specific energy is usable stored energy divided by device mass, commonly in Wh/kg; specific power is deliverable power per the same mass, in W/kg. The definition must specify whether packaging and inactive components count, and the discharge conditions must be comparable. On a Ragone plot with energy horizontal and power vertical, ordinary capacitors occupy high-power/low-energy territory, batteries tend toward higher energy and lower power, and supercapacitors bridge and overlap the two. These are broad regions rather than universal measured boundaries.

High specific energy means long operation for a given mass; high specific power means rapid delivery. Their ratio is a characteristic duration: $t_{\mathrm{h}}=e_s/p_s$ when the preceding units are used, or $t_{\mathrm{s}}=3600e_s/p_s$. The diagonal lines in the sketch keep that unit conversion explicit. A cell's rated energy and peak pulse power need not be available simultaneously throughout discharge.

A material exhibiting piezoelectricity converts cyclic mechanical stress into electric charge; its mechanical and electrical boundary conditions jointly set the extractable energy. piezoelectric ceramics such as lead zirconate titanate are stiff and often have large charge coefficients and electromechanical coupling, but are brittle and can be difficult to match to large-strain flexible motion. piezoelectric polymers such as polyvinylidene fluoride are compliant, light and compatible with flexible large-area films. Their lower charge coefficient is partly offset in voltage-sensitive applications by lower permittivity: the voltage coefficient is $g=d/ϵ$.

Ceramics often favor strong charge generation and coupling; polymers favor flexibility and strain accommodation. The larger open-circuit voltage of a low-capacitance film does not itself imply more useful power. For a simple fixed-stress comparison the electrical energy scale involves $d^2/ϵ$, while coupling also depends on mechanical compliance. At fixed imposed strain, different stiffnesses change the available mechanical energy. resonance tuning, fatigue, mechanical mounting, rectifier losses and load matching can reverse a simplistic material ranking; compare complete devices under the same actual vibration source.

First hold the production rate fixed, so total electric current remains $200\mathrm{A}$. If only the impure anode receives a factor $10^5$ increase in accessible area, its local current density becomes $0.002\mathrm{A}{\mathrm{m}}^{-2}$, far below $j_0=1\mathrm{A}{\mathrm{m}}^{-2}$. Extrapolating the large-current Tafel equation would give a negative anodic overpotential, which is physically the wrong regime. Near equilibrium the Butler-Volmer equation instead gives, in the same conditional two-electron model,The anode activation loss nearly disappears; the 1.20 V ohmic loss does not disappear. The cathode still operates close to its original transport limit if it is unchanged. The conditional kinetic model gives $V_{\mathrm{app}}\simeq 1.3286$ V for anode-only enlargement. Its anodic potential returns close to 0.34 V, making retention of noble silver and gold more favorable, while less noble nickel dissolution and lead sulfate formation remain relevant.

If both active areas increase by $10^5$ and their independent mass-transfer capacity scales by the same factor, both local kinetic losses become small and the conditional prediction is about $1.20005$ V. Porous internal area does not generally multiply an external diffusion-limited flux by $10^5$: pore access, ion depletion, tortuosity, electrical connectivity and slime blockage must be checked. If the specified 200 A/m² is instead held per true enlarged area, total current rises to $2\times 10^7$ A and the unchanged electrolyte resistance would give $1.2\times 10^5$ V, so this is an entirely different operating condition.

For the same refining rate and the same effective kinetic model, the anode-only estimate is therefore $1.3286\mathrm{V}$. The robust conclusion at fixed total current is reduced local activation polarization on the genuinely accessible enlarged surface, with an unchanged ohmic floor unless cell geometry or conductivity also changes. Increased geometric area alone does not prove a proportionate decrease in energy cost or improved purity. The distinct area and current conventions above specify how to interpret the prediction; they must not be mixed in one calculation.

Gas formation is decided by each interfacial electrode potential and reaction kinetics, rather than by comparing the terminal cell voltage directly with 1.23 V. On the conditional model above, the cathode potential is $E_c\simeq 0.34-0.12854=+0.21146$ V versus the standard hydrogen electrode. At unit proton activity and unit hydrogen pressure this is above the hydrogen equilibrium potential, so hydrogen evolution is not favored there. The anode is at $E_a\simeq 1.0188$ V, below the unit-activity oxygen equilibrium potential of 1.23 V, so oxygen evolution is likewise not favored at the intended point. Neither gas is expected in that model, despite a terminal voltage near 2 V; 1.2 V is lost in the electrolyte.

