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An ion channel supplies a water-accessible pathway through the otherwise strongly insulating, hydrophobic lipid bilayer. Moving a hydrated ion directly into that low-permittivity region is energetically expensive. The channel's polar pore and coordinating groups reduce the barrier while retaining selectivity. Transport is passive: the electrochemical potential difference drives net movement, and the channel does not itself pump an ion uphill. For an ideal species of charge number $z$,The net electric current of a pathway permeable only to that species reverses at its Nernst equilibrium potential. An open channel can carry zero net electric current at that voltage even while individual ions cross both ways.

The selectivity filter balances loss of the hydration shell against favorable coordination within the pore. In a typical potassium channel, backbone carbonyl oxygen atoms create appropriately arranged binding sites for a substantially dehydrated potassium ion. The smaller sodium ion is not automatically preferred simply because it fits geometrically: its hydration is stronger and the filter does not compensate its dehydration in the same way. Geometry, flexibility, coordination chemistry and ion occupancy all matter. Sodium-selective and calcium-selective channels use different arrangements, often involving charged side chains; a single rigid-sieve explanation is inadequate.

Rapid passage coexists with transient binding because several nearby sites have comparable free energies and relatively low barriers between them. Multiple ions occupying a pore influence one another through electrostatic interactions, enabling correlated exchange and displacement rather than irreversible trapping at one deep site. High throughput therefore requires favorable passage kinetics as well as strong discrimination against unwanted species. A more tightly bound species can even block a pathway instead of being its fastest permeant.

Gating and selectivity are distinct. A voltage-sensing domain couples membrane potential to activation-gate conformation; ligand-gated ion channels couple binding to opening; mechanosensitive ion channels couple membrane stress to opening. Ion-channel inactivation introduces additional unavailable states. None of these mechanisms alone identifies which ion the open pore prefers.

For $N$ similar independent channels, a local Ohmic approximation with outward-positive electric current isHere open probability controls availability, single-channel electrical conductance controls current per open pore, and $V-E_{\mathrm{rev}}$ is the electrical driving factor in this approximation. Nonlinear current-voltage relations and mixed-ion permeation may require a fuller model. Single-channel recording resolves discrete openings; reversal shifts when ionic concentrations change test selectivity; gating electric currents test sensor motion; structural evidence identifies the pore and its coordinating groups. Together these distinguish rapid selective permeation from the separate process of opening the gate.

In the continuum cylindrical-pore estimate, the electrical resistance is $R_p=\rho \ell /(\pi a^2)$, with radius $a=0.075\mathrm{nm}$, length $\ell =0.5\mathrm{nm}$ and electrical resistivity $\rho =80\Omega \mathrm{cm}=0.80\Omega \mathrm{m}$. ThereforeUnrestricted approach from the reservoirs removes access barriers, but still permits a continuum spreading or access resistance of a circular pore. Including $\rho /(4a)$ on each side givesThus the usual estimate neglecting the access contribution is about $44\mathrm{pS}$, or about $36\mathrm{pS}$ with spreading resistance. A $1.5$-ångström diameter is at the molecular scale, smaller than hydrated ions; bulk electrical resistivity cannot quantitatively describe dehydration, selectivity or discrete-ion transport. These values are the requested continuum estimates, not evidence that such a pore freely passes hydrated potassium ions.

The rising sigmoid shows activation by depolarization: $P_o$ is almost zero at very negative membrane potential, rises steeply around $-20\mathrm{mV}$, and approaches about $0.8$ at positive membrane potential. The graph describes the fraction of time a single ion channel is open, not a continuously variable opening amplitude. A convenient phenomenological fit isHere $V_{1/2}$ refers to half the maximum, about $0.4$, not absolute $P_o=0.5$. The effective gating charge describes the steepness; a sigmoid alone does not identify the microscopic gating sequence or explain why the observed maximum is below one.

A voltage-gated potassium channel has four pore-forming subunits. Each canonical subunit contains an S1–S4 voltage-sensing domain and S5–S6 pore elements. Positive charges in S4 respond to the electric field; their displacement during depolarization changes the energetic preference of sensor conformations. Coupling through the sensor–pore interface favors opening of the intracellular pore gate. The potassium selectivity filter is structurally distinct from this voltage sensor.

Several independent observations support this account. Voltage clamp detects transient gating electric currents before pore conduction, even when ionic conduction is blocked: charge movement precedes opening. Charge-neutralization studies change total gating charge and voltage dependence, with contributions from charged S4 residues and an S2 residue. Accessibility and voltage-sensitive optical measurements support conformational motion rather than permeating potassium itself acting as the sensor. Structural measurements locate the charged sensor beside the pore. These observations constrain a coupled moving-sensor mechanism, while the exact displacement cannot be deduced from this $P_o$ curve alone.^[ref][ref]

In ball-and-chain inactivation, also called N-type inactivation, a flexible intracellular N-terminal region carries an inactivation particle. Once a voltage-gated potassium channel opens, this tethered particle can enter and occlude the inner pore, stopping electric current even while the activation machinery favors an open configuration. Recovery requires the particle to leave, after which the activation gate can return to its resting conformation. This supplies an additional unavailable state, not merely reversal of activation.

The decisive evidence separates necessity from sufficiency. Removing the relevant N-terminal region greatly reduces fast ion-channel inactivation while preserving activation. Applying a peptide corresponding to the missing N-terminal region on the cytoplasmic side restores blocking with similar properties. Changes in the blocking region alter blocking kinetics, and altering the tether affects the rate at which the particle reaches the pore. Open-pore access and competition with intracellular pore blockers support occlusion of the inner cavity. A tether makes the particle's local effective concentration high, explaining rapid intramolecular block.^[ref][ref]

This mechanism is distinct from slower filter-associated C-type inactivation. An N-terminal particle is a mechanism for certain potassium ion channels, not a universal explanation of all ion-channel inactivation.

