(j\right)$ is easy to compare with the Tafel equation. Furthermore, its advantage will be clarified when predicting thermodynamically stable chemical species and their possible reaction by referring to the E-pH diagram. In addition to the above two expressions, we have another expression: polarization resistance curve, $h\left(j\right)$. The $h\left(j\right)$ is not so familiar, but it is possible to express the system. The great advantage of it lies in reduction of parameters, which must be experimentally determined. For example, overpotential, which is not so easy to determine in experimentally, is disappeared due to its constant value. As a result, the $h\left(j\right)$ tends to be a concise expression. These curves have very different forms, but mathematically, they are exactly the same and are interchangeable.

$$j\left(\eta \right)\rightleftarrows E\left(j\right)\rightleftarrows \left\{\begin{array}{c}\mathrm{d}E\left(j\right)/\mathrm{d}j=h\left(j\right)\\ E\left(0\right)=E_{\mathrm{eq}} \left(\mathrm{one} \mathrm{initial} \mathrm{condition}\right)\end{array}\right.$$

(12)

Detailed expressions of $h\left(j\right)$ are shown below. Differentiating Equation (4), we can obtain polarization conductance, $g\left(\eta \right)$:

$$\begin{gathered} g\left(\eta \right)=\frac{\mathrm{d} j\left(\eta \right)}{\mathrm{d} \eta }=\frac{\mathrm{d} \left(j_{\mathrm{a}}\left(\eta \right)+j_{\mathrm{c}}\left(\eta \right)\right)}{\mathrm{d} \eta }=g_{\mathrm{a}}\left(\eta \right)+g_{\mathrm{c}}\left(\eta \right) \\ =f_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)-f_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)-j\left(\eta \right)\left\{\frac{f_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)}{j_{\mathrm{Red},\mathrm{L}}}+\frac{f_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)}{-j_{{\mathrm{Ox}}^{z+},\mathrm{L}}}\right\} \end{gathered}$$

(13)

where,

$$g_{\mathrm{a}}\left(\eta \right)=\frac{\mathrm{d}{j}_{\mathrm{a}}\left(\eta \right)}{\mathrm{d}\eta }=f_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)-j_{\mathrm{a}}\left(\eta \right)\left\{\frac{f_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)}{j_{\mathrm{Red},\mathrm{L}}}+\frac{f_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right\},$$

(14)

$$g_{\mathrm{c}}\left(\eta \right)=\frac{\mathrm{d}{j}_{\mathrm{c}}\left(\eta \right)}{\mathrm{d}\eta }=-f_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)-j_{\mathrm{c}}\left(\eta \right)\left\{\frac{f_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)}{j_{\mathrm{Red},\mathrm{L}}}+\frac{f_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right\}$$

(15)

Using the inverse function relation between $g\left(\eta \right)$ and $h\left(j\right)$, the below will be obtained:

$$\frac{\mathrm{d} j}{\mathrm{d} \eta }\frac{\mathrm{d} \eta }{\mathrm{d} j}=g\left(\eta \right)\frac{\mathrm{d} \left(E\left(j\right)-E_{\mathrm{eq}}\right)}{\mathrm{d} j}=g\left(\eta \right)\frac{\mathrm{d} E\left(j\right)}{\mathrm{d} j}=g\left(\eta \right) h\left(j\right)=1.$$

(16)

The $g\left(\eta \right)$ is a function of $\eta$, but the $h\left(j\right)$ is a function of j. In order to express the $g\left(\eta \right)$ as a function of j, all of $j\left(\eta \right)$, $j_{\mathrm{a}}\left(\eta \right)$, and $j_{\mathrm{c}}\left(\eta \right)$ must be expressed as a function of j. Using the Maclaurin expand series, they are approximated to the linear relations:

$$\begin{array}{l}j\left(\eta \right)=j\left(0\right)+\frac{j^{\prime }\left(0\right)}{1 !}\eta +\frac{j^{″}\left(0\right)}{2 !}{\eta }^2+\cdots \approx g\left(0\right) \eta =g\left(0\right) h\left(0\right) j=j,\end{array}$$

(17)

$$\begin{array}{l}{j}_{\mathrm{a}}\left(\eta \right)=j_{\mathrm{a}}\left(0\right)+\frac{{j_{\mathrm{a}}}^{\prime }\left(0\right)}{1 !}\eta +\frac{j_{\mathrm{a}}^{″}\left(0\right)}{2 !}{\eta }^2+\cdots \approx j_{\mathrm{a}}\left(0\right)+g_{\mathrm{a}}\left(0\right) \eta =j_{\mathrm{a}}\left(0\right)+\frac{g_{\mathrm{a}}\left(0\right)}{g_{\mathrm{a}}\left(0\right)+g_{\mathrm{c}}\left(0\right)} j,\end{array}$$

(18)

$$\begin{array}{l}{j}_{\mathrm{c}}\left(\eta \right)=j_{\mathrm{c}}\left(0\right)+\frac{{j_{\mathrm{c}}}^{\prime }\left(0\right)}{1 !}\eta +\frac{j_{\mathrm{c}}^{″}\left(0\right)}{2 !}{\eta }^2+\cdots \approx j_{\mathrm{c}}\left(0\right)+g_{\mathrm{c}}\left(0\right) \eta =j_{\mathrm{c}}\left(0\right)+\frac{g_{\mathrm{c}}\left(0\right)}{g_{\mathrm{a}}\left(0\right)+g_{\mathrm{c}}\left(0\right)} j\end{array}$$

(19)

By arranging the $\frac{g_{\mathrm{a}}\left(0\right)}{g_{\mathrm{a}}\left(0\right)+g_{\mathrm{c}}\left(0\right)}$ and $\frac{g_{\mathrm{c}}\left(0\right)}{g_{\mathrm{a}}\left(0\right)+g_{\mathrm{c}}\left(0\right)}$, $j_{\mathrm{a}}\left(j\right)$, and $j_{\mathrm{c}}\left(j\right)$ are expressed as:

$$j_{\mathrm{a}}\left(j\right)=j_{\mathrm{a}}\left(0\right)+\left\{{\alpha }_{\mathrm{a}}-\left(\frac{{\alpha }_{\mathrm{a}} j_{\mathrm{a}}\left(0\right)}{j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}} j_{\mathrm{c}}\left(0\right)}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\}j,$$

(20)

$$j_{\mathrm{c}}\left(j\right)=j_{\mathrm{c}}\left(0\right)+\left\{{\alpha }_{\mathrm{c}}+\left(\frac{{\alpha }_{\mathrm{a}} j_{\mathrm{a}}\left(0\right)}{j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}} j_{\mathrm{c}}\left(0\right)}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\}j.$$

(21)

Then, the $h\left(j\right)$ can be expressed as:

$$h\left(j\right)=\frac{1}{f_{\mathrm{a}} j_{\mathrm{a}}\left(j\right)-f_{\mathrm{c}} j_{\mathrm{c}}\left(j\right)-j \left\{f_{\mathrm{a}}{j}_{\mathrm{a}}\left(j\right)/j_{\mathrm{Red},\mathrm{L}}+f_{\mathrm{c}} j_{\mathrm{c}}\left(j\right)/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right\}}.$$

(22)

At the equilibrium state ($j=0$), common expression of $h\left(0\right)$ is obtained:

$$h\left(0\right)=\frac{1}{f_{\mathrm{a}} j_{\mathrm{a}}\left(0\right)-f_{\mathrm{c}} j_{\mathrm{c}}\left(0\right)}=\frac{1}{(f_{\mathrm{a}}+f_{\mathrm{c}})j_{\mathrm{a}}\left(0\right)}=\frac{RT}{ z F}\frac{1}{j_{\mathrm{a}}\left(0\right)}=\frac{0.026}{ z }\frac{1}{j_{\mathrm{a}}\left(0\right)}$$

(23)

We can see that the $h\left(0\right)$ is inversely proportional to $j_{\mathrm{a}}\left(0\right)$ and has a constant of $0.026/z$. In other words, the above equation tells that the $j_{\mathrm{a}}\left(0\right)$ will be easily calculated by reading the $h\left(0\right)$ when the z is already known. Around the equilibrium state ($j\approx 0$), $h\left(j\right)$ can be approximated to linear relation:

$${\left[h\left(j\right)\right]}_{j\approx 0}=h\left(0\right)\left\{1+\left(\frac{2 {\alpha }_{\mathrm{a}} }{j_{\mathrm{Red},\mathrm{L}} }+\frac{2 {\alpha }_{\mathrm{c}} }{j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}} }+\frac{1-2 {\alpha }_{\mathrm{a}}}{j_{\mathrm{a}}\left(0\right)}\right)j\right\}.$$

(24)

We can see that the ${\left[h\left(j\right)\right]}_{j\approx 0}$ is a straight line having an intercept of $h\left(0\right)$ and a slope of $h\left(0\right)\left(\frac{2 {\alpha }_{\mathrm{a}} }{j_{\mathrm{Red},\mathrm{L}}}+\frac{2 {\alpha }_{\mathrm{c}} }{j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}+\frac{1-2 {\alpha }_{\mathrm{a}}}{j_{\mathrm{a}}\left(0\right)}\right)$.

Being in a far polarized state ($\left|\eta \right|\gg 0$), information for each branch reaction can be obtained. The kinetic parameters for anodic branch will be obtained when the system is anodically far-polarized ($\eta \gg 0$):

$$j=j_{\mathrm{a}}\left(j\right)+j_{\mathrm{c}}\left(j\right)\approx j_{\mathrm{a}}\left(j\right) \left(\mathrm{or} j_{\mathrm{c}}\left(j\right)\approx 0\right).$$

(25)

Let $j_{\mathrm{pa}}$ be the minimum $j_{\mathrm{a}}\left(j\right)$ in the above state. The polarization resistance curve for anodic branch, $h_{\mathrm{a}-\mathrm{branch}}\left(j_{\mathrm{a}}\right)$, will be expressed by arranging Equation (22):

$$\begin{gathered} h_{\mathrm{a}-\mathrm{branch}}\left(j_{\mathrm{a}}\right)=h_{\mathrm{a}-\mathrm{branch}}\left(j\right)={\left[h\left(j\right)\right]}_{j_{\mathrm{c}}\left(j\right)\approx 0}\approx \frac{1}{f_{\mathrm{a}} j-j \left\{f_{\mathrm{a}}j/j_{\mathrm{Red},\mathrm{L}}\right\}} \\ =\frac{RT}{{\alpha }_{\mathrm{a}} z F}\left(\frac{1}{ j}+\frac{1}{ j_{\mathrm{Red},\mathrm{L}}-j}\right). \end{gathered}$$

(26)

The second term in the above will be close to zero if j is sufficiently large:

$${\left[\frac{RT}{{\alpha }_{\mathrm{a}} z F}\frac{1}{ j_{\mathrm{Red},\mathrm{L}}-j}\right]}_{j_{\mathrm{pa}}\ll j}={\left[\frac{0.026}{{\alpha }_{\mathrm{a}} z }\frac{1}{ j_{\mathrm{Red},\mathrm{L}}-j}\right]}_{j_{\mathrm{pa}}\ll j}\approx 0.$$

(27)

Then, Equation (26) is approximated to:

$${\left[h\left(j\right)\right]}_{j_{\mathrm{pa}}\ll j}\approx \frac{RT}{{\alpha }_{\mathrm{a}} z F}\frac{1}{ j}=\frac{0.026}{{\alpha }_{\mathrm{a}} z }\frac{1}{ j}.$$

(28)

Arranging the above, the ${\alpha }_{\mathrm{a}} z$ can be calculated as:

$${\alpha }_{\mathrm{a}} z\approx {\left[\frac{0.026}{ h\left(j\right) j}\right]}_{j_{\mathrm{pa}}\ll j} \left(j>0\right).$$

(29)

Similarly, information on the cathodic branch will be obtained when the system is cathodically far-polarized ($\eta \ll 0$):

$$j=j_{\mathrm{a}}\left(j\right)+j_{\mathrm{c}}\left(j\right)\approx j_{\mathrm{c}}\left(j\right) \left(\mathrm{or} j_{\mathrm{a}}\left(j\right)\approx 0\right).$$

(30)

Let $j_{\mathrm{pc}}$ be the maximum $j_{\mathrm{c}}\left(j\right)$ in the above state. Then, the polarization resistance curve for cathodic branch, $h_{\mathrm{c}-\mathrm{branch}}\left(j_{\mathrm{c}}\right)$ can be obtained:

$$h_{\mathrm{c}-\mathrm{branch}}\left(j_{\mathrm{c}}\right)=h_{\mathrm{c}-\mathrm{branch}}\left(j\right)={\left[h\left(j\right)\right]}_{j_{\mathrm{a}}\left(j\right)\approx 0}\approx \frac{1}{-f_{\mathrm{c}} j-j \left\{f_{\mathrm{c}}j/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right\}}=\frac{RT}{{\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}+\frac{1}{j-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right).$$

(31)

For sufficiently large $-j \left(j<0\right)$, the second term will be close to zero:

$${\left[\frac{RT}{{\alpha }_{\mathrm{c}} z F}\frac{1}{j-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right]}_{j\ll j_{\mathrm{pc}}}={\left[\frac{0.026}{{\alpha }_{\mathrm{c}} z }\frac{1}{j-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right]}_{j\ll j_{\mathrm{pc}}}\approx 0.$$

(32)

Then, Equation (31) is approximated to:

$${\left[h\left(j\right)\right]}_{j\ll j_{\mathrm{pc}}}\approx \frac{RT}{{\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}\right)=\frac{0.026}{{\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}\right).$$

(33)

The ${\alpha }_{\mathrm{c}} z$ can be calculated as:

$${\alpha }_{\mathrm{c}} z\approx {\left[\frac{0.026}{ h\left(j\right) \left|j\right|}\right]}_{\left|j_{\mathrm{pc}}\right|\ll \left|j\right|} \left(j<0\right)$$

(34)

Each of $j_{\mathrm{pa}}$ and $j_{\mathrm{pc}}$ will be expressed by arranging Equations (20), (21), (25) and (30):

$$j_{\mathrm{pa}}={\left(\frac{{\alpha }_{\mathrm{c}}}{j_{\mathrm{a}}\left(0\right)}+\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{Red},\mathrm{L}}}-\frac{{\alpha }_{\mathrm{c}}}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1},$$

(35)

$$j_{\mathrm{pc}}=-{\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{a}}\left(0\right)}-\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}}}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1}.$$

(36)

When we can find two inflection points on the curve, a very interesting relationship between Equations (35) and (36) is shown as:

$$\frac{1}{j_{\mathrm{a}}\left(0\right)}=\frac{1}{j_{\mathrm{pa}}}+\frac{1}{-j_{\mathrm{pc}}}.$$

(37)

The same equation had been already established by J.M. Pearson in 1942 in the corrosion field [28].

3.3. The Relationship between the Tafel Extrapolation Method (tem) and H(j)

The Tafel equation is an empirical equation showing the relationship between $\eta \left(\mathrm{usually}, \left|\eta \right|\gg 0\right)$ and j. It is expressed as:

$$\eta =a+b\log \left|j\right|=a+\frac{b}{2.3}\ln \left|j\right|.$$

(38)

Naturally, it is possible to express the Tafel equation in differential form:

$$\frac{\mathrm{d} \eta }{\mathrm{d} j}=\frac{\mathrm{d}\left(E\left(j\right)-E_{\mathrm{eq}}\right)}{\mathrm{d} j} =\frac{\mathrm{d} E\left(j\right)}{\mathrm{d} j}=h\left(j\right)=\frac{b}{2.3}\frac{1}{\left|j\right|}.$$

(39)

We can see that the above is inverse-proportional to $\left|j\right|$ and has a constant of $b/2.3$. Taking the logarithm, the following is obtained:

$$\log h\left(j\right)=\log \frac{b}{2.3}-\log \left|j\right|.$$

(40)

The above expression can give us a clear guide when finding the Tafel region: (1) the differentiated Tafel equation has a linear relationship between the $\log h\left(j\right)$ and $\log \left|j\right|$, and (2) its straight line always has slope of −1. Namely, the linear slope region observed on $\log h\left(j\right) \mathrm{vs}. \log \left|j\right|$ curve exactly corresponds to the Tafel slope region, and vice versa.

$$\frac{\mathrm{d} \log h\left(j\right)}{\mathrm{d}\log \left|j\right|}=-1 \leftrightarrows \mathrm{Tafel} \mathrm{slope}.$$

(41)

The existence of this straight line is extremely helpful in finding the Tafel slope region and in determining its accurate value.

The graphical representation of $h\left(j\right)$ brings direct benefits to our understanding. Example data shown in

Table 4. Example data for graphical representation for three classified reactions are shown: (A) reversible, (B) irreversible, and (C) quasireversible reactions.

Item	$\mathit{j}\left(\mathit{\eta }\right)\mathbf{=}\frac{exp\left({\mathit{f}}_a\mathit{\eta }\right)\mathbf{-}exp\left(\mathbf{-}{\mathit{f}}_c\mathit{\eta }\right)}{\mathbf{1}\mathbf{/}{\mathit{j}}_{\mathbf{0}}\mathbf{+}exp\left({\mathit{f}}_a\mathit{\eta }\right)\mathbf{/}{\mathit{j}}_{Red\mathbf{,}L}\mathbf{+}exp\left(\mathbf{-}{\mathit{f}}_c\mathit{\eta }\right)\mathbf{/}\mathbf{-}{\mathit{j}}_{O{x}^{z\mathbf{+}}\mathbf{,}L}}$			Remarks
	(A): Reversible	(B): Irreversible	(C): Quasi-Reversible
	(j0 ≫ jd)	(j0 ≪ jd)	(j0 ≈ jd)
$j_{\mathrm{Red},\mathrm{L}}/\mathrm{mA}{\mathrm{cm}}^{-2}$	1	1	1
$j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}/\mathrm{mA}{\mathrm{cm}}^{-2}$	−100	−100	−100
$j_{\mathrm{d}}/\mathrm{mA}{\mathrm{cm}}^{-2}$	0.99	0.99	0.99	$j_{\mathrm{d}}=\frac{1}{1/j_{\mathrm{Red},\mathrm{L}}+1/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}$
$j_0/\mathrm{mA}{\mathrm{cm}}^{-2}$	1000	0.001	1
${\alpha }_{\mathrm{a}} \left(-\right)$	0.3	0.3	0.3
${\alpha }_{\mathrm{c} }\left(-\right)$	0.7	0.7	0.7
$z \left(-\right)$	2	2	2
$j_{\mathrm{a}}\left(0\right)/\mathrm{mA}{\mathrm{cm}}^{-2}$	0.989	0.000989	0.497	$j_{\mathrm{a}}\left(0\right)=\frac{1}{1/j_0+1/j_d}$
$h\left(0\right)/\mathrm{mA}{\mathrm{cm}}^{-2}$	0.013	13	0.026

The $j\left(\eta \right)$ of the above three reactions are shown in Figure 8 for reference.

were employed to draw the $h\left(j\right)$ curves.