Without the missing cathodic kinetic information, the exact cathode potential is not fixed, so the hydrogen conclusion is conditional. Locally depleted copper ions, increased current, electrode blocking or strong passivation can move an interface into gas evolution. Actual proton/oxygen/hydrogen activities shift equilibrium through the Nernst equation, and gas nucleation and activation barriers further affect observable bubbles. A reference electrode at each interface and direct gas observation distinguish these effects from an increased solution-resistance drop.

nickel is less noble than copper and can dissolve anodically, but its reduction requires a much more negative cathodic electrode potential. It therefore normally accumulates in solution rather than codepositing with copper. lead can also oxidize, but sulfate can precipitate the resulting lead ions as poorly soluble lead sulfate, contributing to anode slime or passivation. gold normally remains in the solid residue. silver is normally recovered predominantly in anode slime, but its path can include dissolution and reprecipitation or cementation: a permanent refusal of every silver atom to dissolve is not the mechanism. Experiments with controlled Cu–Ag–Pb anodes, a fixed sulfate electrolyte and SEM/EDS plus diffraction analysis explicitly find metallic silver and sulfate-containing slime phases.^[ref]

Selective electrode potential, solution management and physical exclusion of slime permit purer copper deposition. However, the unusually large printed anodic slope prevents an unconditional silver-retention claim here. The preceding calculation places the anode near $0.34+0.6788=1.0188$ V versus the standard hydrogen electrode, above the tabulated ${\mathrm{Ag}}^{+}$/Ag value. silver oxidation is then thermodynamically possible; any silver ions reaching the cathode can be reduced preferentially to copper. Actual activities, precipitation, cementation and transport determine contamination. The requested 99.999 wt% purity cannot be established from the listed standard potentials alone. Monitor anode potential against a reference, assay the electrolyte and deposit, and prevent slime entrainment; do not silently replace the printed 0.295 V/decade with a smaller number to remove this issue.

On the stated $1{\mathrm{m}}^2$ area, the total electric current is $I=200\mathrm{A}$. The known ohmic polarization is $I{R}_{\Omega }=1.20\mathrm{V}$. Applying the supplied anodic Tafel equation givesAssume the quoted limiting current uses the same cathodic area convention. A steady diffusion-film model gives $c_s/c_b=1-j/j_L=1/41$. The Nernst equation then makes the cathodic concentration overpotential magnitudeRelative to the local surface equilibrium potential, the cathode also needs an activation overpotential. Neglecting anodic concentration polarization, the separated-loss accounting isTo calculate that activation term, adopt a single effective two-electron Butler-Volmer equation with constant transfer coefficients satisfying ${\alpha }_a+{\alpha }_c=1$, no anodic concentration polarization and reduction proportional to the surface ${\mathrm{Cu}}^{2+}$ concentration. The supplied anodic Tafel slope then determines the reverse slope throughWith $f=2F/(RT)$ and cathodic overpotential ${\eta }_c$ referenced to bulk equilibrium,Solving at $j_c=-200\mathrm{A}{\mathrm{m}}^{-2}$ gives ${\eta }_c\simeq -0.12854\mathrm{V}$, including both concentration and activation effects. ThusThe $0.04765$ V concentration loss must not be added again. The kinetic assumption matters: the supplied slope is for copper oxidation, and an exchange current density alone does not specify every cathodic current-potential relation. A multistep copper reaction need not obey the effective single-step relation used for this estimate.

The desired net process transfers copper from an impure anode to a purer cathode. In the idealized common-electrolyte approximation, both interfaces use the same ${\mathrm{Cu}}^{2+}$/Cu equilibrium potential. Setting solid copper activities to one and interfacial ion activities equal gives $E_{\mathrm{rev}}=0$ V: subtracting two equal 0.34 V reduction potentials yields zero, rather than 0.34 V.

