1)Qualitative evidence of the complexation reaction

Figure 1below shows the Intensity/Potential curves obtained in a thin layer on platinum with the different reagents.

Figure 1. Current potential curves obtained by thin-layer electrochemistry on a platinum anode, with the various reagents involved in this study. KCl = 0.2 M; r = 0.3 mV/s; pH = 12.6. 1): Residual current (Blank). The same curve (as (1)) is obtained in the presence of Hm 10⁻³ M; 2): Solution of curve (1) without Hm + NiSO₄·6H₂O 10⁻³ M; 3): Solution of curve (2) + Hm 10⁻³ M; 4): Curve resulting from the subtraction of curve (1) from curve (3).

This figure shows the current-potential curves obtained in thin-layer configuration on platinum with the various reagents (residual electrolyte, Ni(II) alone, and Ni(II) + histamine). As already established in [1], the electrolyte and histamine alone give only the platinum-oxidation signal near 0.55 - 0.9 V; adding Ni(II) produces the Ni(II)/Ni(III) and PtO signals together (0.6 - 1 V); and the ternary mixture (KCl + Ni(II) + histamine) generates an additional, lower-potential signal (0.4 - 0.6 V) that is absent from either reagent alone.

Subtracting the blank from the ternary-mixture curve (curve 4) isolates this signal, which we assign, as in [1], to the oxidation of the Hm-Ni(II) adduct formed via equilibrium (1). This qualitative assignment, together with the complementary NMR evidence, was established and discussed in detail in [1] and is not repeated here.

2)Kinetic rate constant of the Ni(III)-HmP decomposition step

In this section, voltammograms were recorded at increasing histamine concentrations (1, 3, 5, and 7 mmol·L⁻¹) at fixed Ni(II) (1 mmol·L⁻¹). The voltammograms obtained are shown in Figure 2.

Figure 2. Voltammograms obtained in thin film on a platinum electrode. KCl = 0.2 mol∙L⁻¹; r = 0.3 mV/s; pH = 12.6; [NiSO₄∙6H₂O] = 1 mmol∙L⁻¹. (1) à (4): [Histamine] respectivement 10⁻³ M, 3∙10⁻³ M, 5∙10⁻³ M et 7∙10⁻³ M.

The Ni(II)-Hm oxidation signal becomes clearly resolved for [Hm] ≥ 3 mmol·L⁻¹ and shifts slightly to lower potential with increasing histamine, consistent with the progressive stabilisation of the complex.

The mass balance for the Ni(III)-HmP species in the thin layer, and its resolution using the pseudo-first-order approximation (NaOH in large excess, so [OH⁻] is effectively constant), follows exactly the treatment derived and justified in our first work [1] (their Equations (11)-(14)); we do not re-derive it here. Applying that treatment to the present voltammograms gives an apparent rate constant k = ( 2 ± 1 ) × 10 − 6 s − 1 , k ′ = 4.5 × 10 − 8 m 3 ⋅ mol − 1 ⋅ s − 1 i.e., the same order of magnitude reported for this decomposition step in our first article [1]. We report this value here explicitly as a consistency check, as it reproduces, rather than newly discovers, the kinetics of the chemical step and is used as the calibration point against which the independent MATLAB simulation is tested in our first studies.

Applying that treatment point by point to the four voltammograms of Figure 2 ([Hm] = 1, 3, 5, 7 mmol∙L⁻¹) gives a set of independent local estimates of the apparent rate constant, k = ( 2 ± 1 ) × 10 − 6 s − 1 (mean ± range), the spread reflecting both the expected [Hm]-independence of this pseudo-first-order constant and the peak-current reproducibility of the technique ΔIpic < 5 % , (three replicate traces per condition) [1]; a further systematic uncertainty of comparable order (~20 - 25%) arises from the thin-layer volume VTL, which enters the underlying mass balance (Equation (10) of our first article) [1], as a direct scaling factor and was itself found to require a 25% correction to reconcile the simulated and experimental current-potential responses.

3)Value of the complexation constantK

The complexation constant K of equilibrium (1) is a distinct physical quantity from the kinetic rate constant k discussed above: K describes the position of the Hm-Ni(II) complexation equilibrium, whereas k describes the rate at which the resulting Ni(III)-HmP adduct subsequently decomposes. The two should not be compared directly and are addressed separately here.

An estimate of K for this equilibrium was obtained in our earlier work [1] from a mass-balance analysis at a histamine-to-nickel ratio of 10:1 (1 mmol∙L⁻¹ Hm, 0.1 mmol∙L⁻¹ Ni(II)), a condition for which the mixture is fully clear (complete dissolution and complexation of Ni(II)). Using the free Ni²⁺ concentration set by the solubility of Ni(OH)₂ at pH 12.6 (Ksp ≈ 10^−14.7, giving [Ni²⁺] ≈ 1.3 × 10⁻¹² mol·L⁻¹), this analysis gives: K ≈ 2.5 × 10⁶, a value corroborated independently, in the same earlier work, by the ≈0.4 V lowering of the oxidation peak potential between free Ni(II) (≈0.9 V) and the Hm-Ni(II) complex (≈0.5 V), which gives exp(F·ΔE/RT) ≈ 5 × 10⁶, the same order of magnitude. Both estimates indicate a strong Hm-Ni(II) affinity. We report this value here as established context from first works [1] rather than as a new result of the present study.