We can see a general tendency that (1) the $j_{\mathrm{Red},\mathrm{L}}$ and $j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$ indicate the upper and lower limit, respectively; (2) the $j_{\mathrm{a}}\left(0\right)$ relates to the awayness between $j_{\mathrm{a}}\left(\eta \right)$ and $j_{\mathrm{c}}\left(\eta \right)$ (not depicted); and (3) the ${\alpha }_{\mathrm{a}}$ and ${\alpha }_{\mathrm{c}}$ affect the symmetry of the curve. The quasireversible $j\left(\eta \right)$ is located between reversible and irreversible and slightly closer to the reversible. This close position is thought to be the reason why “quasi” was given. To illustrate the benefits of the $h\left(j\right)$ expression, each curve is shown in Figure 9 using the same data in Table 3.

The straight lines concerning Equation (41) are drawn with three orange dashed lines. We can see that the Tafel slope are visually held in the wider anodic branch ($5\times {10}^{-3}{ \mathrm{mA} \mathrm{cm}}^{-2}\lesssim j\lesssim {10}^{-1} {\mathrm{mA} \mathrm{cm}}^{-2}$) and the widest cathodic branch ($3\times {10}^{-2} {\mathrm{mA} \mathrm{cm}}^{-2}\lesssim \left|j\right|\lesssim 10 {\mathrm{mA} \mathrm{cm}}^{-2}$) for irreversible reaction (blue). The Tafel slope for the quasireversible reaction (green) is satisfied in the narrow cathodic region ($\approx 12 {\mathrm{mA} \mathrm{cm}}^{-2}$). In the case of reversible $h\left(j\right)$ (red), we can see that the Tafel slope is valid for only one point ($\approx 6 {\mathrm{mA} \mathrm{cm}}^{-2}$). The curve of $\log h\left(j\right) \mathrm{vs}. \log \left|j \right|$ clarified that the TEM is eminently valid for irreversible reaction.

3.4. Kinetic Parameter Determination Using $h\left(j\right)$

Since graphical $j\left(\eta \right)$ can be classified into the above three reactions, $h\left(j\right)$ can also be classified into the three.

3.4.1. Reversible Reaction

The $j_{\mathrm{a}}\left(j\right)$ and $j_{\mathrm{c}}\left(j\right)$ of the reversible reaction are simplified to the below by arranging Equations (20) and (21), respectively:

$$j_{\mathrm{a}}\left(j\right)=j_{\mathrm{d}}+\frac{ j_{\mathrm{Red},\mathrm{L}}}{ j_{\mathrm{Red},\mathrm{L}}-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}j=j_{\mathrm{d}}\left(1+\frac{j}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right),$$

(42)

$$j_{\mathrm{c}}\left(j\right)=-j_{\mathrm{d}}+\frac{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}{ j_{\mathrm{Red},\mathrm{L}}-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}j=-j_{\mathrm{d}}\left(1+\frac{j}{ j_{\mathrm{Red},\mathrm{L}}}\right).$$

(43)

Substituting them into Equation (22), we can obtain the reversible $h\left(j\right)$.

$$h\left(j\right)=\frac{RT}{ z F}\left(\frac{1}{ j_{\mathrm{Red},\mathrm{L}}-j}+\frac{1}{j-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right).$$

(44)

At the equilibrium state ($j=0$),

$$h\left(0\right)=\frac{RT}{ z F}\left(\frac{1}{ j_{\mathrm{Red},\mathrm{L}}}+\frac{1}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)=\frac{0.026}{ z }\frac{1}{j_{\mathrm{d}}}\left(=\frac{0.026}{ z }\frac{1}{j_{\mathrm{a}}\left(0\right)}\right).$$

(45)

If we can read the three values ($h\left(0\right)$, $j_{\mathrm{Red},\mathrm{L}}$, and $j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$) from experiments, we can easily determine z. If that is impossible or ambiguous, there is another way to know it: a curve technique, which is known as the parallel displacement in geometry. The following simultaneous equation discloses the technique:

$$h\left(j+ j_{\mathrm{Red},\mathrm{L}}\right)= \frac{RT}{z F}\left(\frac{1}{-j}+\frac{1}{j+ j_{\mathrm{Red},\mathrm{L}}- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right),$$

(46)

$$\frac{RT}{z F}\frac{1}{j+j_{\mathrm{Red},\mathrm{L}}-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\approx 0.$$

(47)

Substituting the precondition (Equation (47)) into Equation (46), the below equation is obtained:

$$h\left(j+ j_{\mathrm{Red},\mathrm{L}}\right)\approx \frac{RT}{z F}\frac{1}{-j}=\frac{0.026}{z \left|j\right|}\left(j<0\right).$$

(48)

We can see that the above has an inverse proportional relation between $h\left(j\right)$ and $\left|j\right|$, which has the same relation as Equation (39). From the above, we can determine the z:

$$z= \frac{0.026}{\left|j\right|h\left(j+ j_{\mathrm{Red},\mathrm{L}}\right)} \left(j<0\right).$$

(49)

Similarly, the same displacement for another side will lead to the same z:

$$z= \frac{0.026}{j h\left(j+ j_{{\mathrm{Ox}}^{\mathbf{z}+},\mathrm{L}}\right) }= \frac{0.026}{\left|j\right|h\left(j+ j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right) }\left(j>0\right).$$

(50)

Around the equilibrium state ($j\approx 0$), the reversible $h\left(j\right)$ is linearly expressed to:

$${\left[h\left(j\right)\right]}_{j\approx 0}=h\left(0\right)\left\{1+\left(\frac{1}{ j_{\mathrm{Red},\mathrm{L}} }+\frac{1}{ j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)j\right\}.$$

(51)

Arranging Equations (42) and (43), $j_{\mathrm{pa}}$ and $j_{\mathrm{pc}}$ will be expressed as:

$$j_{\mathrm{pa}}=j_{\mathrm{Red},\mathrm{L}}$$

(52)

$$j_{\mathrm{pc}}=j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$$

(53)

We can see that Equation (37) holds true in the reversible system.

3.4.2. Irreversible Reaction

Considering $j_0\ll j_{\mathrm{d}}$, the $j_{\mathrm{a}}\left(j\right)$ and $j_{\mathrm{c}}\left(j\right)$ are simplified below:

$$j_{\mathrm{a}}\left(j\right)=j_0+\left\{{\alpha }_{\mathrm{a}}-j_0\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{Red},\mathrm{L}}}+\frac{-{\alpha }_{\mathrm{c}}}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\} j\approx j_0+{\alpha }_{\mathrm{a}} j,$$

(54)

$$j_{\mathrm{c}}\left(j\right)=j_0+\left\{{\alpha }_{\mathrm{c}}+j_0\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{Red},\mathrm{L}}}+\frac{-{\alpha }_{\mathrm{c}}}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\} j\approx -j_0+{\alpha }_{\mathrm{c}} j.$$

(55)

Substituting them into Equation (22), we can obtain the irreversible $h\left(j\right)$.

$$\begin{array}{c}h\left(j\right)=\frac{ j_{\mathrm{Red},\mathrm{L}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}{f_{\mathrm{c}}\left(j-j_0\right)\left(j-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right) j_{\mathrm{Red},\mathrm{L}}+f_{\mathrm{a}} j_0 j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\left( j_{\mathrm{Red},\mathrm{L}}-j\right)+{\alpha }_{\mathrm{a}} j \left\{f_{\mathrm{c}} ( j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}-j) j_{\mathrm{Red},\mathrm{L}}+f_{\mathrm{a}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\left( j_{\mathrm{Red},\mathrm{L}}-j\right)\right\}}\\ \approx \frac{RT}{ z F}\frac{1}{j_0+\left({\alpha }_{\mathrm{a}}-{\alpha }_{\mathrm{c}}\right) j}=\frac{0.026}{ z }\frac{1}{j_0+\left({\alpha }_{\mathrm{a}}-{\alpha }_{\mathrm{c}}\right) j}.\end{array}$$

(56)

At the equilibrium state ($j=0$),

$$h\left(0\right)=\frac{1}{ f}\frac{1}{j_0}=\frac{RT}{ z F}\frac{1}{j_0}=\frac{0.026}{ z }\frac{1}{j_0}\left(=\frac{0.026}{ z }\frac{1}{j_{\mathrm{a}}\left(0\right)}\right).$$

(57)

Around the equilibrium state ($j\approx 0$), $h\left(j\right)$ is linearly expressed as:

$${\left[h\left(j\right)\right]}_{j\approx 0}=h\left(0\right)\left\{1+\left(\frac{ {\alpha }_{\mathrm{a}} }{j_{\mathrm{Red},\mathrm{L}} }+\frac{{\alpha }_{\mathrm{c}} }{j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}} }+\frac{1-2 {\alpha }_{\mathrm{a}}}{j_0}\right)j\right\}.$$

(58)

At anodically far polarized state ($\eta \gg 0, \mathrm{or} j\approx j_{\mathrm{a}}, \mathrm{or} j_{\mathrm{c}}\approx 0, \mathrm{or} j_{\mathrm{pa}}<j$), anodic $h\left(j\right)$ of the irreversible reaction is expressed:

$${\left[h\left(j\right)\right]}_{\eta \gg 0}=\frac{RT}{ {\alpha }_{\mathrm{a}} z F}\left(\frac{1}{j}+\frac{1}{ j_{\mathrm{Red},\mathrm{L}}-j}\right).$$

(59)

The second term above will be close to zero when $\left(j_{\mathrm{Red},\mathrm{L}}-j\right)\gg 0$:

$$\frac{RT}{ {\alpha }_{\mathrm{a}} z F}\left(\frac{1}{ j_{\mathrm{Red},\mathrm{L}}-j}\right)\approx 0,$$

(60)

Then, Equation (59) is simplified to:

$${\left[h\left(j\right)\right]}_{\eta \gg 0}\approx \frac{RT}{ {\alpha }_{\mathrm{a}} z F}\left(\frac{1}{j}\right)=\frac{0.026}{ {\alpha }_{\mathrm{a}} z }\left(\frac{1}{j}\right).$$

(61)

Similarly, when being cathodically far polarized ($\eta \ll 0, \mathrm{or} j\approx j_{\mathrm{c}},\mathrm{or} j_{\mathrm{a}}\approx 0, \mathrm{or} j<j_{\mathrm{pc}}$), ${\left[h\left(j\right)\right]}_{\eta \ll 0}$ is:

$${\left[h\left(j\right)\right]}_{\eta \ll 0}=\frac{RT}{ {\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}+\frac{1}{j- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right).$$

(62)

When the second term is close to zero:

$$\frac{RT}{ {\alpha }_{\mathrm{c}} z F}\left(\frac{1}{j- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\approx 0.$$

(63)

Then, the far cathodic $h\left(j\right)$ is expressed below, which is the same as Equation (33):

$${\left[h\left(j\right)\right]}_{\eta \gg 0}=\frac{RT}{ {\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}\right).$$

(64)

It is important to notice that irreversible $z$ is the number of electrons transferred in the rate determine step (rds), not $z$ of the whole reaction of Equation (1). Arranging Equations (54) and (55), $j_{\mathrm{pa}}$ and $j_{\mathrm{pc}}$ will be expressed as:

$$j_{\mathrm{pa}}=j_0/{\alpha }_{\mathrm{c}}$$

(65)

$$j_{\mathrm{pc}}=-j_0/{\alpha }_{\mathrm{a}}$$

(66)

We can see that the relation of Equation (37) holds true in the irreversible system.

3.4.3. Quasireversible Reaction

The $j_{\mathrm{a}}\left(j\right)$ and $j_{\mathrm{c}}\left(j\right)$ are expressed as:

$$j_{\mathrm{a}}\left(j\right)=\frac{j_{\mathrm{d}}}{2}+\left\{{\alpha }_{\mathrm{a}}-\frac{j_{\mathrm{d}}}{2}\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}}}{ j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\}j,$$

(67)

$$j_{\mathrm{c}}\left(j\right)=\frac{-j_{\mathrm{d}}}{2}+\left\{{\alpha }_{\mathrm{c}}+\frac{j_{\mathrm{d}}}{2}\left(\frac{{\alpha }_{\mathrm{a}}}{ j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}}}{ j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\}j.$$

(68)

Substituting them into Equation (22), we can obtain the quasireversible:

$$h\left(j\right)=\frac{2 j_{\mathrm{Red},\mathrm{L}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\left(j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}-j_{\mathrm{Red},\mathrm{L}}\right)}{{\alpha }_{\mathrm{a}} j \left(j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}-j_{\mathrm{Red},\mathrm{L}}\right)\left\{ f_{\mathrm{c} }{j}_{\mathrm{Red},\mathrm{L}}\left(j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}-j\right)+f_{\mathrm{a}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\left( j_{\mathrm{Red},\mathrm{L}}-j\right)\right\}+\left(j-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right) j_{\mathrm{Red},\mathrm{L}}\left\{f_{\mathrm{a}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\left(j-j_{\mathrm{Red},\mathrm{L}}\right)+f_{\mathrm{c}}\left(2 j j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}-j j_{\mathrm{Red},\mathrm{L}}-j_{\mathrm{Red},\mathrm{L}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right)\right\}}$$

(69)

The above equation has little practical use due to its overcomplicated expression. Another expression is needed for quasireversible $h\left(j\right)$, which will be discussed in detail in Section 3.6.3. At the equilibrium state ($j=0$):

$$h\left(0\right)=\frac{RT}{ z F}\frac{2}{j_{\mathrm{d}}}\left(=\frac{RT}{ z F}\frac{2}{j_0}\right)=\frac{0.052}{ z j_{\mathrm{d}}}\left(=\frac{0.052}{ z j_0}\right)=\left(\frac{0.026}{ z }\frac{1}{j_{\mathrm{a}}\left(0\right)}\right).$$

(70)

Each of $j_{\mathrm{pa}}$ and $j_{\mathrm{pc}}$ will be expressed as:

$$j_{\mathrm{pa}}={\left(\frac{{\alpha }_{\mathrm{c}}}{j_{\mathrm{d}}}+\frac{1}{j_{\mathrm{Red},\mathrm{L}}}\right)}^{-1}={\left(\frac{1+{\alpha }_{\mathrm{c}}}{j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}}}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1}$$

(71)

$$j_{\mathrm{pc}}=-{\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{d}}}+\frac{1}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1}=-{\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{Red},\mathrm{L}}}+\frac{1+{\alpha }_{\mathrm{a}}}{-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1}$$

(72)

It can be seen that the relationship in Equation (37) is satisfied in the quasireversible reaction. Since Equations (23) and (37) are valid for all reactions, the follow is a common relation:

$$j_{\mathrm{a}}\left(0\right)=\frac{R T}{ z F}\frac{1}{h\left(0\right)}=\frac{0.026}{ z h\left(0\right) }=\frac{1}{1/j_{\mathrm{pa}}+1/-j_{\mathrm{pc}}}.$$

(73)

3.5. Graphical Determination of $j_{\mathrm{a}}\left(0\right)$

In addition to the above algebraic way, it is worth glancing at the geometric way. This section shows that the reason why the graphical estimation of $j_{\mathrm{a}}\left(0\right)$ is possible and how to use it. The common $h\left(0\right)$ at the equilibrium state is expressed as:

$$h\left(0\right)=\frac{R T}{ z F}\frac{1}{j_{\mathrm{a}}\left(0\right)}=\frac{0.026}{ z }\left(\frac{1}{j_0}+\frac{1}{ j_{\mathrm{Red},\mathrm{L}}}+\frac{1}{- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right).$$

(74)

We can regard the $h\left(0\right)$ as a function of $j_0$ because three value of z, $j_{\mathrm{Red},\mathrm{L}}$, and $j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$ are usually constant. Here, let us consider the new function, $h_{\mathrm{a}0}\left(j_0\right)$, which can be expressed below:

$$h_{\mathrm{a}0}\left(j_0\right) =\frac{0.026}{ z }\left(\frac{1}{j_0}+\frac{1}{ j_{\mathrm{Red},\mathrm{L}}}+\frac{1}{- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)=\frac{0.026}{ z }\left(\frac{1}{j_0}+\frac{1}{ j_{\mathrm{d}}}\right).$$

(75)

Using the above relation, each $h_{\mathrm{a}0}\left(j_0\right)$ in the (A), (B), and (C) is shown as:

$${\left[h_{\mathrm{a}0}\left(j_0\right) \right]}_{j_0\gg j_{\mathrm{d}}}\simeq \frac{0.026}{ z }\left(\frac{1}{ j_{\mathrm{Red},\mathrm{L}}}+\frac{1}{- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)=\frac{0.026}{ z }\left(\frac{1}{ j_{\mathrm{d}}}\right): \mathrm{constant} \mathrm{for} \mathrm{reversible};$$

(76)

$${\left[h_{\mathrm{a}0}\left(j_0\right) \right]}_{j_0\ll j_{\mathrm{d}}}\approx \frac{0.026}{ z }\left(\frac{1}{j_0}\right): \mathrm{variable} \mathrm{for} \mathrm{irreversible};$$

(77)

$${\left[h_{\mathrm{a}0}\left(j_0\right) \right]}_{j_0\approx j_{\mathrm{d}}}\approx {\left[\frac{0.026}{ z }\left(\frac{1}{j_0}+\frac{1}{j_{0\mathrm{d}}}\right)\right]}_{j_0\approx j_{\mathrm{d}}}=\frac{0.052}{z j_0}\left(=\frac{0.052}{ z j_{\mathrm{d}} }\right): \mathrm{variable} \mathrm{for} \mathrm{quasireversible}$$

(78)

The curves of Equations (76)–(78) are shown with pink chain line in Figure 10 together with the $h\left(j\right)$ in Figure 9.

The $h_{\mathrm{a}0}\left(j_0\right)$, which is a function of $j_0$, is drawn in pink with a chain curve. The employed data for drawings are the same data shown in Table 3. Three pink chain lines of $j_{\mathrm{Red},\mathrm{L}}$ and $\left|j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right|$ are added for easy readings.