A strictly thermodynamic refining voltage need not be exactly zero. At equal dissolved-ion activity, the required reversible applied voltage isDilution of copper in the impure phase can therefore require positive reversible work. Weight percentage alone does not specify that phase's thermodynamic activity or impurity molar composition. The zero result is the intended pure-solid approximation, not a precise chemical-potential measurement from the stated weight purities.

A La Mer diagram plots the concentration of growth species against time. Label the equilibrium solubility $C_s$ and a higher effective nucleation threshold $C_n$. Generate growth species until the concentration briefly exceeds $C_n$; a short burst nucleation interval then creates seeds. Consumption brings the concentration below $C_n$ while leaving it above $C_s$, allowing existing particles to grow without substantial continuing nucleation. Rapid mixing, a short seed-formation interval and a common growth period help produce a narrow particle size distribution. Suppress secondary nucleation and aggregation, then quench or consume the remaining growth species when the desired size is reached. The threshold is an operational kinetic scale, not a sharp new equilibrium phase boundary.

Reducing the solid-liquid interfacial tension lowers the classical barrier, which scales as ${\gamma }^3$ at fixed supersaturation. More seeds can then form at a smaller supersaturation; at fixed total precipitated mass, more seeds generally mean a smaller mean final particle size. A narrower distribution is not guaranteed: if nucleation continues throughout growth, particles have different ages and the distribution can broaden. A brief well-separated burst and uniform subsequent growth are the controlling requirements, not low interfacial tension alone. The historical model concerns separated nucleation/growth in sulfur hydrosols, rather than a universal demonstrated mechanism for every nanoparticle synthesis.^[ref]

In a sufficiently supersaturated vapor, making liquid lowers bulk Gibbs free energy, but making a small droplet creates a costly liquid-vapor interface. The volume contribution scales as $r^3$ and the interfacial cost as $r^2$. Identifying a positive bulk driving force density $|\Delta g_v|$ and interfacial tension $\gamma$ gives $A=4|\Delta g_v|/3$ and $B=4\gamma$. In the supplied notation,At $r_{\ast }$, $d^2\Delta G/d{r}^2=-2\pi B<0$: this is the barrier summit. Subcritical droplets tend to shrink, whereas supercritical droplets can grow by reducing Gibbs free energy. Fluctuations must cross the barrier before bulk condensation proceeds, especially in a clean vapor without effective sites for heterogeneous nucleation. A temperature just below 100 °C implies a supersaturated vapor only with a suitable pressure, such as a vapor maintained near one atmosphere; temperature alone does not specify the driving force. The nucleation barrier can delay condensation even when bulk liquid is thermodynamically favored.

Approximate the reaction enthalpy and entropy as constant over the measured interval, so ${\Delta }_r{G}^{\circ }={\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }$. A linear least squares fit of all three free energies against temperature givesand henceThe negative entropy change is reasonable because oxygen gas is incorporated into solids. The fit residuals are $+0.129,-0.257,+0.129\mathrm{kJ}\mathrm{mo}{\mathrm{l}}_{{\mathrm{O}}_2}^{-1}$, so the data support this approximate straight-line description. Extrapolating its intercept to zero temperature is a way of estimating a constant local enthalpy, not evidence that the heat capacities vanish or that the same phases persist to zero temperature. More temperatures and repeated heating/cooling measurements would test curvature and equilibration.

The solid electrolyte conducts oxide ions. Equilibrating oxygen at the two interfaces couples one ${\mathrm{O}}_2$ molecule to four electrons. Taking the positive measured electromotive force as the air-reference potential relative to the Ni/NiO side, the Nernst equation isAssume air at one bar with $p_{\mathrm{air}}/p^{\circ }=0.21$, pure solid phases of unit thermodynamic activity, and $p^{\circ }=1\mathrm{bar}$. For the reaction normalized to one mole of oxygen,The negative sign is physically consistent with a strongly reducing Ni/NiO equilibrium relative to air. Using the supplied $R$ and $F$ givesThe tabulated free energies are per mole of ${\mathrm{O}}_2$, not per mole of NiO. Values normalized to one mole of NiO would be half as large. A reliable oxygen concentration cell measurement requires reversible interfaces, calibrated uniform temperature, a known reference oxygen pressure and predominantly ionic rather than electronic conduction through yttria-stabilized zirconia. Air is not pure oxygen: dropping the $RT\ln 0.21$ term changes the inferred entropy.