本工具是可复核的一维电化学教学模型，不是特定材料、氧化还原对或仪器的标定预测器。独立实现审核状态见下方，AI 审核不等同于电化学专家认证。
平面宏电极上的可逆单电子反应 O + e⁻ ⇌ R。两组分采用同一扩散系数 D，体相初始浓度 cₒ = C*、cᵣ = 0。加入过量支持电解质使电迁移可忽略，溶液静止并忽略对流。反应速度足够快，电极表面始终满足 Nernst 平衡。本版没有可调 Butler–Volmer 速率常数或传递系数。
内部全部采用 SI：D 为 m²/s，距离 m，时间 s，面积 m²，浓度 mol/m³，电流 A，温度 K。界面上 1 mM = 1 mol/m³；面积以 cm² 输入，换算为 m²。电位相对同一个理想参比，以 E⁰′ 为可调形式电位。
An inspectable 1D electrochemical teaching model, not a calibrated predictor for a material, redox couple or instrument. Review status appears below; AI review is not certification by an electrochemistry expert.
A reversible one-electron reaction O + e⁻ ⇌ R at a planar macroelectrode. Both species have the same diffusivity D. Initially cₒ = C* and cᵣ = 0 in bulk. Excess supporting electrolyte permits neglect of migration; the solution is quiescent and convection is omitted. Electron transfer is fast enough to maintain Nernst equilibrium at the surface. Adjustable Butler–Volmer rate constants and transfer coefficients are not implemented.
The solver uses SI: D in m²/s, distance m, time s, area m², concentration mol/m³, current A and temperature K. UI input 1 mM equals 1 mol/m³; cm² is converted to m². Potentials refer to one ideal reference, with adjustable formal potential E⁰′.
∂cₒ/∂t = D ∂²cₒ/∂x² ; cᵣ = C* − cₒ
cₒ(0,t) = C* / {1 + exp[−F(E − E⁰′)/(RT)]}
cₒ(L,t) = C* ; cₒ(x,0) = C*
i = −FA D (∂cₒ/∂x)₀
E: E⁰′ + 0.350 → E⁰′ − 0.350 → E⁰′ + 0.350 V
tmax = 1.400 / ν ; L = 8√(D tmax)
F = 96485.33212 C/mol ; R = 8.314462618 J/(mol K)
x 轴从工作电极指向电解液。还原消耗 O，因此正浓度梯度对应负电流；氧化电流为正。初始高电位使表面 O 近似为 C*，但求解器仍施加精确 Nernst 边界。总浓度恒定来自相同 D、互补反应通量及相同初始/体相总浓度，不能无条件扩展到不同扩散系数。
采用守恒有限体积离散和后向 Euler 隐式时间推进，700 个均匀单元、2800 个时间步。未知量在单元中心；两侧 Dirichlet 边界位于半网格之外，所以边界通量为 2D(c边界−c首单元)/Δx。三对角线由 Thomas 算法求解。界面浓度剖面只显示靠电极的 0–0.38L，但求解域为 0–L。逐步检查总量变化与两边界通量积分之间的残差。
预计算一个完整物理周期后同步回放曲线、读数、组分与三维视图；播放速度只影响观看时长。每个球形标记的颜色按其法向位置的浓度采样，数量与轨迹不代表真实分子。浓度剖面为工作电极法向的独立放大，三电极硬件为通用示意；相机运动不改变物理模型。
自动测试核对 Nernst 正负号、平衡点、无梯度零通量、全离散质量守恒、浓度有界性、还原/氧化电流符号、峰间距与 Randles–Ševčík 峰电流，以及 C*、√D 与 √ν 标度。以 T = 298.15 K、D = 10⁻⁹ m²/s、ν = 0.1 V/s、A = 1 cm²、C* = 1 mM 为基准，理想峰值为下式；数值峰误差门槛 3.5%，峰间距门槛 52–69 mV。有限网格和后向 Euler 会产生数值误差。
The positive x direction points from the working electrode into electrolyte. Reduction consumes O, so a positive concentration gradient corresponds to negative current; oxidation current is positive. At the initial high potential surface O is nearly C*, but the exact Nernst boundary is applied. Constant total concentration follows from equal diffusivities, complementary reaction fluxes and equal initial/bulk totals; it must not be generalized to unequal diffusivities.
Conservative finite volumes with implicit backward Euler, 700 uniform cells and 2800 time steps. Unknowns are cell-centred; each Dirichlet boundary lies half a cell away, giving boundary flux 2D(cboundary−cfirst)/Δx. Thomas elimination solves the tridiagonal system. Profiles display only 0–0.38L near the electrode; the computational domain is 0–L. Every step checks total inventory change against both integrated boundary fluxes.
A complete physical cycle is precomputed and replayed with synchronized plots, readouts and 3D composition. Playback speed changes viewing duration only. Marker colours sample local normal concentrations; counts and trajectories do not represent real molecules. The normal section has a separate magnified scale and hardware is generic. Camera motion does not alter the model.
Automated tests check Nernst sign and midpoint, zero flux at uniform equilibrium, fully discrete mass conservation, bounded concentrations, reduction/oxidation signs, peak separation and the Randles–Ševčík peak current, plus C*, √D and √ν scaling. The benchmark uses T = 298.15 K, D = 10⁻⁹ m²/s, ν = 0.1 V/s, A = 1 cm² and C* = 1 mM. Numerical peak tolerance is 3.5% and peak separation must lie within 52–69 mV. A finite grid and backward Euler introduce numerical error.
|iₚ| = 0.4463 FA C* √(FDν/RT)
独立 AI 交叉审核已复核有限体积边界符号、Thomas 求解与电流方向；350/1400、700/2800、1000/5600 三组空间/时间网格的阴极峰分别为 −268.63765、−268.64344、−268.64483 µA，峰间距为 58、57.5、57.5 mV，单步质量残差小于 3.89×10⁻¹⁹ mol/m²。独立审核未发现实质科学阻塞；不等同于人工电化学专家认证。
不包含迁移、对流、欧姆降、双电层充电电流、有限电子转移动力学、吸附、耦合化学反应或电极边缘效应。域长按每个扫描周期重新设置，不能解释为真实电池容器尺寸。L 大于有限时间内主要扩散尺度，仍是半无限扩散的有限域近似。本工具不读取或拟合实验数据。
Independent AI cross-review re-derived finite-volume boundary signs, Thomas elimination and current direction. Spatial/time grids 350/1400, 700/2800 and 1000/5600 gave cathodic peaks −268.63765, −268.64344 and −268.64483 µA, separations 58, 57.5 and 57.5 mV, and one-step mass residuals below 3.89×10⁻¹⁹ mol/m². No material scientific blocker was identified. This is not human electrochemistry expert certification.
Migration, convection, ohmic drop, double-layer charging, finite electron-transfer kinetics, adsorption, coupled chemical reactions and electrode-edge effects are excluded. Domain length is reset for each scan and is not a real cell-vessel dimension. It exceeds the main diffusion scale over the finite cycle but remains a finite-domain approximation to semi-infinite diffusion. No experimental data import or fitting is performed.
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