The $h_{\mathrm{a}0}\left(j_0\right)$ curve and vertical lines of $j_{\mathrm{Red},\mathrm{L}}$ and $\left|j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right|$ are shown in Figure 10 with a pink chain curve. In the irreversible reaction, the horizontal straight line of $h\left(0\right)$($=13 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$) meets the $h_{\mathrm{a}0}\left(j_0\right)$ at a point $\left(\mathrm{B}1\right)$ and two vertical lines of $j_{\mathrm{Red},\mathrm{L}}$ and $\left|j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right|$ at (B2) and (B3), respectively. Three crossed points can be graphically read as:

$$j_0\approx {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{B}1\right)$$

(79)

$$j_{\mathrm{Red},\mathrm{L}}=1 {\mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{B}2\right)$$

(80)

$$-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}=100 {\mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{B}3\right)$$

(81)

Substituting the above readings into Equation (7), we can calculate the $j_{\mathrm{a}}\left(0\right)$:

$$j_{\mathrm{a}}\left(0\right)=\frac{1}{1/j_0+1/ j_{\mathrm{Red},\mathrm{L}}+1/- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}=\frac{1}{1/\left(\approx {10}^{-3}\right)+1/1+1/100} \approx {10}^{-3}{ \mathrm{mA} \mathrm{cm}}^{-2} \left(=j_0\right).$$

(82)

Employing the $j_{\mathrm{a}}\left(0\right)$ and the $h\left(0\right)$, the z can be confirmed:

$$z=\frac{RT}{F}\frac{1}{j_{\mathrm{a}}\left(0\right) h\left(0\right)}=\frac{0.026}{\left({10}^{-3}\right)\left(13\right)}=2.$$

(83)

Similarly, we can obtain the reversible $j_{\mathrm{a}}\left(0\right)$ by reading three crossed points of (A1), (A2), and (A3):

$$\begin{array}{c}{j}_0\to \infty : \cong \left(\mathrm{A}1\right), \mathrm{the} h\left(0\right) \mathrm{never} \mathrm{crosses} \mathrm{the} \mathrm{Equation} \left(75\right) \mathrm{curve},\\ \mathrm{but} \mathrm{is} \mathrm{asymptotic} \mathrm{to} \mathrm{it}.\end{array}$$

(84)

$$j_{\mathrm{Red},\mathrm{L}}=1{ \mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{A}2\right).$$

(85)

$$-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}=100{ \mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{A}3\right).$$

(86)

Then,

$$j_{\mathrm{a}}\left(0\right)=\frac{1}{1/\left(\infty \right)+1/\left(1\right)+1/\left(+100\right)}\approx 0.99{ \mathrm{mA} \mathrm{cm}}^{-2} \left(=j_{\mathrm{d}}\right).$$

(87)

Using the above, the z is calculated:

$$z=\frac{RT}{F}\frac{1}{j_{\mathrm{a}}\left(0\right) h\left(0\right)}=\frac{0.026}{\left(0.99\right)\left(0.013\right)}\approx 2.$$

(88)

For the quasireversible reaction, the crossed points are (C1), (C2), and (C3):

$$j_0\approx 1{ \mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{C}1\right)$$

(89)

$$j_{\mathrm{Red},\mathrm{L}}=1{ \mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{C}2\right)$$

(90)

$$-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}=100{ \mathrm{mA} \mathrm{cm}}^{-2}: \mathrm{cross} \mathrm{point} \left(\mathrm{C}3\right)$$

(91)

Then,

$$j_{\mathrm{a}}\left(0\right)=\frac{1}{1/\left(\approx 1\right)+1/\left(1\right)+1/\left(+100\right)}\approx 0.498{ \mathrm{mA} \mathrm{cm}}^{-2} \left(\approx \frac{j_{\mathrm{d}}}{2}\right).$$

(92)

Therefore,

$$z=\frac{RT}{F}\frac{1}{j_{\mathrm{a}}\left(0\right) h\left(0\right)}=\frac{0.026}{\left(0.498\right)\left(0.026\right)}\approx 2.$$

(93)

As was so often the case after experiments had finished, one of $j_{\mathrm{Red},\mathrm{L}}$ or $j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$ may be lost or cannot be read with clarity. In this case, graphical estimation is particularly helpful. No appearance of $j_{\mathrm{Red},\mathrm{L}}$ or $\left|j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right|$ usually means that their contribution to Equation (7) can be neglected due to the characteristic of harmonic mean. In short, the larger value is smaller or neglectable contribution.

3.6. Physical Factors Influenced on $h\left(j\right)$

The physical factors such as effective area and solution resistance sometimes distort $E\left(j\right)$ and always prevent us from quantitative approach. The following two factors are important and discussed in this paper.

3.6.1. Effective Area of Electrode

As a matter of fact, what was actually obtained in the experiments was always the total net current, $J\left(\eta \right)$. It consists of the anodic branch current, $J_{\mathrm{a}}\left(\eta \right)$, and cathodic branch current, $J_{\mathrm{c}}\left(\eta \right)$, expressed as:

$$J\left(\eta \right)=J_{\mathrm{a}}\left(\eta \right)+J_{\mathrm{c}}\left(\eta \right).$$

(94)

Using the geometric surface of the electrode, S, the $J\left(\eta \right)$ is normalized to the net current density, $j\left(\eta \right)$:

$$j\left(\eta \right)=J\left(\eta \right)/S.$$

(95)

When, for instance, gas adsorption layers or reaction deposits are present on the electrode surface, the effective area for the reaction will shrink. The $j\left(\eta \right)$ having $J_{\mathrm{a}}\left(\eta \right)$ with the effective anode area, $S_{\mathrm{a}}$, and $J_{\mathrm{c}}\left(\eta \right)$ with the effective cathode area, $S_{\mathrm{c}}$, it is arranged and expressed as:

$$j\left(\eta \right)=\frac{J_{\mathrm{a}}\left(\eta \right)+J_{\mathrm{c}}\left(\eta \right)}{S}=\frac{S_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)}{S}+\frac{S_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)}{S}=w_{\mathrm{a}} j_{\mathrm{a}}\left(\eta \right)+w_{\mathrm{c}} j_{\mathrm{c}}\left(\eta \right)=j_{\mathrm{A}}\left(\eta \right)+j_{\mathrm{C}}\left(\eta \right),$$

(96)

$$j_{\mathrm{A}}\left(\eta \right)=\frac{w_{\mathrm{a}} \exp \left(f_{\mathrm{a}}\eta \right)}{1/j_0+\exp \left(f_{\mathrm{a}}\eta \right)/j_{\mathrm{Red},\mathrm{L}}+\exp \left(-f_{\mathrm{c}}\eta \right)/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}} (>0),$$

(97)

$$j_{\mathrm{C}}\left(\eta \right)=\frac{-w_{\mathrm{c}} \exp \left(-f_{\mathrm{c}}\eta \right)}{1/j_0+\exp \left(f_{\mathrm{a}}\eta \right)/j_{\mathrm{Red},\mathrm{L}}+\exp \left(-f_{\mathrm{c}}\eta \right)/-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}} (<0).$$

(98)

where, the $w_{\mathrm{a}}\left(=S_{\mathrm{a}}/\mathrm{S}\right)$ and $w_{\mathrm{c}}\left(=S_{\mathrm{c}}/\mathrm{S}\right)$ are weighting factors that have suitably weighted values in proportion to the surface of anode and cathode, respectively ($0<w_{\mathrm{a}}, w_{\mathrm{c}}\le 1$) [29]. The $j_{\mathrm{A}}\left(\eta \right)$ and $j_{\mathrm{C}}\left(\eta \right)$ are effective anodic and cathodic branch current density, respectively. It is important to note that $E\left(0\right)$ when $w_{\mathrm{a}}\ne w_{\mathrm{c}}$ is different from $E\left(0\right)$ when $w_{\mathrm{a}}=w_{\mathrm{c}}\left(=1\right)$. When the overpotential having $w_{\mathrm{a}}\ne w_{\mathrm{c}}$ is termed as ${\eta }_{\mathrm{w}}$, the equilibrium state is expressed as:

$$j\left({\eta }_{\mathrm{w}}\right)=j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)+j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)=0.$$

(99)

Arranging the above,

$$\frac{w_{\mathrm{a}}}{w_{\mathrm{c}}}=\frac{-j_{\mathrm{c}}\left({\eta }_{\mathrm{w}}\right)}{j_{\mathrm{a}}\left({\eta }_{\mathrm{w}}\right)}=\frac{\exp \left(-f_{\mathrm{c}} {\eta }_{\mathrm{w}}\right)}{\exp \left(f_{\mathrm{a}} {\eta }_{\mathrm{w}}\right)}=\exp \left(-f {\eta }_{\mathrm{w}}\right).$$

(100)

The above shows that the relation between $w_{\mathrm{a}}/w_{\mathrm{c}}$ and ${\eta }_{\mathrm{w}}$ is mutually dependent and restricted by the above equation. Taking into account $w_{\mathrm{a}}/w_{\mathrm{c}}$, Equation (22) is arranged and expressed as:

$$h\left(j\right)=\frac{1}{g_{\mathrm{A}}\left(\eta \right)+g_{\mathrm{C}}\left(\eta \right)}=\frac{1}{f_{\mathrm{a}} j_{\mathrm{A}}\left(j\right)-f_{\mathrm{c}} j_{\mathrm{C}}\left(j\right)-j \left\{\frac{f_{\mathrm{a}} j_{\mathrm{A}}\left(j\right)}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}+\frac{f_{\mathrm{c}} j_{\mathrm{C}}\left(j\right)}{-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right\}},$$

(101)

where

$$g_{\mathrm{A}}\left(\eta \right)=\frac{\mathrm{d} j_{\mathrm{A}}\left(\eta \right)}{\mathrm{d} \eta }=f_{\mathrm{a}} j_{\mathrm{A}}\left(j\right)-j_{\mathrm{A}}\left(j\right) \left\{\frac{f_{\mathrm{a}} j_{\mathrm{A}}\left(j\right)}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}+\frac{f_{\mathrm{c}} j_{\mathrm{C}}\left(j\right)}{-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right\},$$

(102)

$$g_{\mathrm{C}}\left(\eta \right)=\frac{\mathrm{d} j_{\mathrm{C}}\left(\eta \right)}{\mathrm{d} \eta }=-f_{\mathrm{c}} j_{\mathrm{C}}\left(j\right)-j_{\mathrm{C}}\left(j\right) \left\{\frac{f_{\mathrm{a}} j_{\mathrm{A}}\left(j\right)}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}+\frac{f_{\mathrm{c}} j_{\mathrm{C}}\left(j\right)}{-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right\} ,$$

(103)

$$j\left(\eta \right)=j\left({\eta }_{\mathrm{w}}\right)+\frac{j^{\prime }\left({\eta }_{\mathrm{w}}\right)}{1 !}\left(\eta -{\eta }_{\mathrm{w}}\right)+\frac{j^{″}\left({\eta }_{\mathrm{w}}\right)}{2 !}{\left(\eta -{\eta }_{\mathrm{w}}\right)}^2+\cdots \approx g\left({\eta }_{\mathrm{w}}\right) \left(\eta -{\eta }_{\mathrm{w}}\right)=j,$$

(104)

$$j_{\mathrm{A}}\left(j\right)\approx j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)+\frac{ g_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}{ g_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)+g_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)} j=j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)+\left\{{\alpha }_{\mathrm{a}}-\left(\frac{{\alpha }_{\mathrm{a}} j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}} j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)}{-w_{\mathrm{c} }{j}_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\}j ,$$

(105)

$$\begin{gathered} j_{\mathrm{C}}\left(j\right)\approx j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)+\frac{g_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)}{g g_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)+g_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)} j \\ =j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)+\left\{{\alpha }_{\mathrm{c}}+\left(\frac{{\alpha }_{\mathrm{a}} j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}} j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)}{-w_{\mathrm{c} }{j}_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)\right\}j \end{gathered}$$

(106)

The $h\left(0\right)$ is expressed as:

$$h\left(0\right)=\frac{1}{f_{\mathrm{a}} j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)-f_{\mathrm{c}} j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)}=\frac{1}{f j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}=\frac{RT}{z F}\frac{1}{ j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}\left(=\frac{RT}{z F}\frac{1}{ -j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)}\right).$$

(107)

The ${\left[h\left(j\right)\right]}_{j\approx 0}$ is approximated to the linear relation:

$${\left[h\left(j\right)\right]}_{j\approx 0}=h\left(0\right)\left\{1+\left(\frac{2 {\alpha }_{\mathrm{a}}}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}} }+\frac{2 {\alpha }_{\mathrm{c}}}{ w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}+\frac{1-2 {\alpha }_{\mathrm{a}}}{j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}\right)j\right\}.$$

(108)

where,

$$j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)=-j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)=w_{\mathrm{a}} j_{\mathrm{a}}\left({\eta }_{\mathrm{w}}\right)=-w_{\mathrm{c}} j_{\mathrm{c}}\left({\eta }_{\mathrm{w}}\right)=\frac{w_{\mathrm{a}}{}^{{\alpha }_{\mathrm{c}}} w_{\mathrm{c}}{}^{{\alpha }_{\mathrm{a}}}}{1/j_0+{\left(\frac{w_{\mathrm{c}}}{w_{\mathrm{a}}}\right)}^{{\alpha }_{\mathrm{a}}}/ j_{\mathrm{Red},\mathrm{L}}+{\left(\frac{w_{\mathrm{a}}}{w_{\mathrm{c}}}\right)}^{{\alpha }_{\mathrm{c}}}/\left(-j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right)}.$$

(109)

We can see that the ${\left[h\left(j\right)\right]}_{j\approx 0}$ is approximated to a straight line having an intercept of $h\left(0\right)$ and a slope of $h\left(0\right)\left(\frac{2 {\alpha }_{\mathrm{a}}}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}} }+\frac{2 {\alpha }_{\mathrm{c}}}{ w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}+\frac{1-2 {\alpha }_{\mathrm{a}}}{j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}\right)$.

Graphical representation is helpful to see the influence of $w_{\mathrm{a}}/w_{\mathrm{c}}$ on the $h\left(j\right)$ shape. Employing reversible reaction data in Table 3, three reversible $j\left(\eta \right)$ curves were drawn as an example. Reversible $j\left(\eta \right)$, having ratios of $w_{\mathrm{a}}/w_{\mathrm{c}}=1$ (as a reference), $w_{\mathrm{a}}/w_{\mathrm{c}}=0.1/0.9$, and $w_{\mathrm{a}}/w_{\mathrm{c}}=0.6/0.8$, are shown in Figure 11. Their $h\left(j\right)$ are shown in Figure 12.

We can see that not only the $j\left(\eta \right)$ but also $h\left(j\right)$ is strongly influenced by the $w_{\mathrm{a}}/w_{\mathrm{c}}$ ratio. Equation (100) will be useful when estimating the $w_{\mathrm{a}}/w_{\mathrm{c}}$ ratio. It is necessary to pay careful attention when determining the kinetic parameters using the reversible ${\left[h\left(j\right)\right]}_{j\approx 0}$. Namely, graphical determination of ${\alpha }_{\mathrm{a}}$ and ${\alpha }_{\mathrm{c}}$ of reversible reaction is very difficult. The reason is that the line slope when $w_{\mathrm{a}}/w_{\mathrm{c}}=1$ and 0.75 ($=0.6/0.8$) is too small to determine. Even if it were possible, a 4–5 decimal place reading is needed for accurate determination.

In the case of the irreversible system having $w_{\mathrm{a}}/w_{\mathrm{c}}\ne 1$, the information on the anodic branch will be possible when being in a far polarized state ($\left|\eta \right|\gg 0$):

$$j=j_{\mathrm{A}}\left(j\right)+j_{\mathrm{C}}\left(j\right)\approx j_{\mathrm{A}}\left(j\right) \left(\mathrm{or} j_{\mathrm{C}}\left(j\right)\approx 0\right).$$

(110)

Let $j_{\mathrm{pA}}$ be the minimum $j_{\mathrm{A}}\left(j\right)$ in the above state. The $h_{\mathrm{A}-\mathrm{branch}}\left(j_{\mathrm{A}}\right)$ can be obtained:

$$h_{\mathrm{A}-\mathrm{branch}}\left(j_{\mathrm{A}}\right)=h_{\mathrm{A}-\mathrm{branch}}\left(j\right)={\left[h\left(j\right)\right]}_{j_{\mathrm{pA}}<j}\approx \frac{1}{f_{\mathrm{A}} j-j \left\{f_{\mathrm{A}} j /\left(w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}\right)\right\}}=\frac{RT}{{\alpha }_{\mathrm{a}} z F}\left(\frac{1}{ j}+\frac{1}{ w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j}\right).$$

(111)

The second term will be close to zero when $\left(w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j\right)\gg 0$:

$${\left[\frac{RT}{{\alpha }_{\mathrm{a}} z F}\frac{1}{ w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j}\right]}_{j_{\mathrm{pA}}<j\ll w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}={\left[\frac{0.026}{{\alpha }_{\mathrm{a}} z }\frac{1}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j}\right]}_{j_{\mathrm{pA}}<j\ll w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}\approx 0.$$

(112)

Then, Equation (111) is approximated as:

$${\left[h\left(j\right)\right]}_{j_{\mathrm{pA}}<j\ll w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}\approx \frac{RT}{{\alpha }_{\mathrm{a}} z F}\frac{1}{ j}=\frac{0.026}{{\alpha }_{\mathrm{a}} z }\frac{1}{ j}.$$

(113)

The value of ${\alpha }_{\mathrm{a}} z$ can be calculated by readings of $h\left(j\right)$ and j:

$${\alpha }_{\mathrm{a}} z\approx {\left[\frac{0.026}{ h\left(j\right) j}\right]}_{j_{\mathrm{pA}}<j\ll w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}} (j>0).$$

(114)

Similarly, ${\alpha }_{\mathrm{c}} z$ of the irreversible cathodic branch will be obtained when the system is cathodically far-polarized ($\eta \ll 0$):

$$j=j_{\mathrm{A}}\left(j\right)+j_{\mathrm{C}}\left(j\right)\approx j_{\mathrm{C}}\left(j\right) \left(\mathrm{or} j_{\mathrm{A}}\left(j\right)\approx 0\right).$$

(115)

Let $j_{\mathrm{pC}}$ be the maximum $j_{\mathrm{C}}\left(j\right)$ in the above state. Then, the $h_{\mathrm{C}-\mathrm{branch}}\left(j_{\mathrm{C}}\right)$ can be obtained:

$$h_{\mathrm{C}-\mathrm{branch}}\left(j_{\mathrm{C}}\right)=h_{\mathrm{C}-\mathrm{branch}}\left(j\right)={\left[h\left(j\right)\right]}_{j<j_{\mathrm{pC}}}\approx \frac{1}{-f_{\mathrm{c}} j-j \left\{f_{\mathrm{c}} j/\left(-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right)\right\}}=\frac{RT}{{\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}+\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right).$$

(116)

When $\left(j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right)\gg 0$, the second term will be close to zero:

$${\left[\frac{RT}{{\alpha }_{\mathrm{c}} z F}\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right]}_{w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\ll j<j_{\mathrm{pC}}}={\left[\frac{0.026}{{\alpha }_{\mathrm{c}} z }\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right]}_{w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\ll j<j_{\mathrm{pC}}}\approx 0.$$

(117)

Then, Equation (116) is approximated as:

$${\left[h\left(j\right)\right]}_{w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\ll j<j_{\mathrm{pC}}}\approx \frac{RT}{{\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}\right)=\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{-j}\right).$$

(118)

The value of ${\alpha }_{\mathrm{c}} z$ can be calculated as:

$${\alpha }_{\mathrm{c}} z\approx {\left[\frac{0.026}{ h\left(j\right) \left|j\right|}\right]}_{\left|j_{\mathrm{pC}}\right|\left.〈\left|j\right|\ll \right|w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}|} (j<0).$$

(119)

It is important to notice that the $w_{\mathrm{a}}$ and $w_{\mathrm{c}}$ can influence diffusion currents ($j_{\mathrm{Red},\mathrm{L}}$ and $j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$) but not charge current ($j_0$). Arranging Equations (105), (106), (110), and (115), $j_{\mathrm{pA}}$ and $j_{\mathrm{pC}}$ will be expressed as:

$$j_{\mathrm{pA}}={\left(\frac{{\alpha }_{\mathrm{c}}}{j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}+\frac{{\alpha }_{\mathrm{a}}}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}-\frac{{\alpha }_{\mathrm{c}}}{-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1},$$

(120)

$$j_{\mathrm{pC}}={\left(\frac{{\alpha }_{\mathrm{a}}}{j_{\mathrm{C}}\left({\eta }_{\mathrm{w}}\right)}-\frac{{\alpha }_{\mathrm{a}}}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}+\frac{{\alpha }_{\mathrm{c}}}{-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)}^{-1}.$$

(121)

Each of $j_{\mathrm{pA}}$ and $j_{\mathrm{pC}}$ is graphically characterized as inflection points on $h\left(j\right)$. We can obtain the relationship between $j_{\mathrm{pA}}$ and $j_{\mathrm{pC}}$ from Equations (120) and (121):

$$\frac{1}{j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)}=\frac{1}{j_{\mathrm{pA}}}+\frac{1}{-j_{\mathrm{pC}}}$$

(122)

We can see that the above relation is essentially same as Equation (37).