At the working pH of 12.6, Ni(II) is present predominantly as solid/colloidal Ni(OH)₂ in the absence of histamine, with a residual free-Ni²⁺ concentration of ≈10⁻¹² mol·L⁻¹ set by the solubility product [1]; this precipitate dissolves progressively as the histamine-to-nickel ratio increases, becoming fully dissolved and complexed for [ Hm ] / [ Ni ( II ) ] ≥ 10 [1]. Histamine, whose amine pKa values (≈5.8 and ≈9.4) [22]-[24] lie well below the working pH, is essentially fully deprotonated under these conditions. These assumptions justify treating the Hm-Ni(II) complexation equilibrium (1) as essentially complete at high [Hm]/[Ni(II)] ratios and neglecting the free-Ni²⁺ contribution in the kinetic mass balance of Section III.B.

A more rigorous, concentration-resolved determination of K specific to the conditions of Figure 2 (1 mmol·L⁻¹ Ni(II), 1-7 mmol·L⁻¹ Hm) could be obtained from the dependence of the complex oxidation peak potential Ep on log[Hm], using the Lingane-type relation:

where the slope of Ep vs. log[Hm] gives the Hm:Ni stoichiometry p, and the intercept gives K.

This refinement is left for future work, pending tabulation of the four Ep values of Figure 2 (currently described only qualitatively in the text); it would be expected, given the same underlying equilibrium, to yield a value consistent with the ≈10⁶ order of magnitude established in our first study [1].

To test independently whether the kinetics established in our first article [1] are compatible with a first-principles description of the coupled electrochemical-chemical (EC) process, the governing differential equation for the Ni(III)-HmP concentration was solved numerically:

d C d t = k 1 ( C 0 − C ) exp ( k 2 ( k 3 + r t ) ) − k 4 C with k₁ - k₄ composite constants

derived from the independently known physicochemical parameters of the system (see Table 1 below), integrated with the ODE45 solver in MATLAB. Simulations were run over a range of trial values of the chemical rate constant k until the simulated current matched the experimental voltammetric response (Figure 3).

The composite parameters are k 1 = S ⋅ k 0 / V T L ( s − 1 ) , k 2 = α n lim F / ( R T ) ( V − 1 ) , k 3 = E 0 ′ − E I = 0 ( mV ) , and k 4 = k ( s − 1 ) , the scanned chemical rate constant, where E I = 0 is the initial potential of the voltammetric sweep, chosen experimentally low enough that the Faradaic current is essentially zero at t = 0. For the base-case parameters of Table 1 ( E 0 ′ = 600 mV, E I = 0 = 225 mV, n = 3, α = 0.24, T = 25˚C), k 1 = 1.0352 × 10 − 6 s − 1 , k 2 = 28.0575 V − 1 , k 3 = 375 mV . The model was integrated with the initial condition C (t = 0) = 0, C⁰ = 5 mM, scan rate r = 0.3 mV·s⁻¹, and V_TL = 25 µL (base case; corrected by 25% for Figure 5), over the potential window plotted in Figure 3 (200 - 1000 mV, i.e.t ∈ [0, 2667 s] at r = 0.3 mV·s⁻¹), using MATLAB’s ODE45 solver (variable-step Runge-Kutta 4(5)) with RelTol = 1 × 10⁻⁶ and AbsTol = 1 × 10⁻⁹ (mM), tightened from the ODE45 defaults to resolve the sharp, exponentially amplified peak in C₀ − C(t) and confirmed by a tolerance-halving convergence check.

Among the trial values of k explored, ksim was selected as the value for which the simulated C 0 − C curve reached the maximum expected conversion ( C 0 − C = 5 mM ) within the timescale of the potential sweep; the resulting current-potential response was then compared, as an independent check, with the experimental voltammogram (Figure 4 and Figure 5).

Here, C(t) denotes the concentration of Ni(II)-bound histamine not yet converted through the coupled electrochemical-chemical cycle, C₀ = 5 mM is the total initial concentration engaged in the cycle (Table 1), and C 0 − C is the cumulative concentration of histamine consumed; complete conversion therefore corresponds to C → 0 .

Table 1. Experimental data.

The principal objective is to determine the value of the rate constant k for which complete depletion of histamine is achieved, i.e., when the concentration of histamine reaches C = 0 , corresponding to C 0 − C = 5   mM .

The variable C used above to denote the Ni(III)-HmP adduct concentration in this single-step mass balance is distinct from the cumulative histamine-conversion variable C introduced in Section III.B.

Figure 3. Theoretical concentration curves of C 0 − C for histamine obtained from MATLAB simulations.

The best agreement, corresponding to complete histamine consumption over the timescale of the experiment, was obtained for: ksim = 2.8 × 10⁻⁶.

This value is of the same order of magnitude as, and within experimental uncertainty of, the k = ( 2 ± 1 ) × 10 − 6   s − 1 , obtained independently by thin-layer voltammetry in [1]. This numerical reproduction of an experimentally established rate constant, from a first-principles kinetic model built independently of the voltammetric fitting procedure of our first work [1], is the principal new result of this study.

Using ksim = 2.8 × 10 − 6 , the current I was recalculated from the electrochemical rate expression.

I = n F S k 0 C ′ exp ( α n lim F R T ) × ( E I = 0 − E 0 ′ + r t ) and compared to the experimental voltammogram (Figure 4 below).

Figure 4. Comparison of the experimental results with the theoretical results obtained from the simulation for k = 2.8 × 10 − 6 .

The two curves agree well overall, but with a ≈ 100 mV shift in peak potential and a slightly higher experimental peak current, attributable to the sensitivity of thin-layer measurements to small volumetric uncertainties, and to a likely overestimation of the effective thin-layer volume in the model. Correcting this volume by 25% substantially improves the agreement (Figure 5 below), supporting the validity of the simulation for describing histamine oxidation kinetics under these conditions.

Figure 5. Comparison of the experimental results with the theoretical results obtained from the simulation for k = 2.8 × 10 − 6 , with correction of the thin-layer volume.