3.6.2. Solution Resistance

In the practical analyses, all $h\left(j\right)$ must be compensated with physical factors such as oxide film resistance and solution resistance. For instance, the cathodic branch $h\left(j\right)$ of the irreversible reaction must be compensated by oxide film resistance. It is expressed as:

$${\left[h\left(j\right)\right]}_{j<j_{\mathrm{pC}}}=\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{-j}+\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{f}}$$

(123)

The experimentally obtained curve always contains the solution resistance ${\left(l/\kappa \right)}_{\mathrm{s}}$.

$$\begin{gathered} {\left[h_{\exp }\left(j\right)\right]}_{j<j_{\mathrm{pC}}}={\left[h\left(j\right)\right]}_{j<j_{\mathrm{pC}}}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{s}}=\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{-j}+\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{f}}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{s}} \\ =\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{-j}+\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}. \end{gathered}$$

(124)

where, ${\left(l/\kappa \right)}_{\mathrm{c}}$ is the total cathodic polarization resistance $\left(= {\left(l/\kappa \right)}_{\mathrm{f}}+{\left(l/\kappa \right)}_{\mathrm{s}}\right)$. The above consists of three terms: $\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{-j}\right)$, $\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)$, and ${\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}$. The first term is related to the charge transfer process; the second term, to the diffusion transfer process; and the third term, to the physical factor. Since the ${\left(l/\kappa \right)}_{\mathrm{c}}$ is a constant value, it will graphically emerge as a horizontal line when $\left|j\right|$ becomes large:

$${\left[h_{\exp }\left(j\right)\right]}_{\left|j_{\mathrm{pC}}\right|<\left(\left|j\right|\to \mathrm{large}\right)}={\left[\frac{0.026}{{\alpha }_{\mathrm{c}} z }\left(\frac{1}{-j}+\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}\right]}_{\left|j\right|\to \mathrm{large}}\gtrsim {\left(l/\kappa \right)}_{\mathrm{c}}$$

(125)

Employing the above inequality, we can graphically estimate the value of ${\left(l/\kappa \right)}_{\mathrm{c}}$ as a horizontal line (an asymptote line). In order to visualize the relation between ${\left(l/\kappa \right)}_{\mathrm{c}}$ and the curve shape, the cathodic branch of the $h_{\exp }\left(j\right)$ ($w_{\mathrm{c}}=1$) was drawn using example data of ${\left(l/\kappa \right)}_{\mathrm{c}}={10}^{-3},{10}^{-2}$, and ${10}^{-1}{ \mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$. Their concrete expressions are shown below. Their curves are shown in Figure 13.

$$h_1\left(j\right)={\left[h_{\mathrm{C}-\mathrm{branch}}\left(j\right)+0\right]}_{\left|j\right|\to \mathrm{large}}={\left[0.019\left(\frac{1}{-j}+\frac{1}{j+100}\right)+0\right]}_{\left|j\right|\to \mathrm{large}}>0 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$$

(126)

$$h_2\left(j\right)={\left[h_{\mathrm{C}-\mathrm{branch}}\left(j\right)+{10}^{-3}\right]}_{\left|j\right|\to \mathrm{large}}={\left[0.019\left(\frac{1}{-j}+\frac{1}{j+100}\right)+{10}^{-3}\right]}_{\left|j\right|\to \mathrm{large}}>{10}^{-3} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$$

(127)

$$h_3\left(j\right)={\left[h_{\mathrm{C}-\mathrm{branch}}\left(j\right)+{10}^{-2}\right]}_{\left|j\right|\to \mathrm{large}}={\left[0.019\left(\frac{1}{-j}+\frac{1}{j+100}\right)+{10}^{-2}\right]}_{\left|j\right|\to \mathrm{large}}\gtrsim {10}^{-2} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$$

(128)

$$h_4\left(j\right)={\left[h_{\mathrm{C}-\mathrm{branch}}\left(j\right)+{10}^{-1}\right]}_{\left|j\right|\to \mathrm{large}}={\left[0.019\left(\frac{1}{-j}+\frac{1}{j+100}\right)+{10}^{-1}\right]}_{\left|j\right|\to \mathrm{large}}\cong {10}^{-1} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$$

(129)

Examining the relation between the curve shape and the horizontal lines, we can see a clear tendency that the $h\left(j\right)$ curve with the larger ${\left(l/\kappa \right)}_{\mathrm{c}}$ has the wider horizontal region. In addition to that, we can also see a tendency between ${\left(l/\kappa \right)}_{\mathrm{c}}$ and the Tafel slope region; the larger ${\left(l/\kappa \right)}_{\mathrm{c}}$, the narrower the Tafel slope region. The latter tendency gives us empirical advice that, to obtain the errorless Tafel parameters, we should employ an environment having scanty oxide film resistance and small electrolyte resistance. The above discussion is also valid for reversible and quasireversible reactions.

3.6.3. The $h_{\exp }\left(j\right)$ in the Whole Current Range

When the curve shape of $h\left(j\right)$ largely depends on the degree of polarization, division of $h\left(j\right)$ is needed for accurate analysis. Using $j_{\mathrm{pA}}$ and $j_{\mathrm{pC}}$, we can divide $h_{\exp }\left(j\right)$ into three parts:

$${\left[h_{\exp }\left(j\right)\right]}_{j_{\mathrm{pA}}<j<w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}=h_{\mathrm{A}-\mathrm{branch}}\left(j\right)+{\left(l/\kappa \right)}_{\mathrm{a}}=\frac{RT}{{\alpha }_{\mathrm{a}} z F}\left(\frac{1}{ j}+\frac{1}{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j}\right)+{\left(l/\kappa \right)}_{\mathrm{a}}$$

(130)

$${\left[h_{\exp }\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}={\left[h\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}+\left(l/\kappa \right)$$

(131)

$${\left[h_{\exp }\left(j\right)\right]}_{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}<j<j_{\mathrm{pC}}}=h_{\mathrm{C}-\mathrm{branch}}\left(j\right)+{\left(l/\kappa \right)}_{\mathrm{c}}=\frac{RT}{{\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}+\frac{1}{j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+{\left(l/\kappa \right)}_{\mathrm{c}}$$

(132)

${\left[h\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ in Equation (131) is almost identical to the ${\left[h\left(j\right)\right]}_{j\approx 0}$ in Equation (108), but we often encounter that the linear (the first order) approximation is graphically impossible. In this case, the second or third order approximation is necessary. Since the high-order approximation generally becomes complicated, the use of ${\left[h\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ will be helpful and effective. Its content is detailed as:

$${\left[h\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}=\frac{1}{\left(1/{\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}+1/{\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}\right)}.$$

(133)

where the ${\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ and ${\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ is anodic and cathodic branch polarization resistance, respectively. They can be expressed by considering the following graphical relation:

$${\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}=\left\{\begin{array}{c}{h}_{\mathrm{A}}\left(j_{\mathrm{pA}}\right)=h\left(j_{\mathrm{pA}}\right) \\ h_{\mathrm{A}}\left(j_{\mathrm{pC}}\right)\to \infty \end{array}\right.$$

(134)

$${\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}=\left\{\begin{array}{c}{h}_{\mathrm{C}}\left(j_{\mathrm{pC}}\right)=h\left(j_{\mathrm{pC}}\right) \\ h_{\mathrm{C}}\left(j_{\mathrm{pA}}\right)\to \infty \end{array}\right.$$

(135)

Since ${\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ is the deformed curve of $h_{\mathrm{A}-\mathrm{branch}}\left(j_{\mathrm{A}}\right)$ and ${\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ is that of $h_{\mathrm{C}-\mathrm{branch}}\left(j_{\mathrm{C}}\right)$, the following expressions are available in the experimental reality:

$${\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}=\frac{RT}{{\alpha }_{\mathrm{a}} z F}\frac{w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}\left(j_{\mathrm{pA}}-j_{\mathrm{pC}}\right)}{j_{\mathrm{pA}}\left(w_{\mathrm{a}}{j}_{\mathrm{Red},\mathrm{L}}-j_{\mathrm{pC}}\right)}\left(\frac{1}{j-j_{\mathrm{pC}}}+\frac{1}{w_{\mathrm{a}}{j}_{\mathrm{Red},\mathrm{L}}-j}\right)$$

(136)

$${\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}=\frac{RT}{{\alpha }_{\mathrm{c}} z F}\frac{w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\left(j_{\mathrm{pA}}-j_{\mathrm{pC}}\right)}{j_{\mathrm{pC}}\left(j_{\mathrm{pA}}-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\right)}\left(\frac{1}{j_{\mathrm{pA}}-j}+\frac{1}{ j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)$$

(137)

The advantage of the above expressions is easy embodiment because the $j_{\mathrm{pA}}$ and $j_{\mathrm{pC}}$ can be directly read as inflection points appeared on $h_{\exp }\left(j\right)$. Fortunately, the $h_{\exp }\left(j\right)$ of reversible reaction can be expressed as a single equation shown below:

$${\left[h_{\exp }\left(j\right)\right]}_{w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\le j\le w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}}=\frac{RT}{ z F}\left(\frac{1}{ w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j}+\frac{1}{ j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+\left(l/\kappa \right)$$

(138)

Of course, the reversible $h_{\exp }\left(j\right)$ can also be obtained using Equation (133). The reason is that Equation (138) can be obtained by only substituting $j_{\mathrm{pA}}=w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}$ and $j_{\mathrm{pC}}=w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}$. After all, the $h_{\exp }\left(j\right)$ for all reactions can be summarized as follows;

$${\left[h_{\exp }\left(j\right)\right]}_{j_{\mathrm{pA}}\le j\le w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}} }=\frac{RT}{ {\alpha }_{\mathrm{a}} z F}\left(\frac{1}{ j}+\frac{1}{ w_{\mathrm{a}} j_{\mathrm{Red},\mathrm{L}}-j}\right)+{\left(l/\kappa \right)}_{\mathrm{a}}$$

(139)

$${\left[h_{\exp }\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}} }=\frac{1}{1/{\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}} }+1/{\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}} }}+\left(l/\kappa \right)$$

(140)

where, ${\left[h_{\mathrm{A}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ is Equation (136), and ${\left[h_{\mathrm{C}}\left(j\right)\right]}_{j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}}$ is Equation (137).

$${\left[h_{\exp }\left(j\right)\right]}_{w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}\le j\le j_{\mathrm{pC}}}=\frac{RT}{ {\alpha }_{\mathrm{c}} z F}\left(\frac{1}{-j}+\frac{1}{ j-w_{\mathrm{c}} j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}\right)+{\left(l/\kappa \right)}_{\mathrm{c}}$$

(141)

Here, at practical curve analysis, we can use a few distinctive characteristics on the $h_{\exp }\left(j\right)$: (1) If we can observe a straight line having a slope of −1, it is an appearance of irreversible process corresponding to rds. (2) When a vertical line is found, its current corresponds to a limiting diffusion current; the anodic vertical line corresponds to the limiting diffusion of reductant, Red, and the cathodic vertical line to that of oxidant, ${\mathrm{Ox}}^{\mathrm{z}+}$. (3) When we can find a horizontal line at large |j|, its value corresponds to physical resistance.

3.7. Determination of Stable Chemical Species on $E_{\exp }\left(j\right)$ Using E-pH Diagram

The stable species appearing on $E_{\exp }\left(j\right)$ was examined using the E-pH diagram [30]. Since H₂ and CO have been directly injected into the test solution, two E-pH diagrams must be considered. The E-pH diagram of CO/CO₂ system superimposed by the H₂/H⁺ system is shown in Figure 14.

Three equilibrium lines, an upper blue dashed line ($ⓑ$) of $E_{\mathrm{eq}}\left({\mathrm{H}}_2\mathrm{O}/{\mathrm{O}}_2\right)=1.23-0.059 \mathrm{pH}$, a lower blue dashed line ($ⓐ$) of $E_{\mathrm{eq}}\left({\mathrm{H}}_2/{\mathrm{H}}^{+}\right)=-0.059 \mathrm{pH}$, and a lowest orange dashed line of $E_{\mathrm{eq}}\left(\mathrm{CO}/{\mathrm{CO}}_2\right)=-0.10-0.059 \mathrm{pH}$ are depicted in Figure 14. A red vertical line shows experimental tracks of $E_{\exp }\left(j\right)$. The reason why all curves of $E_{\exp }\left(j\right)$ follow along the red vertical line is that a 0.5 mol dm⁻³ H₂SO₄ has remained at constant $\left(\approx \mathrm{pH} 0.3\right)$ due to the very strong acid. Therefore, the thermodynamically stable chemical species can only appear on the red vertical line. Their chemical species are H₂, H₂O (rich H⁺), CO, and CO₂. The reactions discussed in this paper were restricted to the redox reactions related to the four species. There is an opinion that the derivatives must be also discussed due to the unstable CO in the water [30]. For instance, they said that the following chemical reaction is necessary to be discussed:

$$\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}=\mathrm{HCOOH}.$$

(142)

Furthermore, they said that, since the HCOOH is also unstable [30], the below oxidation reactions are possible when being anodically polarized:

$$\mathrm{HCOOH}+{\mathrm{H}}_2\mathrm{O}\to {\mathrm{H}}_2{\mathrm{CO}}_3+2{\mathrm{H}}^{+}+2\mathrm{e},$$

(143)

or

$$\mathrm{HCOOH}\to {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2\mathrm{e}.$$

(144)

Although carbon derivatives such as HCOOH and H₂CO₃ are considered as chemical attendants in actual operating environments, this paper limits to the redox reactions among four species (H₂, H⁺, CO, and CO₂) for the sake of simple and essential discussion.

3.8. Curve Analysis for Reversible HER in Environment (I)

Stable chemical species and their redox reactions appearing on $E_{\exp }\left(j\right)$ in H₂ + H₂SO₄ solution were examined. Since the E-pH diagram showed that the stable species at the state $\left(\mathrm{pH}\approx 0.3, E_{\mathrm{ocp}}\approx 0 \mathrm{V}\right)$ are H₂ and H⁺, the redox reaction is the reversible HER:

$${\mathrm{H}}_2\rightleftarrows 2{\mathrm{H}}^{+}+2\mathrm{e}$$

(145)

The reciprocal redox reaction occurs at E_eq; the injected H₂ is oxidized to H⁺, and at the same time, the H⁺ is reduced to H₂. The $h\left(j\right)$ belongs properly to the reaction of the category (A). The analysis will be carried out employing curve techniques in this category.

3.8.1. Estimation of $w_{\mathrm{a}}$ and $w_{\mathrm{c}}$

First, the estimation of $w_{\mathrm{a}}$ and $w_{\mathrm{c}}$ is examined. The representative tracks of $\left(\mathrm{A}\right)\to \left(\mathrm{B}\right)\to \left(\mathrm{C}\right)\to \left(\mathrm{D}\right)\to \left(\mathrm{E}\right)\to \left(\mathrm{B}\right)$ in Figure 2 was employed. The polarizing start point (A) stays on the line $ⓐ$, on which the reversible HER occurs. When $\eta <0$, the cathodic branch as her will be increased. Since H⁺ abundantly exists in the solution, the her exponentially increases with increase of $\left|\eta \right|$. When $\eta \ll 0$, the system will be overwhelmed by her (point (B)). When $\eta >0$, on the contrary, the anodic branch as hor can be observed at $\left(\mathrm{C}\right)\to \left(\mathrm{D}\right)$. Due to the poor solubility of H₂, the anodic current of hor will reach a certain limit. This vertical line has its root in the limiting diffusion current of H₂, $j_{{\mathrm{H}}_2,\mathrm{L}}$. We can calculate the theoretical $j_{{\mathrm{H}}_2,\mathrm{L}}$ based on two suppositions: (1) a moderate agitation using magnetic stirrer (${\delta }_{{\mathrm{H}}_2}$ dependincg on agitation. It was estimated to $\approx 0.01 \mathrm{cm}$ in this experiment [2,6,7,31]), and (2) the test solution in the H₂-saturated state ($\approx {10}^{-3} {\mathrm{mol} \mathrm{dm}}^{-3}$) [5,32]. Their calculations are shown as:

$$\begin{gathered} j_{{\mathrm{H}}_2,\mathrm{L}}=z F \frac{D_{{\mathrm{H}}_2}}{{\delta }_{{\mathrm{H}}_2}}{\left[{\mathrm{H}}_2\right]}_{\mathrm{sat}.}\approx \left(2\right)\left(96.5\times {10}^3{\mathrm{A} \mathrm{s} \mathrm{mol}}^{-1}\right)\left(\frac{3.8\times {10}^{-5}{\mathrm{cm}}^2{ \mathrm{s}}^{-1}}{0.01\mathrm{cm}}\right)\left({10}^{-3} {\mathrm{mol} \mathrm{dm}}^{-3}\right) \\ \approx 0.73 {\mathrm{mA} \mathrm{cm}}^{-2}. \end{gathered}$$

(146)

Table 1 tells that the $w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}$ as an anodic vertical line was:

$$w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}\approx 0.7 {\mathrm{mA} \mathrm{cm}}^{-2}.$$

(147)

Then, $w_{\mathrm{a}}$ can be calculated as:

$$w_{\mathrm{a}}=\frac{w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}}{j_{{\mathrm{H}}_2,\mathrm{L}}}=\frac{0.7 {\mathrm{mA} \mathrm{cm}}^{-2}}{0.74 {\mathrm{mA} \mathrm{cm}}^{-2}}=0.96.$$

(148)

On the other hand, we can estimate the $w_{\mathrm{a}}/w_{\mathrm{c}}$ ratio by employing Equation (100). Substituting ${\eta }_{\mathrm{w}}=0 \mathrm{V}$ into Equation (100) returns the following:

$$\frac{w_{\mathrm{a}}}{w_{\mathrm{c}}}=\exp \left(\frac{-z F}{RT} {\eta }_{\mathrm{w}}\right)=\exp \left(\frac{-2}{0.026}\times 0\right)=1.$$

(149)

The above two results can lead to the conclusion of $w_{\mathrm{a}}=w_{\mathrm{c}}=0.96$. Using $w_{\mathrm{c}}=0.96$, we can estimate the $w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}$:

$$\begin{gathered} w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}=-w_{\mathrm{c}} z F \frac{D_{{\mathrm{H}}^{+}}}{{\delta }_{{\mathrm{H}}^{+}}}{\left[{\mathrm{H}}^{+}\right]}_{\mathrm{bulk}} \\ \approx -\left(0.96\right)\left(1\right)\left(96.5\times {10}^{3 }{\mathrm{A} \mathrm{s} \mathrm{mol}}^{-1}\right)\left(\frac{7\times {10}^{-5}{ \mathrm{cm}}^2{ \mathrm{s}}^{-1}}{ 0.01 \mathrm{cm}}\right)\left({10}^{-0.3} {\mathrm{mol} \mathrm{dm}}^{-3}\right)=-325 {\mathrm{mA} \mathrm{cm}}^{-2}. \end{gathered}$$

(150)

Unfortunately, we cannot find the actual $w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}$ experimentally because it is over-scaled in Figure 2.

3.8.2. Determination of z for Reversible HER

Using the readings in Table 1, rough estimation of z is possible. The $z$ is calculated by Equation (45), which is modified by $w_{\mathrm{a}}$, $w_{\mathrm{c}}$, and $l/\kappa$:

$$z=\frac{R T}{F}\frac{1}{h_{\exp }\left(0\right)-l/\kappa }\left(\frac{1}{w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}}+\frac{2}{-w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)\approx \frac{0.026}{2.2\times {10}^{-2}-{10}^{-3}}\left(\frac{1}{0.7}+\frac{2}{325}\right)= 1.8.$$

(151)

If the 1.8 can be rounded to an integer, the z becomes 2. In order to obtain the more accurate z, the parallel displacement technique will be helpful. Employing Equation (48), we can plot the relation between $\log \left\{h\left(j+w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}\right)-l/\kappa \right\}$ and $\log \left|j\right|$. The results of all curves are shown in Figure 15.

We can find that there is a current region satisfying the Tafel slope, which is depicted as the green-toned region ($4{ \mathrm{mA} \mathrm{cm}}^{-2}\lesssim \left|j\right|\lesssim 8{ \mathrm{mA} \mathrm{cm}}^{-2}$). The relation between the calculated z and the green-toned $\left|j\right|$ were plotted. The results are shown in Figure 16.

We can see that the HER occurred on Pt electrode has exactly z = 2. In this calculation, the precondition of Equation (47) has been confirmed in advance:

$${\left[\frac{RT}{z F}\frac{2}{j+w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}-w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}}\right]}_{4\lesssim \left|j\right|\lesssim 8 }=\frac{0.026}{2}\frac{2}{\left(4~8\right)+0.7+325}\lesssim {10}^{-4.1}.$$

(152)

It is acceptable in the margin of error that the approximation of $\lesssim {10}^{-4.1}$ is $\approx 0$.

3.8.3. Confirmation of $j_{\mathrm{A}}\left(0\right)\approx j_{\mathrm{d}}$ for Reversible HER

When $w_{\mathrm{a}}=w_{\mathrm{c}}=0.96$, the $j_{\mathrm{d}}$ is calculated to:

$$j_{\mathrm{d}}=\frac{1}{1/\left(w_{\mathrm{a} }{j}_{{\mathrm{H}}_2,\mathrm{L}}\right)+1/\left(-w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}\right)}=\frac{1}{1/0.7+1/325}\approx 0.70 {\mathrm{mA} \mathrm{cm}}^{-2}.$$

(153)

Another way is possible by employing Equation (122):

$${\left[j_{\mathrm{A}}\left({\eta }_{\mathrm{w}}\right)\right]}_{{\eta }_{\mathrm{w}}=0}={\left[j_{\mathrm{A}}\left(j\right)\right]}_{j=0}=\frac{1}{{\frac{1}{j}}_{\mathrm{pA}}+\frac{1}{-j_{\mathrm{pC}}}}=\frac{1}{\frac{1}{\left(w_{\mathrm{a} }{j}_{{\mathrm{H}}_2,\mathrm{L}}\right)}+\frac{1}{\left(-w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}\right)}}=\frac{1}{\frac{1}{0.7}+\frac{1}{325}}\approx 0.70 {\mathrm{mA} \mathrm{cm}}^{-2}.$$

(154)

We can see that $j_{\mathrm{A}}\left(0\right)$ for reversible HER is confirmed to $0.70 {\mathrm{mA} \mathrm{cm}}^{-2}$, and it is exactly the same as the $j_{\mathrm{d}}$ ($=w_{\mathrm{a} }{j}_{{\mathrm{H}}_2,\mathrm{L}}$). In addition to the above, there is another way to obtain the $j_{\mathrm{A}}\left(0\right)$. Its value can be calculated by substituting the experimental readings in Table 1 into Equation (109):

$$j_{\mathrm{A}}\left(0\right)=\frac{{0.96}^{{\alpha }_{\mathrm{c}}} {0.96}^{{\alpha }_{\mathrm{a}}}}{1/j_0+ {\left(0.96/0.96\right)}^{{\alpha }_{\mathrm{a}}}/ 0.74 +{\left(0.96/0.96\right)}^{{\alpha }_{\mathrm{c}}}/ 339}\approx \frac{{\left(0.96\right)}^1 }{1/\infty + {\left(1\right)}^{{\alpha }_{\mathrm{a}}}/ 0.74 +{\left(1\right)}^{{\alpha }_{\mathrm{c}}}/ 339}=0.71{ \mathrm{mA} \mathrm{cm}}^{-2}.$$

(155)

We can see that the above $j_{\mathrm{A}}\left(0\right)$ well agrees with the above two results.

3.8.4. Determination of Kinetic Parameters for Reversible HER

It is reported that the Tafel slope of reversible HER is $\approx 30{ \mathrm{mV} \mathrm{decade}}^{-1}$, although the reported values have a scattered tendency [2,3,33]. It has been also pointed out that reliable value will be obtained in a low overpotential region [33,34]. Ambiguous problems have been remained when employing the TEM. The employment of $h\left(j\right)$ can bring the clear conclusion to $b=30{ \mathrm{mV} \mathrm{decade}}^{-1}$. Since the slope of the differentiated Tafel equation (Equation (39)) is identical with the slope as a result of the parallel displacement (Equation (48)), $\mathrm{d} \eta /\mathrm{d} j \approx h\left(j+w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}\right)$. Namely,

$$\frac{\mathrm{d} \eta }{\mathrm{d} j}=\frac{b}{2.3}\frac{1}{\left|j\right|}={\left[\frac{0.026}{z \left|j\right| }\right]}_{z=2}$$

(156)

From the above, the b is as below:

$$b=2.3\times 0.013=0.0299 {\mathrm{V} \mathrm{decade}}^{-1}\approx 30 {\mathrm{mV} \mathrm{decade}}^{-1}.$$

(157)

Using $h\left(j\right)$, it is conclusively proved that the Tafel slope of reversible HER is proved to be $30 {\mathrm{mV} \mathrm{decade}}^{-1}$.

3.8.5. Agreement between $E_{\exp }\left(j\right)$ and $E_{\mathrm{th}}\left(j\right)$

Employing three data ($l/\kappa ={10}^{-3} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$, $w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}=0.7 {\mathrm{mA} \mathrm{cm}}^{-2}$, and $w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}=-325 {\mathrm{mA} \mathrm{cm}}^{-2}$), the $h_{\mathrm{th}}\left(j\right)$ is shown as:

$$h_{\mathrm{th}}\left(j\right)=h_{\mathrm{rev}}\left(j\right)+\frac{l}{\kappa }=\frac{R T}{z F }\left(\frac{1}{w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}} - j}+\frac{2}{ j- w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)+\frac{l}{\kappa }=\frac{0.026}{2 }\left(\frac{1}{0.7 - j}+\frac{2}{ j+325}\right)+{10}^{-3}.$$

(158)

The above $h_{\mathrm{th}}\left(j\right)$ was plotted on the $h_{\exp }\left(j\right)$ in Figure 3. The result is shown in Figure 17.

We can see that the red $h_{\mathrm{th}}\left(j\right)$ is on the gray $h_{\exp }\left(j\right)$. In addition to the case of $h\left(j\right)$, it is also necessary to confirm the agreement between $E_{\exp }\left(j\right)$ and $E_{\mathrm{th}}\left(j\right)$. The $E_{\mathrm{th}}\left(j\right)$ can be obtained by solving the following differentiated equation under an initial condition:

$$\frac{\mathrm{d} E_{\mathrm{th}}\left(j\right)}{\mathrm{d} j}=h_{\mathrm{th}}\left(j\right)=\frac{0.026}{2 }\left(\frac{1}{0.7 - j}+\frac{2}{ j+325}\right)+{10}^{-3}$$

(159)

$$E_{\mathrm{th}}\left(0\right)=0 \mathrm{V} \left(\mathrm{initial} \mathrm{condition}\right)$$

(160)

The above solution is as below:

$$E_{\mathrm{th}}\left(j\right)=0.013 \ln \frac{{\left(j+325\right)}^2}{0.7 - j }\frac{0.7}{{325}^2}+{10}^{-3}j$$

(161)

The concrete $E_{\mathrm{th}}\left(j\right)$ is drawn with red on the gray $E_{\exp }\left(j\right)$. The result is shown in Figure 18.

The result that $E_{\mathrm{th}}\left(j\right)$ considerably overlapped with $E_{\exp }\left(j\right)$ is conclusive evidence for the valid approximation.

When the $E\left(j\right)$ was embodied, $\eta$ can be expressed as a function of j:

$$\eta =E-E_{\mathrm{eq}}=E\left(j\right)-E\left(0\right)=0.013\ln \frac{0.7}{{325}^2}\frac{{(j+325)}^2}{\left(0.7-j\right)}+{10}^{-3} \left(=\eta \left(j\right)\right).$$

(162)

The relation between $\left|\eta \right|$ and $\left|j\right|$ is shown in Figure 19.

Figure 19 shows that small overpotential of $\left|\eta \right|\approx 1 \mathrm{mV}$ can flow $\left|j\right|\approx 0.1 {\mathrm{mA} \mathrm{cm}}^{-2}$ to both of branches. In a case of $\left|\eta \right|\approx 10 \mathrm{mV}$, a current deference between the branches takes place; $\approx 0.6 {\mathrm{mA} \mathrm{cm}}^{-2}$ flowing to the anodic branch and $\approx -2 {\mathrm{mA} \mathrm{cm}}^{-2}$ to the cathodic branch. We can see that the current flowing to the cathodic branch is easier than that of the anodic branch in the case of HER.

3.9. Curve Analysis of $h_{\exp }\left(j\right)$ in Environment (II)

Figure 1 shows that the $E_{\mathrm{ocp}}$ was shifted to noble potential ($0.26~0.34 \mathrm{V}$) when CO was added into the H₂ + H₂SO₄ solution. The stable species at this potential are H⁺ and CO₂. The injected H₂ and CO will be oxidized to H⁺ and CO₂ when they come to contact with Pt:

$${\mathrm{H}}_2\to 2{\mathrm{H}}^{+}+2\mathrm{e}.$$

(163)

$$\mathrm{CO} +{ \mathrm{H}}_2\mathrm{O}\to {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2\mathrm{e},$$

(164)

Although the CO/CO₂ redox reaction is usually slow, it will be stimulated by the Pt catalysis and raise to a reasonable rate [30]. Being cathodically far polarized, Figure 4 shows that $\left|j\right|$ exponentially increases and reaches (G). Since (G) is located at −0.5 V, the E-pH diagram shows that the stable species are changed to H₂ and CO from the previous H⁺ and CO₂. At (G), the following cathodic reactions will take place:

$${\mathrm{H}}_2\leftarrow 2{\mathrm{H}}^{+}+2\mathrm{e},$$

(165)

$$\mathrm{CO} +{ \mathrm{H}}_2\mathrm{O}\leftarrow {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2\mathrm{e}.$$

(166)

The former reaction can vigorously occur due to sufficient supply of H⁺ and the latter poorly occur due to insufficient supply of CO₂. Reversing potential, the anodic current was observed around (H). Since this point is in the stable region of CO₂ and H⁺, the H₂ and CO (actually H_ad and CO_ad, discussed later) will be oxidized to H⁺ and CO₂ again.

$${\mathrm{H}}_{\mathrm{ad}}\to {\mathrm{H}}^{+}+\mathrm{e},$$

(167)

$${\mathrm{CO}}_{\mathrm{ad} }+{ \mathrm{H}}_2\mathrm{O}\to {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2\mathrm{e}.$$

(168)

Since the repeated reactions shown above influence on the curve shape, their $h\left(j\right)$ will become complicated. In order to be simple $h\left(j\right)$, the route of $\left(\mathrm{F}\right)\to \left(\mathrm{G}\right)\to \left(\mathrm{H}\right)\to \left(\mathrm{I}\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{K}\right)$ in Figure 5 was broadly divided into three parts: (1)$\left(\mathrm{F}\right)\to \left(\mathrm{G}\right)$ (downward); (2) $\left(\mathrm{G}\right)\to \left(\mathrm{H}\right)\to \left(\mathrm{I}\right)$ (upward); and (3) $\left(\mathrm{I}\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{K}\right)$ (downward). If necessary, some of them were further separated into smaller parts.

3.9.1. Analysis of $\left(\mathrm{F}\right)\to \left(\mathrm{G}\right)$

The $h_{\exp }\left(j\right)$ of downward $\left(\mathrm{F}\right)\to \left(\mathrm{G}\right)$ is picked up and shown again in Figure 20 together with the $E_{\exp }\left(j\right)$.

(a)

Point Analysis of $\left(\mathrm{F}\right)$

Since the stable chemical species at (F) are CO₂ and H⁺, the injected H₂ and CO are changed to H⁺ and CO₂. The redox reactions related to them are shown as:

$${\mathrm{H}}_2\rightleftarrows 2{\mathrm{H}}^{+}+2\mathrm{e} \left(j_1,E_{\mathrm{eq}1}\right),$$

(169)

$$\mathrm{CO} +{ \mathrm{H}}_2\mathrm{O}\rightleftarrows {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2\mathrm{e} \left(j_2,E_{\mathrm{eq}2}\right).$$

(170)

Since $E_{\mathrm{eq}1}\left(=0 \mathrm{V}\right)>E_{\mathrm{eq}2}\left(=-0.1 \mathrm{V}\right)$, the mixed system will be built up. The following reactions having mixed potential ($E_{\mathrm{m}}, E_{\mathrm{eq}2}<E_{\mathrm{m}}<E_{\mathrm{eq}1})$ will proceed:

$${\mathrm{H}}_2\leftarrow 2{\mathrm{H}}^{+}+2\mathrm{e} \left(j_{\mathrm{c}1},E_{\mathrm{eq}1}\right),$$

(171)

$$\mathrm{CO} +{ \mathrm{H}}_2\mathrm{O}\to {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2\mathrm{e} \left(j_{\mathrm{a}2},E_{\mathrm{eq}2}\right).$$

(172)

The net current at $E_{\mathrm{m}}$ is expressed as:

$$j= j_{\mathrm{A}}+ j_{\mathrm{C}}=w_{{\mathrm{a}}_1} j_{{\mathrm{a}}_1}+w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}+w_{{\mathrm{a}}_2} j_{{\mathrm{a}}_2}+w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}\approx w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}+w_{{\mathrm{a}}_2} j_{{\mathrm{a}}_2}=0.$$

(173)

In the actual state, the $j_{{\mathrm{c}}_1}$ is prevented from attending because the solution is saturated with continuous H₂ bubbling and there is no space of the H₂ solubility. As a result, the her hardly occurs.

$$w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}\approx 0.$$

(174)

In the other hand, Equation (172) can occur because the continuous injections of H₂ and CO will purge the ${\mathrm{CO}}_2$ from the solution. There is a space for the CO₂ solubility. Its solubility can be estimated using the Nernst equation:

$$E_{\mathrm{eq}2}=-0.10-0.059 \mathrm{pH}+0.03 \log \frac{p_{{\mathrm{CO}}_2}}{p_{\mathrm{CO}}}$$

(175)

The relation between $\mathrm{CO}$ and ${\mathrm{CO}}_2$ is calculated at $E_{\mathrm{eq}2}=0.32 \mathrm{V} \left(\approx E_{\mathrm{ocp}}\right)$:

$$\frac{p_{{\mathrm{CO}}_2}}{p_{\mathrm{CO}}}={10}^{\frac{0.32+0.10+0.059\left(0.3\right)}{0.03}}\approx {10}^{14}$$

(176)

The thermodynamic answer tells us that the solution at (F) contains huge amounts of ${\mathrm{CO}}_2$. However, the huge amount does not mean infinite CO₂ solubility. We know that here is a solubility restriction; the actual ${\mathrm{CO}}_2$ solubility is up to its saturation [32]:

$${\left[{\mathrm{CO}}_2\right]}_{\mathrm{bulk}}\le {\left[{\mathrm{CO}}_2\right]}_{\mathrm{sat}}\approx {10}^{-1.5} {\mathrm{mol} \mathrm{dm}}^{-3}.$$

(177)

The CO₂ in the solution is probably over-saturated. As a result, the Pt electrode at (F) will be entirely surrounded with a plenty of CO and CO₂.

(b)

Part analysis of $\left(\mathrm{F}\right) \to \left(\mathrm{F}1\right)$

The reaction occurring on $\left(\mathrm{F}\right) \to \left(\mathrm{F}1\right) \to \left(\mathrm{F}{1}^{\prime }\right)$ is Equation (172). It is the cathodic branch of the reversible reaction of Equation (170). A clear vertical line observed on the way of $\left(\mathrm{F}1\right) \to \left(\mathrm{F}{1}^{\prime }\right)$ is a sign of the CO₂ reduction reaction having a diffusion process. We can analyze Equation (172) using $h\left(j\right)$. In order to compare the theoretical $h_{\mathrm{th}}\left(j\right)$ with the experimental $h_{\exp }\left(j\right)$, the $h_{\mathrm{th}}\left(j\right)$ of the stationary state can be expressed as below:

$$h_{\mathrm{th}}\left(j\right)= \frac{RT}{z F}\left(\frac{1}{w_{\mathrm{a}2} j_{\mathrm{CO},\mathrm{L}}-j}+\frac{1}{j-w_{\mathrm{c}2 }{j}_{{\mathrm{CO}}_2,\mathrm{L}}}\right)+\frac{l}{\kappa }.$$

(178)

Employing $w_{\mathrm{c}2 }{j}_{{\mathrm{CO}}_2,\mathrm{L}}=-1.5\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$ (cathodic vertical line, $\left(\mathrm{F}1\right)$ in Figure 5) and $w_{\mathrm{a}2} j_{\mathrm{CO},\mathrm{L}}=1.2\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$ (anodic vertical line, $\left(\mathrm{I}1\right)$ in Figure 21 shown later), $l/\kappa \approx 80 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$ (estimated using actual value of $h_{\exp }\left(0\right)$). It is expressed concretely as:

$$h_{\mathrm{th}}\left(j\right)= \frac{0.026}{2}\left(\frac{1}{1.2\times {10}^{-3}-j}+\frac{1}{j+1.5\times {10}^{-3}}\right)+80.$$

(179)

The cathodic part shown above is drawn on the gray $\left(\mathrm{F}\right) \to \left(\mathrm{F}1\right)$ with pink in Figure 21, in which the gray $h_{\exp }\left(j\right)$ is shown for comparison.

The large value of $l/\kappa \left(=80 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2\right)$ suggests that the Pt electrode surface was heavily covered with adsorption CO₂ and CO. In addition to the large $l/\kappa$, there is other evidence relating to the heavy coverage. Considering the actually observed values of CO and CO₂, the $w_{\mathrm{a}}$ and $w_{\mathrm{c} }$ can be roughly estimated using the following $j_{\mathrm{CO},\mathrm{L}}$ and $j_{{\mathrm{CO}}_2,\mathrm{L}}$:

$$\begin{gathered} j_{\mathrm{CO},\mathrm{L}}=zF \frac{D_{\mathrm{CO}}}{{\delta }_{\mathrm{CO}}}{\left[\mathrm{CO}\right]}_{\mathrm{sat}.}\approx \left(2\right)\left(96.5\times {10}^{3 }{\mathrm{A} \mathrm{s} \mathrm{mol}}^{-1}\right)\left(\frac{{10}^{-5} {\mathrm{cm}}^2 {\mathrm{s}}^{-1}}{ 0.01 \mathrm{cm}}\right)\left({10}^{-3} {\mathrm{mol} \mathrm{dm}}^{-3}\right) \\ =0.19 {\mathrm{mA} \mathrm{cm}}^{-2}, \end{gathered}$$

(180)

$$\begin{gathered} j_{{\mathrm{CO}}_2,\mathrm{L}}=-zF \frac{D_{{\mathrm{CO}}_2}}{{\delta }_{{\mathrm{CO}}_2}}{\left[{\mathrm{CO}}_2\right]}_{\mathrm{sat}.}\approx -\left(2\right)\left(96.5\times {10}^{3 }{\mathrm{A} \mathrm{s} \mathrm{mol}}^{-1}\right)\left(\frac{{10}^{-5}{ \mathrm{cm}}^2{ \mathrm{s}}^{-1}}{ 0.01 \mathrm{cm}}\right)\left({10}^{-1.5} {\mathrm{mol} \mathrm{dm}}^{-3}\right) \\ =-6.1 {\mathrm{mA} \mathrm{cm}}^{-2}. \end{gathered}$$

(181)

Each of $w_{\mathrm{a}2}$ and $w_{\mathrm{c}2}$ can be calculated using the above results:

$$w_{\mathrm{a}2}= \frac{w_{\mathrm{a}2} j_{\mathrm{CO},\mathrm{L}}}{ j_{\mathrm{CO},\mathrm{L}}}= \frac{1.2\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}}{ 0.19 {\mathrm{mA} \mathrm{cm}}^{-2}}={10}^{-2.2},$$

(182)

$$w_{\mathrm{c}2}=\frac{w_{\mathrm{c}2}{j}_{{\mathrm{CO}}_2,\mathrm{L}}}{ j_{{\mathrm{CO}}_2,\mathrm{L}}}=\frac{-1.5\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}}{-6.1 {\mathrm{mA} \mathrm{cm}}^{-2}}={10}^{-3.6}.$$

(183)

Small $w_{\mathrm{a}2}\left(={10}^{-2.2}\right)$ and $w_{\mathrm{c}2}\left(={10}^{-3.6}\right)$ give confirmatory evidence that the active surface is almost lost. Close observation of Figure 5 shows that all $h_{\exp }\left(j\right)$ have peculiar humps around (F2). Considering the humps, it is natural to divide the $\left(\mathrm{F}1\right) \to \left(\mathrm{G}\right)$ into two parts: $\left(\mathrm{F}{1}^{\prime }\right) \to \left(\mathrm{F}2\right)$ and $\left(\mathrm{F}2\right) \to \left(\mathrm{G}\right)$.

(b1)

Part Analysis of $\left(\mathrm{F}{1}^{\prime }\right) \to \left(\mathrm{G}\right)$

Since $E\left(j\right)$ in Figure 20 shows that the stable chemical species of $\left(\mathrm{F}{1}^{\prime }\right) \to \left(\mathrm{G}\right)$ are CO and H₂, the following two cathodic reactions can occur competitively:

$${\mathrm{H}}_2\leftarrow 2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-} \left(j_{\mathrm{c}1},E_{\mathrm{eq}1}\right),$$

(184)

$$\mathrm{CO}+{ \mathrm{H}}_2\mathrm{O}\leftarrow {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-} \left(j_{\mathrm{c}2},E_{\mathrm{eq}2}\right).$$

(185)

(b2)

Part analysis of $\left(\mathrm{F}{1}^{\prime }\right) \to \left(\mathrm{F}2\right)$

The j in this cathodic region can be expressed as:

$$j= j_{\mathrm{A}}+ j_{\mathrm{C}}\approx j_{\mathrm{C}}=w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}+w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}.$$

(186)

Considering that CO₂ is consumed and poorly supplied, it will be acceptable that the $j_{\mathrm{c}2}$ (=$j_{{\mathrm{CO}}_2,\mathrm{L}}$) is almost constant current. The $j_{\mathrm{c}1}$ can be arranged as:

$$j_{\mathrm{c}1}=\left(j-w_{\mathrm{c}2} j_{\mathrm{c}2}\right)/w_{\mathrm{c}1}=\left(j-w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}\right)/w_{\mathrm{c}1}.$$

(187)

Then, the $h\left(j\right)$ is expressed as:

$$h\left(j\right)=\frac{\mathrm{d} E}{\mathrm{d} j}=\frac{\mathrm{d} E}{\mathrm{d} \left(w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}+w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}\right)}=\frac{1}{w_{{\mathrm{c}}_1} \mathrm{d}{j}_{{\mathrm{c}}_1}/\mathrm{d}E +w_{{\mathrm{c}}_2}\mathrm{d}{j}_{{\mathrm{c}}_2}/\mathrm{d}E}=\frac{1}{w_{{\mathrm{c}}_1}/h_{{\mathrm{c}}_1}(j_{{\mathrm{c}}_1})+w_{{\mathrm{c}}_2}/h_{{\mathrm{c}}_2}(j_{{\mathrm{c}}_2})}=\frac{1}{w_{\mathrm{c}1}}{h}_{\mathrm{c}1}(j_{\mathrm{c}1}).$$

(188)

The above means that the $h\left(j\right)$ of $\left(\mathrm{F}{1}^{\prime }\right) \to \left(\mathrm{F}2\right)$ is identical to $h_{\mathrm{c}1}(j_{\mathrm{c}1})/w_{\mathrm{c}1}$. Since the $h_{\mathrm{c}1}(j_{\mathrm{c}1})$ is cathodic part of reversible reaction, the $h_{\mathrm{th}}\left(j\right)$ is:

$$h_{\mathrm{th}}\left(j\right)=\frac{1}{w_{\mathrm{c}1}}{h}_{\mathrm{c}1}(j_{\mathrm{c}1})+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}} =\frac{1}{w_{\mathrm{c}1}}\left\{\frac{RT}{ z F}\left(\frac{1}{- j_{\mathrm{c}1}}+\frac{2}{ j_{\mathrm{c}1}-w_{\mathrm{c}1} j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)\right\}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}$$

(189)

Substituting Equation (187) into the above:

$$h_{\mathrm{th}}\left(j\right)=\frac{RT}{ z F}\left(\frac{1}{w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}-j}+\frac{2}{j-w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}-{w_{\mathrm{c}1}}^2 j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}.$$

(190)

Employing $w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}=-1.5\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$, $w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}+{w_{\mathrm{c}1}}^2 j_{{\mathrm{H}}^{+},\mathrm{L}}\approx -1.5\times {10}^{-2} {\mathrm{mA} \mathrm{cm}}^{-2}$ (vertical line extrapolated from (F2)), and ${\left(l/\kappa \right)}_{\mathrm{c}}\approx {10}^{-3} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$, the $h_{\mathrm{th}}\left(j\right)$ is embodied:

$$h_{\mathrm{th}}\left(j\right)=\frac{0.026}{ 2 }\left(\frac{1}{-1.5\times {10}^{-3}-j}+\frac{2}{j+1.5\times {10}^{-2}}\right)+{10}^{-3}.$$

(191)

The above $h_{\mathrm{th}}\left(j\right)$ having pink is added to the gray $\left(\mathrm{F}1\right) \to \left(\mathrm{F}2\right)$ in Figure 21. We can see that the $h_{\mathrm{th}}\left(j\right)$ is on the $h_{\exp }\left(j\right)$. The large decrease of ${\left(l/\kappa \right)}_{\mathrm{c}}$ from the previous $80 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$ to ${10}^{-3} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$ is due to the departure of adsorption CO and CO₂ from the electrode surface. It is originated from the H₂ evolution and the disappearance of CO₂ by the consumption reaction of Equation (185).

(b3)

Part analysis of $\left(\mathrm{F}2\right) \to \left(\mathrm{G}\right)$

Since a straight line having a slope of −1 was observed on $\left(\mathrm{F}2\right) \to \left(\mathrm{G}\right)$, we can regard it as an irreversible reaction having rds. Therefore, the total number of reactions occurring on $\left(\mathrm{F}2\right) \to \left(\mathrm{G}\right)$ are three. They are mutually competitive reactions.

$${\mathrm{H}}_2\leftarrow 2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\left(j_{\mathrm{c}1},E_{\mathrm{eq}1}\right),$$

(192)

$$\mathrm{CO} +{ \mathrm{H}}_2\mathrm{O} \leftarrow {\mathrm{CO}}_2+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-} \left(j_{\mathrm{c}2},E_{\mathrm{eq}2}\right),$$

(193)

$${\mathrm{H}}_{\mathrm{ad}} \leftarrow {\mathrm{H}}^{+}+{\mathrm{e}}^{-}\left(j_{\mathrm{c}3}, E_{\mathrm{eq}3}, \mathrm{rds}, \mathrm{disacussed} \mathrm{later}\right).$$

(194)

We can calculate the ${\alpha }_{\mathrm{c}3} z_3$ using the straight line data of $0.3 \lesssim \left|j\right|\lesssim 1 {\mathrm{mA} \mathrm{cm}}^{-2}$:

$${\alpha }_{\mathrm{c}3} z_3=\frac{0.026}{\left|j\right| {\left[h_{\exp }\left(j\right)-{10}^{-3}\right]}_{0.3 \lesssim \left|j\right|\lesssim 1}}\approx 0.62.$$

(195)

Needless to say, the following precondition has been confirmed in advance:

$$\frac{0.026}{{\alpha }_{\mathrm{c}3} z_3}\frac{1}{j- w_{\mathrm{c}3}{j}_{{\mathrm{H}}^{+},\mathrm{L}}}={\left[\frac{0.026}{0.7 }\frac{1}{j+ >100}\right]}_{\begin{array}{c}0.3 \lesssim \left|j\right|\lesssim 1\\ \end{array}}<{10}^{-3.4} \left(\approx 0\right).$$

(196)

Although the $|w_{\mathrm{c}3} j_{{\mathrm{H}}^{+},\mathrm{L}}|$ cannot be observed in Figure 5, it is certainly $>100 {\mathrm{mA} \mathrm{cm}}^{-2}$. The obtained values of ${\alpha }_{\mathrm{c}3}$ and $z_3$ must simultaneously satisfy the following restrictions:

$${\alpha }_{\mathrm{c}3} z_3=0.62$$

(197)

$$z_3=1, 2, \dots \left(\mathrm{integer}\right)$$

(198)

$$0<{\alpha }_{\mathrm{c}3}<1$$

(199)

The most preferable combination is ${\alpha }_{\mathrm{c}3}=0.62$ and $z_3=1$, because if $z_3$ = 2, Equation (194) becomes reversible reaction. The result of $z_3=1$ means that the rds is one electron transferred reaction. When the conjugate reductant of the H⁺ is H_ad, it is the well-known reaction having the Tafel–Volmer mechanism:

$${\mathrm{H}}_{\mathrm{ad}}\leftarrow {\mathrm{H}}^{+}+\mathrm{e}$$

(200)

The below reaction (Heyrovsky mechanism) is also possible if the result is z =1, only:

$${\mathrm{H}}_2\leftarrow {\mathrm{H}}^{+}+{\mathrm{H}}_{\mathrm{ad}}+\mathrm{e}$$

(201)

The net current in this track is shown as:

$$j= j_{\mathrm{A}}+j_{\mathrm{C}}\approx j_{\mathrm{C}}=w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}+w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}+w_{{\mathrm{c}}_3} j_{{\mathrm{c}}_3}$$

(202)

Taking into account of the limited $w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}\left(=w_{{\mathrm{c}}_1} j_{{\mathrm{H}}^{+},\mathrm{L}}\right)$ and the constant $w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}\left(=w_{{\mathrm{c}}_2} j_{{\mathrm{CO}}_2,\mathrm{L}}\right)$, the $h_{\mathrm{th}}\left(j\right)$ is shown as:

$$h_{\mathrm{th}}\left(j\right)=\frac{1}{w_{{\mathrm{c}}_1}/h_{{\mathrm{c}}_1}(j_{{\mathrm{c}}_1)}+w_{{\mathrm{c}}_2}/h_{{\mathrm{c}}_2}(j_{{\mathrm{c}}_2})+w_{{\mathrm{c}}_3}/h_{{\mathrm{c}}_3}(j_{{\mathrm{c}}_3})}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}=\frac{1}{w_{{\mathrm{c}}_3}}{h}_{{\mathrm{c}}_3}\left( j_{{\mathrm{c}}_3}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}.$$

(203)

Arranging Equation (202), $j_{{\mathrm{c}}_3}$ is shown as:

$$j_{{\mathrm{c}}_3}=\left(j-w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}-w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}\right)/w_{{\mathrm{c}}_3}$$

(204)

Similarly, the $h_{\mathrm{th}}\left(j\right)$ is expressed as:

$$\begin{gathered} h_{\mathrm{th}}\left(j\right)=\frac{1}{w_{{\mathrm{c}}_3}}\left\{\frac{RT}{ {\alpha }_{\mathrm{c}3} z_3 F}\left(\frac{1}{- j_{{\mathrm{c}}_3}}+\frac{1}{ j_{{\mathrm{c}}_3}-w_{{\mathrm{c}}_3} j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)\right\}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}} \\ =\frac{1}{w_{{\mathrm{c}}_3}}\left\{\frac{RT}{ {\alpha }_{\mathrm{c}3} z_3 F}\left(\frac{1}{-\frac{j-w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}-w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}}{w_{{\mathrm{c}}_3}}}+\frac{1}{\frac{j-w_{{\mathrm{c}}_1} j_{{\mathrm{c}}_1}-w_{{\mathrm{c}}_2} j_{{\mathrm{c}}_2}}{w_{{\mathrm{c}}_3}}-w_{{\mathrm{c}}_3} j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)\right\}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}} \\ =\frac{0.026}{ {\alpha }_{\mathrm{c}3} z_3}\left(\frac{1}{w_{\mathrm{c}1} j_{{\mathrm{H}}^{+},\mathrm{L}}+w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}-j}+\frac{1}{j-w_{\mathrm{c}1} j_{{\mathrm{H}}^{+},\mathrm{L}}-w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}-{w_{\mathrm{c}3}}^2 j_{{\mathrm{H}}^{+},\mathrm{L}}}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}. \end{gathered}$$

(205)

Employing the assumed value of $w_{\mathrm{c}1} j_{{\mathrm{H}}^{+},\mathrm{L}}+$ $w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}\approx -5\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$ and $|{w_{\mathrm{c}3}}^2 j_{{\mathrm{H}}^{+},\mathrm{L}}|>100 {\mathrm{mA} \mathrm{cm}}^{-2}$, $h_{\mathrm{th}}\left(j\right)$ can be embodied as:

$$h_{\mathrm{th}}\left(j\right)=\frac{0.026}{ 0.62 }\left(\frac{1}{-5\times {10}^{-3}-j}+\frac{1}{j + 5\times {10}^{-3} + >100}\right)+{10}^{-3}\approx 0.042\left(\frac{1}{-5\times {10}^{-3}-j}\right)+{10}^{-3}.$$

(206)

The pink curve above is added to the gray $\left(\mathrm{F}2\right) \to \left(\mathrm{G}\right)$. We can see that the $h_{\mathrm{th}}\left(j\right)$ is almost on the $h_{\exp }\left(j\right)$, except around (G). This exception may be caused by a fact that the ${\left(l/\kappa \right)}_{\mathrm{c}}$ around (G) is not always constant and probably changeable by the vigorous H₂ evolution.

3.9.2. Analysis of $\left(\mathrm{G}\right) \to \left(\mathrm{H}\right) \to \left(\mathrm{I}\right)$

The representative $h_{\exp }\left(j\right)$ of upward track of $\left(\mathrm{G}\right) \to \left(\mathrm{H}\right) \to \left(\mathrm{I}\right)$ is picked up and shown in Figure 22 together with its $E_{\exp }\left(j\right)$.

Considering the zigzag curve around (G1), it is natural to divide $\left(\mathrm{G}\right) \to \left(\mathrm{H}\right) \to \left(\mathrm{I}\right)$ into two parts: $\left(\mathrm{G}\right) \to \left(\mathrm{G}1\right)$ and $\left(\mathrm{G}1\right) \to \left(\mathrm{H}\right) \to \left(\mathrm{I}\right)$.

(a)

Part analysis of $\left(\mathrm{G}\right) \to \left(\mathrm{G}1\right)$

This part consists of the same cathodic reactions of $\left(\mathrm{F}2\right) \to \left(\mathrm{G}\right)$, but the electrons pumped into the system are gradually decreased this time. Similarly, as before, the ${\alpha }_{\mathrm{c}3} z_3$ can be calculated using the actual current range indicating a line slope of −1. The result is shown as:

$${\alpha }_{\mathrm{c}3} z_3=\frac{0.026}{\left|j\right| {\left[h_{\exp }\left(j\right)-{10}^{-3}\right]}_{0.1 \lesssim \left|j\right|\lesssim 1}}\approx 0.55.$$

(207)

The above result of 0.55 is different from the previous result of 0.62. Clear reasons are unknown, but one of them must be due to change of electrode surface after or before being attacked by her. In an example case of ${\alpha }_{\mathrm{c}3}\approx 0.6$ and $w_{\mathrm{c}1} j_{{\mathrm{H}}^{+},\mathrm{L}}+w_{\mathrm{c}2} j_{{\mathrm{CO}}_2,\mathrm{L}}\approx -1.5\times {10}^{-2} {\mathrm{mA} \mathrm{cm}}^{-2}$, the previous Equation (206) is changed to the following $h_{\mathrm{th}}\left(j\right)$:

$$h_{\mathrm{th}}\left(j\right)=\frac{0.026}{ 0.6 }\left(\frac{1}{-1.5\times {10}^{-2}-j}+\frac{1}{j + 1.5\times {10}^{-2} + >100}\right)+{10}^{-3}\approx 0.043\left(\frac{1}{-1.5\times {10}^{-2}-j}\right)+{10}^{-3}.$$

(208)

The above is drawn in pink in Figure 23 in which the gray $h_{\exp }\left(j\right)$ is shown for comparison.

A similar result that the $h_{\mathrm{th}}\left(j\right)$ was almost on the $h_{\exp }\left(j\right)$ except around (G) was obtained.

(b)

Analysis of $\left(\mathrm{G}1\right) \to \left(\mathrm{H}\right) \to \left(\mathrm{I}1\right)$

Considering the zigzag behavior in the cathodic region and the vertical line in the anodic region, the reactions are supposed to be a mixture of Equations (192)–(194). In this paper, the main reactions are assumed as the below, which are irreversible reactions:

$${\mathrm{CO}}_{\mathrm{ad}}+{ \mathrm{H}}_2\mathrm{O} \to {\mathrm{CO}}_2+{\mathrm{H}}^{+}+{ \mathrm{H}}_{\mathrm{ad}} +\mathrm{e} \left(j_{\mathrm{a}4},E_{\mathrm{eq}4}\right),$$

(209)

$${\mathrm{H}}_2\leftarrow {\mathrm{H}}^{+}+{ \mathrm{H}}_{\mathrm{ad}}+\mathrm{e} \left(j_{\mathrm{c}5},E_{\mathrm{eq}5}\right).$$

(210)

Since the above system is a mixed system having large $h_{\exp }\left(0\right)$, each branch reaction is categorized to the irreversible. The $j_{\mathrm{a}}\left(0\right)$ of irreversible reaction almost equals to $j_0$, the $j_0$ can be calculated using the common Equation (57).

$$j_0=\frac{0.026}{z \left( h_{\exp }\left(0\right)-l/\kappa \right)}=\frac{0.026}{\left(1\right)\times \left( 43-{10}^{-3}\right)}=6\times {10}^{-4} {\mathrm{mA} \mathrm{cm}}^{-2}.$$

(211)

Employing Equation (56), $h_{\mathrm{th}}\left(j\right)$ of the anodic branch of Equation (209) (${\alpha }_{\mathrm{a}4}\approx 0.28$ by trial and error) and the cathodic branch of Equation (210) (${\alpha }_{\mathrm{c}5}\approx 0.6$) are embodied as:

$${\left[h_{\mathrm{th}}\left(j\right)\right]}_{j>0}={\left[h_{\mathrm{irrev}}\left(j\right)\right]}_{j>0}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}}=\frac{0.026}{z_4 }\left\{\frac{1}{ j_0+\left({\alpha }_{\mathrm{a}4}-{\alpha }_{\mathrm{c}4}\right) j}\right\}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}}=\frac{0.026}{1 }\left(\frac{1}{6\times {10}^{-4} -0.45 j}\right)+{10}^{-3},$$

(212)

$${\left[h_{\mathrm{th}}\left(j\right)\right]}_{j<0}={\left[h_{\mathrm{irrev}}\left(j\right)\right]}_{j<0}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}=\frac{0.026}{z_5 }\left\{\frac{1}{ j_0+\left({\alpha }_{\mathrm{a}5}-{\alpha }_{\mathrm{c}5}\right) j}\right\}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}=\frac{0.026}{1 }\left(\frac{1}{6\times {10}^{-4}-0.2 j}\right)+{10}^{-3}.$$

(213)

The above $h_{\mathrm{th}}\left(j\right)$ with pink is added to Figure 23. We can see that the pink $h_{\mathrm{th}}\left(j\right)$ is on the gray $h_{\exp }\left(j\right)$.

(c)

Analysis of $\left(\mathrm{I}1\right) \to \left(\mathrm{I}2\right)$

The stable chemical species of $\left(\mathrm{I}1\right) \to \left(\mathrm{I}2\right)$ are H⁺ and CO₂. Since two clear vertical lines were observed at (I1) and (I2), these anodic reactions relate to two limiting diffusion processes. The candidate reactions are:

$${\mathrm{CO}}_{\mathrm{ad}}+{ \mathrm{H}}_2\mathrm{O} \to {\mathrm{CO}}_2+{\mathrm{H}}^{+}+{ \mathrm{H}}_{\mathrm{ad}} +\mathrm{e} \left(j_{\mathrm{a}4},E_{\mathrm{eq}4}\right),$$

(214)

$${\mathrm{H}}_2 \to {\mathrm{H}}^{+}+{ \mathrm{H}}_{\mathrm{ad}}+\mathrm{e} \left(j_{\mathrm{a}5},E_{\mathrm{eq}5}\right),$$

(215)

$${\mathrm{H}}_2 \to 2{\mathrm{H}}^{+}+2\mathrm{e} \left(j_{\mathrm{a}2},E_{\mathrm{eq}2}\right).$$

(216)

The above j is expressed as:

$$j= j_{\mathrm{A}}+ j_{\mathrm{C}}\approx j_{\mathrm{A}}=w_{\mathrm{a}2} j_{\mathrm{a}2 }+w_{\mathrm{a}4} j_{\mathrm{a}4}+w_{\mathrm{a}5} j_{\mathrm{a}5 }.$$

(217)

It is impossible to distinguish $j_{\mathrm{a}5}$ from $j_{\mathrm{a}2}$, but the observed $j_{{\mathrm{H}}_2,\mathrm{L}}$ is a summation of them. Considering the constant values of $w_{\mathrm{a}4} j_{\mathrm{a}4}+w_{\mathrm{a}5} j_{\mathrm{a}5 }=w_{\mathrm{a}4} j_{{\mathrm{CO}}_{\mathrm{ad}},\mathrm{L}}+w_{\mathrm{a}5} j_{{\mathrm{H}}_2,\mathrm{L}}\left(\approx 1.3\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}\right)$, the $j_{\mathrm{a}2}$ is arranged as:

$$j_{\mathrm{a}2}=\left(j-w_{\mathrm{a}4} j_{{\mathrm{CO}}_{\mathrm{ad}},\mathrm{L}}-w_{\mathrm{a}5} j_{{\mathrm{H}}_2,\mathrm{L}}\right)/w_{\mathrm{a}2}=\left(j-1.3\times {10}^{-3}\right)/w_{\mathrm{a}2}.$$

(218)

Similarly, the $h\left(j\right)$ is expressed as:

$$h\left(j\right)=\frac{h_{\mathrm{a}2}(j_{\mathrm{a}2})}{w_{\mathrm{a}2}}=\frac{h_{\mathrm{a}2}\left(\left(j-1.3\times {10}^{-3}\right)/w_{\mathrm{a}2}\right)}{w_{\mathrm{a}2}}.$$

(219)

Since the above $h_{\mathrm{a}2}(j_{\mathrm{a}2})$ is the anodic branch of the reversible HERher, its $h_{\mathrm{th}}\left(j\right)$ is:

$$\begin{gathered} h_{\mathrm{th}}\left(j\right)=\frac{h_{\mathrm{a}2}\left(\left(j-1.3\times {10}^{-3}\right)/w_{\mathrm{a}2}\right)}{w_{\mathrm{a}2}}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}}=\frac{1}{ w_{\mathrm{a}2} }\frac{R T}{ z_2 F }\left(\frac{1}{\left(j-1.3\times {10}^{-3}\right)/w_{\mathrm{a}2}}+\frac{2}{ w_{\mathrm{a}2} j_{{\mathrm{H}}_2,\mathrm{L}}-\left(j-1.3\times {10}^{-3}\right)/w_{\mathrm{a}2} }\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}} \\ \approx \frac{0.026}{ 2 }\left(\frac{1}{j-1.3\times {10}^{-3}}+\frac{2}{ {w_{\mathrm{a}2}}^2 j_{{\mathrm{H}}_2,\mathrm{L}}+1.3\times {10}^{-3}-j}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}}. \end{gathered}$$

(220)

For an example, when ${w_{\mathrm{a}2}}^2 j_{{\mathrm{H}}_2,\mathrm{L}}\approx 2\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$, and ${\left(l/\kappa \right)}_{\mathrm{a}}\approx 140 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$, the above is embodied as:

$$h_{\mathrm{th}}\left(j\right)=\frac{0.026}{2}\left(\frac{1}{ j-1.3\times {10}^{-3}}+\frac{2}{ 3.3\times {10}^{-3}-j}\right)+140.$$

(221)

The above is drawn with pink in Figure 20. We can see that the pink $h_{\mathrm{th}}\left(j\right)$ is on the gray $h_{\exp }\left(j\right)$. The large increase of ${\left(l/\kappa \right)}_{\mathrm{a}}$ from ${10}^{-3} {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$ to $140 {\mathrm{k}\mathsf{\Omega } \mathrm{cm}}^2$ is probably due to the residue H_ad and the newly produced H_ad by Equations (214) and (215).

(d)

Analysis of $\left(\mathrm{I}2\right) \to \left(\mathrm{I}{2}^{\prime }\right)$

The stable chemical species of $\left(\mathrm{I}2\right) \to \left(\mathrm{I}{2}^{\prime }\right)$ are H⁺ and CO₂, too. It is characteristic that three-digit drop of $h_{\exp }\left(j\right)$ was observed. Considering the significant decrease, it is apparent that the H_ad consuming reaction had occurred. The oxidation reaction of H_ad should be newly added to the previous reactions:

$${\mathrm{H}}_{\mathrm{ad}} \to {\mathrm{H}}^{+}+{\mathrm{e}}^{-}\left(i_{\mathrm{a}3},E_{\mathrm{eq}3}\right),$$

(222)

The j is:

$$j= j_{\mathrm{A}}+ j_{\mathrm{C}}\approx j_{\mathrm{A}}=w_{\mathrm{a}2} j_{\mathrm{a}2}+w_{\mathrm{a}3} j_{\mathrm{a}3}+w_{\mathrm{a}4} j_{\mathrm{a}4}+w_{\mathrm{a}5} j_{\mathrm{a}5}$$

(223)

Similarly, when the vertical value at (I2) is the constant value of $w_{\mathrm{a}2} j_{\mathrm{a}2}+w_{\mathrm{a}4} j_{\mathrm{a}4}+w_{\mathrm{a}5} j_{\mathrm{a}5}=\left(w_{\mathrm{a}2}+w_{\mathrm{a}5}\right)j_{{\mathrm{H}}_2,\mathrm{L}}+w_{\mathrm{a}4} j_{{\mathrm{CO}}_{\mathrm{ad}},\mathrm{L}}\left(\approx 3.2\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}\right)$, the $j_{\mathrm{a}3}$ is expressed as:

$$j_{\mathrm{a}3}=\left(j-3.2\times {10}^{-3}\right)/w_{\mathrm{a}3}.$$

(224)

Since the $h_{\mathrm{a}3}(j_{\mathrm{a}3})$ is the anodic branch resistance of irreversible reaction, the $h_{\mathrm{th}}\left(j\right)$ is expressed when ${\left(l/\kappa \right)}_{\mathrm{a}}\approx {10}^{-3} {\mathrm{k}\Omega \mathrm{cm}}^2$ and ${\alpha }_{\mathrm{a}3} z_3\approx 0.7$:

$$\begin{array}{ll}{h}_{\mathrm{th}}\left(j\right)& =\frac{h_{\mathrm{a}3}\left(\left(j-3.2\times {10}^{-3}\right)/w_{\mathrm{a}3}\right)}{w_{\mathrm{a}3}}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}} .\\ & =\frac{R T}{ {\alpha }_{\mathrm{a}3} z_3 F }\left(\frac{1}{j-3.2\times {10}^{-3}}+\frac{1}{ w_{\mathrm{a}3}{}^2 j_{{\mathrm{H}}_{\mathrm{ad}},\mathrm{L}}+3.2\times {10}^{-3}-j}\right)+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}}\end{array}$$

(225)

Under assumption of

$$\frac{1}{ w_{\mathrm{a}3}{}^2 j_{{\mathrm{H}}_{\mathrm{ad}},\mathrm{L}}-j+3.2\times {10}^{-3} }\approx 0,$$

(226)

Equation (225) will be approximated to:

$$h_{\mathrm{th}}\left(j\right)\approx \frac{0.026}{ 0.7}\frac{1}{j-3.2\times {10}^{-3}}+{10}^{-3}.$$

(227)

The above $h_{\mathrm{th}}\left(j\right)$ of $\left(\mathrm{I}2\right)\to \left(\mathrm{I}{2}^{\prime }\right)$ is drawn with pink in Figure 23. We can see that the pink $h_{\mathrm{th}}\left(j\right)$ is on the gray $h_{\exp }\left(j\right)$. The large drop of ${\left(l/\kappa \right)}_{\mathrm{a}}$ to ${10}^{-3} {\mathrm{k}\Omega \mathrm{cm}}^2$ from the previous $140 {\mathrm{k}\Omega \mathrm{cm}}^2$ is due to complete disappearance of H_ad.

3.9.3. Analysis of $\left(\mathrm{I}{2}^{\prime }\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{K}\right)$

The representative $h_{\exp }\left(j\right)$ of downward track of $\left(\mathrm{I}{2}^{\prime }\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{K}\right)$ is picked up and shown in Figure 24 together with its $E_{\exp }\left(j\right)$.

The reactions in this track should be the same reactions as that of $\left(\mathrm{G}\right)\to \left(\mathrm{I}{2}^{\prime }\right)$. We can obtain the numerical $h\left(j\right)$ using the same equations but different data. Similarly, it is necessary to divide $\left(\mathrm{I}{2}^{\prime }\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{K}\right)$ into five parts: $\left(\mathrm{I}{2}^{\prime }\right)\to \left(\mathrm{I}3\right)$, $\left(\mathrm{I}3\right)\to \left(\mathrm{I}4\right)$, $\left(\mathrm{I}4\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{J}1\right)$, $\left(\mathrm{J}1\right)\to \left(\mathrm{J}{1}^{\prime }\right)$, and $\left(\mathrm{J}{1}^{\prime }\right)\to \left(\mathrm{K}\right)$. The $h_{\mathrm{th}}\left(j\right)$ divided into five parts is shown in Figure 25 with pink together with the gray $h_{\exp }\left(j\right)$ curve.

(a)

Part analysis of $\left(\mathrm{I}{2}^{\prime }\right)\to \left(\mathrm{I}3\right)$

Using the same as Equation (227) but different data, the $h_{\mathrm{th}}\left(j\right)$ of $\left(\mathrm{I}{2}^{\prime }\right)\to \left(\mathrm{I}3\right)$ is embodied as:

$$h_{\mathrm{th}}\left(j\right)\approx \frac{0.026}{ 0.5}\frac{1}{j-1.8\times {10}^{-3}}+{10}^{-3}$$

(228)

The overlap between the $h_{\mathrm{th}}\left(j\right)$ and the $h_{\exp }\left(j\right)$ shows the result of good approximation. The different data used are two: ${\alpha }_{\mathrm{a}3}=0.5$ (from the previous value of 0.7) and $\left(w_{\mathrm{a}2}+w_{\mathrm{a}5}\right)j_{{\mathrm{H}}_2,\mathrm{L}}+w_{\mathrm{a}4} j_{{\mathrm{CO}}_{\mathrm{ad}},\mathrm{L}}\approx 1.8\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$ (from the previous value of $3.2\times {10}^{-3} {\mathrm{mA} \mathrm{cm}}^{-2}$). Clear reasons for the differences are unknown, but one of them may be due to the appearance or disappearance of H_ad.

(b)

Part analysis of $\left(\mathrm{I}3\right)\to \left(\mathrm{I}4\right)$

The $h_{\mathrm{th}}\left(j\right)$ in this region is the same as Equation (221), but different data.

$$h_{\mathrm{th}}\left(j\right)=\frac{0.026}{0.5}\left(\frac{1}{j-0.35\times {10}^{-3}}+\frac{1}{2.2\times {10}^{-3}-j}\right)+70$$

(229)

The reason for change of $z=2$ to $z=0.5$ is by curve fitting result. The actual reaction may be different from Equations (214)–(216). The further experiments are needed. The above is drawn with pink in Figure 25. We can see that the pink $h_{\mathrm{th}}\left(j\right)$ is on the gray $h_{\exp }\left(j\right)$. The decrease of ${\left(l/\kappa \right)}_{\mathrm{a}}$ to $70 {\mathrm{k}\Omega \mathrm{cm}}^2$ from $140 {\mathrm{k}\Omega \mathrm{cm}}^2$ is probably due to the decrease of the H_ad.

(c)

Part analysis of $\left(\mathrm{I}4\right)\to \left(\mathrm{J}\right)\to \left(\mathrm{J}1\right)$

The $h_{\mathrm{th}}\left(j\right)$ in this region is the same as Equations (212) and (213), but different data. Similarly, $j_0$ can be calculated as:

$$j_0=\frac{0.026}{z \left( h_{\exp }\left(j\right)-l/\kappa \right)}=\frac{0.026}{\left(1\right)\times \left( 190-{10}^{-3}\right)}=1.4\times {10}^{-4} {\mathrm{mA} \mathrm{cm}}^{-2}.$$

(230)

Employing Equation (56) and the curve fitting, $h_{\mathrm{th}}\left(j\right)$ of the anodic branch (${\alpha }_{\mathrm{a}4}\approx 0.2)$ and the cathodic branch (${\alpha }_{\mathrm{c}5}=0.995$) are expressed as:

$${\left[h_{\mathrm{th}}\left(j\right)\right]}_{j>0}={\left[h_{\mathrm{irrev}}\left(j\right)\right]}_{j>0}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{a}}=\frac{0.026}{1 }\left(\frac{1}{1.4\times {10}^{-4} -0.6 j}\right)+{10}^{-3},$$

(231)

$${\left[h_{\mathrm{th}}\left(j\right)\right]}_{j<0}={\left[h_{\mathrm{irrev}}\left(j\right)\right]}_{j<0}+{\left(\frac{l}{\kappa }\right)}_{\mathrm{c}}=\frac{0.026}{1 }\left(\frac{1}{1.4\times {10}^{-4}-0.99 j}\right)+{10}^{-3}.$$

(232)

The above $h_{\mathrm{th}}\left(j\right)$ with pink is added to Figure 25. We can see that the pink $h_{\mathrm{th}}\left(j\right)$ is on the gray $h_{\exp }\left(j\right)$. Around (J1), there was a gap. This gap is caused by irreversible reactions having values of ${\alpha }_{\mathrm{a}3} =0.6$ and ${\alpha }_{\mathrm{a}5} \approx 0.995$.

(d)

Part analysis of $\left(\mathrm{J}{1}^{\prime }\right)\to \left(\mathrm{K}\right)$

Using the same Equation (208) and the same data:

$$h_{\mathrm{th}}\left(j\right)=\frac{0.026}{ 0.6 }\left(\frac{1}{-1.5\times {10}^{-2}-j}+\frac{1}{j + 1.5\times {10}^{-2} + >100}\right)+{10}^{-3}\approx 0.043\left(\frac{1}{-1.5\times {10}^{-2}-j}\right)+{10}^{-3}$$

(233)

The above $h_{\mathrm{th}}\left(j\right)$ curves considerably overlapped with the $h_{\exp }\left(j\right)$ curves.

3.10. Analysis of Quasireversible HER (Environment (III))

When the CO injection is stopped, the test solution will be gradually occupied by H₂. Complete restoration to the reversible reaction will depend on whether the Pt surface is thoroughly clean or not.

3.10.1. Estimation of $w_{\mathrm{a}}$ at CO Stopped

Figure 6 shows the shape changes of $E_{\exp }\left(j\right)$ when the CO injection is stopped. We can see that the first $E_{\exp }\left(j\right)$ followed the irregular track of (M) → (N) → (O)→ (P), but the second and subsequent curve converged to the regular track of (P) → (Q) → (R). Considering the solution saturated with H₂, main reaction is the HER:

$${\mathrm{H}}_2\rightleftarrows 2{ \mathrm{H}}^{+} +2\mathrm{e} \left(j_1,E_{\mathrm{eq}1}\right).$$

(234)

The cathodic branch curve shape in Figure 6 is very similar to that in Figure 2, so we can deduce that the cathodic branch contains reversible her process. Focusing on the anodic branch, on the other hand, an apparent tendency of sequential changes of $j_{\mathrm{a}1}$ was observed, especially at the potential range of $0.3 \mathrm{V}\le E\le 0.7 \mathrm{V}$. Careful reading leads to the fact that the value of $j_{\mathrm{a}1}$ at 0.7 V increased with the increase of cycle number (N). The increased $w_{\mathrm{a}}$ can be calculated as:

$$w_{\mathrm{a}}=\frac{{\left[w_{\mathrm{a}} j_{\mathrm{a}1}\right]}_{\mathrm{at} 0.7 \mathrm{V}}}{j_{{\mathrm{H}}_2,\mathrm{L}}}=\frac{3\times {10}^{-2} ~5\times {10}^{-2}}{0.74 }=0.04~0.07$$

(235)

The increasing $w_{\mathrm{a}}$ as increasing N is another sign that the hor gradually recovers from the contaminated state. We can see that there is an interesting fact when calculating $j_{\mathrm{A}}\left(0\right)$. Comparing the $h_{\exp }\left(0\right)\approx 0.02{ \mathrm{k}\Omega \mathrm{cm}}^2$ in Figure 3 against the $h_{\exp }\left(0\right)\approx 0.03~0.06{ \mathrm{k}\Omega \mathrm{cm}}^2$ in Figure 7, the $j_{\mathrm{A}}\left(0\right)$ ratio between them is calculated below:

$$\frac{j_{\mathrm{A}}{\left(0\right)}_{\mathrm{in} \mathrm{Figure} 3}}{j_{\mathrm{A}}{\left(0\right)}_{\mathrm{in} \mathrm{Figure} 7}}=\frac{{\left[\frac{R T}{z F}\frac{1}{\left(h_{\exp }\left(0\right)-l/\kappa \right)}\right]}_{\mathrm{in} \mathrm{Figure} 3}}{{\left[\frac{R T}{z F}\frac{1}{\left(h_{\exp }\left(0\right)-l/\kappa \right)}\right]}_{\mathrm{in} \mathrm{Figure} 7}}=\frac{{\left(h_{\exp }\left(0\right)-l/\kappa \right)}_{\mathrm{in} \mathrm{Figure} 7}}{{\left(h_{\exp }\left(0\right)-l/\kappa \right)}_{\mathrm{in} \mathrm{Figure} 3}}=\frac{\left(0.03~0.06\right)-{10}^{-3}}{0.022-{10}^{-3}}=1.4~2.8$$

(236)

When the above ratio is averaged and assumed to 2, it is the same ratio of $j_{\mathrm{A}}\left(0\right)$ between the reversible $\left(j_{\mathrm{A}}\left(0\right)\approx j_{\mathrm{d}}\right)$ and the quasireversible $\left(j_{\mathrm{A}}\left(0\right)\approx j_{\mathrm{d}}/2\right)$. Taking into account the experimental facts, the $j_{\mathrm{A}}\left(0\right)$ ratio and the curve shape resembling, the reaction in the environment (III) can be categorized to the quasireversible. This means that the CO-contaminated Pt surface is gradually changed to be clean after several N.

3.10.2. Determination of Kinetic Parameters in the CO-Stopped Solution

In previous sections, we discussed that redox reaction in the H₂SO₄ + H₂ + CO-CO solution is the quasireversible. Some curves are probably on the way of the recovery process from the irreversible to the reversible. Figure 26 shows the enlarged current region at $j\approx 0$. Arrows and numbers of (1)~(12) are the polarization direction and the N, except for the first cycle.

It is clear that the linear relation holds in the range of $\left|j\right|\le 0.02 {\mathrm{mA} \mathrm{cm}}^{-2}$. The values of ${\alpha }_{\mathrm{c}}$ and z are calculated by employing the $h_{\exp }\left(0\right)$, the calculated slopes, and $j_{\mathrm{A}}\left(0\right)=0.46 {\mathrm{mA} \mathrm{cm}}^{-2}$ (by graphical determination, detailed in later 3.10.3):

$$z=\frac{R T}{F}\frac{1}{j_{\mathrm{A}}\left(0\right)\left(h_{\exp }\left(0\right)-l/\kappa \right)}=\frac{0.026}{\left(0.46\right)\left(h_{\exp }\left(0\right)-0.001\right)}$$

(237)

$$\begin{gathered} \mathrm{slope}=\left(h_{\exp }\left(0\right)-\frac{l}{\kappa }\right)\left(\frac{2 {\alpha }_{\mathrm{a}}}{w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}} }+\frac{2 {\alpha }_{\mathrm{c} }}{ w_{\mathrm{c}}{j}_{{\mathrm{H}}^{+},\mathrm{L}}}+\frac{1-2 {\alpha }_{\mathrm{a}}}{j_{\mathrm{A}}\left(0\right)}\right) \\ \approx \left(h_{\exp }\left(0\right)-{10}^{-3}\right)\left(\frac{2\left(1-{\alpha }_{\mathrm{c} }\right)}{0.7 }+\frac{2 {\alpha }_{\mathrm{c} }}{-325}+\frac{1-2\left(1-{\alpha }_{\mathrm{c} }\right)}{0.46}\right) \end{gathered}$$

(238)

The relation between the calculated values (${\alpha }_{\mathrm{c}}$ and z) and the N was plotted. The results are shown in Figure 27.

We can see that the ${\alpha }_{\mathrm{c}}$ and the z are about 0.5 and 2 at N = 12. In detail, we can find an interesting fact that the ${\alpha }_{\mathrm{c}}$ of the odd N was smaller than that of the even N. The reason is not so clear, but the ${\alpha }_{\mathrm{c}}$ is changeable and may be deeply influenced by polarization history. Further confirmations are needed.

3.10.3. Graphical Determination of j_A(0)

Employing the determination way of $j_{\mathrm{A}}\left(0\right)$ discussed in Section 3.5, we can graphically read its value. One of the tracks (as a representative; (P)–(Q)–(R) of N = 12) was employed. It is the green curve shown in Figure 28.

The $h_{\mathrm{A}0}\left(j_0\right)$ of Equation (75) and the $w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}\left(=0.7 {\mathrm{mA} \mathrm{cm}}^{-2}\right)$ are added to Figure 28 with pink chain curve. The horizontal straight line of $h_{\exp }\left(0\right)-{10}^{-3}$($\approx 0.03 {\mathrm{k}\Omega \mathrm{cm}}^2$ at N = 12) meets the $h_{\mathrm{A}0}\left(j_0\right)$ at $\left(\mathrm{C}1\right)$ and the $w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}$ at $\left(\mathrm{C}2\right)$. Three crossed points are shown as:

$$j_0\approx 1.4 {\mathrm{mA} \mathrm{cm}}^{-2}:\left(\mathrm{C}1\right),$$

(239)

$$w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}=0.7 {\mathrm{mA} \mathrm{cm}}^{-2}: \left(\mathrm{C}2\right),$$

(240)

$$\left|w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}\right|: \left(\mathrm{C}3\right) (\mathrm{cannot} \mathrm{observed}, \mathrm{assumed} \mathrm{to}>{10}^2 {\mathrm{mA} \mathrm{cm}}^{-2}).$$

(241)

Substituting the above readings into Equation (7), we can calculate the $j_{\mathrm{A}}\left(0\right)$:

$$j_{\mathrm{A}}\left(0\right)=\frac{1}{1/j_0+1/ j_{\mathrm{Red},\mathrm{L}}+1/- j_{{\mathrm{Ox}}^{\mathrm{z}+},\mathrm{L}}}=\frac{1}{1/\left(\approx 1.4\right)+1/0.70+1/>{10}^2}\approx 0.46 {\mathrm{mA} \mathrm{cm}}^{-2}.$$

(242)

Employing the $j_{\mathrm{A}}\left(0\right)$ and the $h_{\exp }\left(0\right)$, the z can be roughly calculated to:

$$z=\frac{RT}{F}\frac{1}{j_{\mathrm{A}}\left(0\right) h\left(0\right)}=\frac{0.026}{\left(0.46\right)\left(0.03\right)}=1.77\approx 2.$$

(243)

The rough agreement of $j_{\mathrm{A}}\left(0\right)\left(\approx 0.46 {\mathrm{mA} \mathrm{cm}}^{-2}\right)$ with $j_{\mathrm{d}}/2$ $\left(=0.70{ \mathrm{mA} \mathrm{cm}}^{-2}/2=0.35{ \mathrm{mA} \mathrm{cm}}^{-2}\right)$ can also lead to the conclusion that the redox reaction in a H₂SO₄ + H₂ + CO-CO solution is a quasireversible reaction.

3.10.4. Agreement between E_exp(j) and E_th(j)

Employing ${\alpha }_{\mathrm{c}}=0.6$, $l/\kappa ={10}^{-3} {\mathrm{k}\Omega \mathrm{cm}}^2$, ${\left(l/\kappa \right)}_{\mathrm{a}}={10}^{-3} {\mathrm{k}\Omega \mathrm{cm}}^2$,${\left(l/\kappa \right)}_{\mathrm{c}}={10}^{-3} {\mathrm{k}\Omega \mathrm{cm}}^2$, $w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}=0.7 {\mathrm{mA} \mathrm{cm}}^{-2}$, and $|w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}|>{10}^2 {\mathrm{mA} \mathrm{cm}}^{-2}\left(\approx 325 {\mathrm{mA} \mathrm{cm}}^{-2}\right)$, the quasireversible $h_{\mathrm{th}}\left(j\right)$ is calculated using Equations (133)–(137). The calculated result is shown below:

$$h_{\mathrm{th}}\left(j\right)=h_{\mathrm{quasi}}\left(j\right)+\frac{l}{\kappa }=\frac{0.0024}{16.5 - j}+\frac{0.018}{0.53 - j}+\frac{0.014}{j+4.60 }+\frac{0.0063}{j+338}+1.5\times {10}^{-3}.$$

(244)

Taking into account the realistic restrictions, (1) $h_{\mathrm{th}}\left(j\right)$ has two terms corresponding to the anodic and cathodic branchs, (2) the numerator is near to 0.013 (=0.026/2), and (3) $h_{\mathrm{th}}\left(0\right)=h_{\exp }\left(0\right)\left(\approx 0 {\mathrm{k}\Omega \mathrm{cm}}^2\right)$, the above $h_{\mathrm{th}}\left(j\right)$ is approximated as below:

$$h_{\mathrm{th}}\left(j\right)=0.013\left(\frac{1}{0.53 - j}+\frac{1}{j+4.60 }\right)+1.5\times {10}^{-3}.$$

(245)

The above $h_{\mathrm{th}}\left(j\right)$ above was plotted on the $h_{\exp }\left(j\right)$ in Figure 3. The result is shown in Figure 29.

We can see that the red $h_{\mathrm{th}}\left(j\right)$ is almost on the gray $h_{\exp }\left(j\right)$. In addition to the $h\left(j\right)$ curve agreement, it is needed to confirm the agreement between $E_{\exp }\left(j\right)$ and $E_{\mathrm{th}}\left(j\right)$. In the case of quasireversible reaction, it is necessary for the complete agreement that the $E_{\mathrm{th}}\left(j\right)$ is divided into three parts. They are obtained by solving the following differentiated equations under initial conditions:

(1)

$w_{\mathrm{c}} j_{{\mathrm{H}}^{+},\mathrm{L}}\le j\le j_{\mathrm{pC}} \left(-325 {\mathrm{mA} \mathrm{cm}}^{-2}\le j\le -1.74 {\mathrm{mA} \mathrm{cm}}^{-2}\right)$

$$\frac{\mathrm{d} E_{\mathrm{th}}\left(j\right)}{\mathrm{d} j}=h_{\mathrm{th}}\left(j\right)=0.027 \ln \left(\frac{1}{0.437-j}+\frac{1}{ j+322}\right)+{10}^{-3},$$

(246)

$$E_{\mathrm{th}}\left(j_{\mathrm{pC}}\right)=E_{\mathrm{th}}\left(-1.74\right)=-0.028 \mathrm{V} \left(\mathrm{an} \mathrm{initial} \mathrm{condition}\right).$$

(247)

The above solution is as below:

$$E_{\mathrm{th}}\left(j\right)=-0.16+0.027 \ln \left(\frac{j + 325}{0.437 - j }\right)+{10}^{-3} j.$$

(248)

(2)

$j_{\mathrm{pC}}\le j\le j_{\mathrm{pA}}$$\left(-1.74 {\mathrm{mA} \mathrm{cm}}^{-2}\le j\le 0.437 {\mathrm{mA} \mathrm{cm}}^{-2}\right)$

$$\frac{\mathrm{d} E_{\mathrm{th}}\left(j\right)}{\mathrm{d} j}=h_{\mathrm{th}}\left(j\right)=-0.028+0.013 \ln \left(\frac{j+4.60 }{0.53-j}\right)+1.5\times {10}^{-3},$$

(249)

$$E_{\mathrm{th}}\left(j\right)=0 \mathrm{V} \left(\mathrm{an} \mathrm{initial} \mathrm{condition}\right).$$

(250)

The solution is:

$$\begin{gathered} E_{\mathrm{th}}\left(j\right)=-0.063-0.0024 \ln \left(16.5-j\right)-0.018 \ln \left(0.53-j\right) \\ +0.014 \ln \left(j+4.60\right)+0.063 \ln \left(j+338\right)+1.5\times {10}^{-3}j. \end{gathered}$$

(251)

(3)

$j_{\mathrm{pA}}\le j\le w_{\mathrm{a}} j_{{\mathrm{H}}_2,\mathrm{L}}$$\left(0.437 {\mathrm{mA} \mathrm{cm}}^{-2}\le j\le 0.70 {\mathrm{mA} \mathrm{cm}}^{-2}\right)$

$$\frac{\mathrm{d} E_{\mathrm{th}}\left(j\right)}{\mathrm{d} j}=h_{\mathrm{th}}\left(j\right)=0.046\left(\frac{1}{0.70-j}+\frac{1}{ 1.74+j}\right)+{10}^{-3},$$

(252)

$$E_{\mathrm{th}}\left(j_{\mathrm{pA}}\right)=E_{\mathrm{th}}\left(0.437\right)=0.024 \mathrm{V} \left(\mathrm{an} \mathrm{initial} \mathrm{condition}\right).$$

(253)

The result is shown as:

$$E_{\mathrm{th}}\left(j\right)=-0.074+0.046 \ln \left(\frac{1.74+j}{0.70- j }\right)+{10}^{-3} j.$$

(254)

The concrete $E_{\mathrm{th}}\left(j\right)$ is drawn with red on the gray $E_{\exp }\left(j\right)$. The result is shown in Figure 30.

The curve technique using $h\left(j\right)$ shows that the cathodic part of $E_{\mathrm{th}}\left(j\right)$ completely overlaps with $E_{\exp }\left(j\right)$. In contrast, the anodic part of $E_{\mathrm{th}}\left(j\right)$ shows almost a line compared to the curve of $E_{\exp }\left(j\right)$. When considering $w_{\mathrm{a}}$ as a function of $\eta$ or time, the anodic part of $E_{\mathrm{th}}\left(j\right)$ will probably show better agreement with $E_{\exp }\left(j\right)$. Some mentioned above will be further studied and reported elsewhere.

4. Experimental Section

4.1. Specimens

Two platinum wires (99.98% Pt, NILACO Ltd., Tokyo, Japan) were employed for working and counter electrodes. Each wire was 0.3 mm in diameter and 50 cm in length. The electrodes were formed on a spiral shape. The counter electrode had a geometric area of 4.7 cm². The working electrode had an exposure area of 3 cm² by masking an insulating area with silicon resin. They were ultrasonically cleaned in warm acetone and immersed in a HNO₃ solution for 180 s at an ambient temperature. They were rinsed thoroughly with deionized water before every test.

4.2. Test Solution

Chemical grade sulfuric acid (98% H₂SO₄, Wako Pure Chemical Industries, Tokyo, Japan), high purity hydrogen gas (99.99999% H₂, Air-Water Inc., Tokyo, Japan), and carbon monoxide gas (99.95% CO, Air-Water Inc., Tokyo, Japan) were employed. The test solution was saturated with H₂ and injected with CO under atmospheric pressure. The solution was deionized water having 1 dm³. During experiments, the solution was always maintained in the H₂-saturated condition by continuous H₂ bubbling. The solution pH and electrolyte conductivity of the test solution were about 0.3 and ≥20 S m⁻¹, respectively. The dissolved oxygen concentration in the solution was always <0.4 ppm.

4.3. Measurements

Measurements were carried out under the H₂-saturated solution at around 298 K. The CO gas was injected/stopped in a 0.5 mol dm⁻³ H₂SO₄ solution. An automatic electrochemical instrument (Hokuto Denko Inc., HZ7000, Tokyo, Japan) was employed for CV. The Ag/AgCl electrode in the saturated KCl solution (DKK-TOA Co., HS-305D, Tokyo, Japan) was used as a reference electrode. All of the electrode potentials were converted to V vs. SHE and simplified to V in this paper unless otherwise noted. The distance between the working and reference electrodes was <1 cm. The E(j) curve was not corrected for the jR drop because the physical factors such as solution resistance can be compensated using the curve technique, which was detailed in Section 3.6.2. A slow scan rate of 0.3 mV s⁻¹ was selected for the steady state measurement because the E(j) unaffected by time is supposed to be the steady state curve [5,34]. Since the resulting data had frequent scattering tendency, smoothing was necessary. Using a commercial software (Igor Pro 6, 6.2.2.0, WaveMetrics, Lake Osawego, OR, USA, 2011), the experimental E(j) was mathematically smoothed. Arranging mathematical formulas, commercially available software (Mathematica ver.10) was used.

5. Conclusions

In order to analyze the CO poisoning effect, a few curve techniques were employed to investigate the hydrogen electrode reaction (HER = hor + her) occurring on the Pt electrode in a 0.5 mol dm⁻³ H₂SO₄ solution saturated with H₂ when CO-injected or not. Using the curve techniques, the followings were confirmed: HER before CO injection showed typical reversible reaction, having z = 2. After CO injection, her was changed to an irreversible reaction, having z = 1 and ${\alpha }_{\mathrm{c} }\approx 0.6$. When CO injection was stopped, HER gradually changed to quasireversible from an irreversible reaction. The $h\left(j\right)$ of HER before/after the CO injection were analyzed not only algebraically but also graphically. It was found that HER poisoned with CO would change to a reversible, irreversible, or quasireversible reaction depending on CO content in the solution. It was also found that the kinetic parameters transferring among reversible, irreversible, and quasireversible could be determined by employing $h\left(j\right)$.