In a classical action potential, delayed activation of voltage-gated potassium channels supplies outward electric current that drives repolarization. In the Hodgkin-Huxley model, for example, $I_K={\bar{g}}_K{n}^4(V-E_K)$ and $\dot{n}={\alpha }_n(V)(1-n)-{\beta }_n(V)n$. Increasing the opening rate at depolarized membrane potential, other properties fixed, recruits outward electric current earlier: the action potential generally becomes narrower and may have a smaller peak. Slower opening delays repolarization and broadens the action potential.

Slower closing after repolarization retains excess potassium electrical conductance, lengthening the after-hyperpolarization and often the interval of reduced excitability. Faster closing shortens this tail and permits earlier recovery. If activation or the available electrical conductance is insufficient, normal repolarization may fail; changing rates can also change the steady open fraction. Duration is therefore determined by the whole balance of inward and outward electric currents, not by a closing rate in isolation.

An ion channel can disturb cellular electrical activity by changing unitary electrical conductance, open probability, gating kinetics or the number delivered to the cell membrane. A useful organizing relation isHere outward electric current is positive. Altering one channel's biophysical behavior changes the balance governing membrane potential and hence the timing or recurrence of action potentials.

Long QT syndrome, type 3. Certain disease-associated SCN5A variants impair fast ion-channel inactivation of the cardiac voltage-gated sodium channel NaV1.5. Instead of vanishing promptly after the upstroke, a small persistent inward sodium electric current remains during the plateau and repolarization. Because $V<E_{\mathrm{Na}}$ in this phase, the residual electric current is inward and opposes the outward electric currents that would terminate the cardiac action potential. The ventricular action potential lasts longer, extending the QT interval on the electrocardiogram. Excessively prolonged or uneven repolarization permits early afterdepolarizations and dangerous ventricular rhythm disturbances.

Genetic association identifies the channel, while voltage clamp distinguishes defective ion-channel inactivation from a mere increase in channel abundance. Recordings of disease-associated channels show persistent or bursting openings; coupling measured mutant sodium electric current into an otherwise modeled cardiac membrane through dynamic clamp tests whether that kinetic defect prolongs the action potential. The phenotype is variant-specific: not every SCN5A variant increases late electric current or causes long QT syndrome.^[ref][ref]

Myotonia congenita. CLCN1 encodes the skeletal-muscle ClC-1 chloride channel, which normally supplies a large resting stabilizing electrical conductance. Because the chloride reversal potential is near the resting membrane potential, that pathway resists small depolarizing disturbances. Loss of functional channels, reduced electrical conductance, or a shift in opening toward more positive membrane potential reduces the resting electrical shunt. Potassium accumulation in the transverse-tubule space during activity can then produce larger depolarizing afterpotentials and trigger further muscle action potentials after the intended stimulus has ceased. Continued electrical firing sustains contraction and delays relaxation: this is myotonia, rather than a primary inability of the contractile apparatus to relax.

Patient-family linkage and variant analysis connect the disease to ClC-1. Functional recordings of disease-associated channels demonstrate shifted voltage dependence and reduced availability near resting membrane potential; single-channel and noise measurements also resolve two conductive protopores and their gating behavior. Different variants can act through different mechanisms, so a gating shift is an example rather than an explanation of every case.^{[ref][ref][ref]}

These examples connect contrasting current defects to their consequences: persistent inward sodium electric current prolongs cardiac action potentials, whereas loss of stabilizing resting chloride electrical conductance permits repetitive skeletal-muscle action potentials. Both causal chains require functional channel measurements in addition to a disease-associated sequence change.

G is the diaryl ether $(2-\mathrm{N}{\mathrm{O}}_2{\mathrm{C}}_6{\mathrm{H}}_4)-\mathrm{O}-(2-\mathrm{C}{\mathrm{H}}_2\mathrm{C}(=\mathrm{O}){\mathrm{N}}_{\mathrm{morph}}{\mathrm{C}}_6{\mathrm{H}}_4)$. Here $N_{morph}$ denotes the nitrogen of the intact morpholine ring; the phenoxy oxygen and phenylacetamide chain are ortho on their original ring. potassium carbonate first generates phenoxide from F. Its oxygen attacks the fluorine-bearing carbon of 1-fluoro-2-nitrobenzene, forming a Meisenheimer complex. Delocalization of its negative charge into the ortho nitro group stabilizes this addition intermediate. Fluoride elimination restores aromaticity and completes nucleophilic aromatic substitution.

This first step forms an O–aryl bond, rather than immediately arylating the less acidic alpha carbon. The neutral product has formula ${\mathrm{C}}_{18}{\mathrm{H}}_{18}{\mathrm{N}}_2{\mathrm{O}}_5$, obtained from the two substrates by net loss of HF. The connectivity is shown beside B in the independently drawn diagram above. The amide-substrate O-to-C migration strategy has direct primary precedent.^[ref]

Potassium intercalates between graphite layers and donates electron density to the graphitic pi system. Occupation of formerly less populated electronic states raises the mobile-carrier density, increasing electrical conductivity under the stated comparison.

The gain is electronic doping of the carbon framework, rather than a claim that potassium strengthens every C-C bond or simply acts as a separate wire through the sample. Intercalation also changes layer separation and scattering, so conductivity depends on both carrier population and mobility.

The material is potassium graphite, a graphite intercalation compound.

$\mathrm{A}=\mathrm{K}[{\mathrm{B}}_5{\mathrm{H}}_8]$ is nido; $\mathrm{B}=\mathrm{K}[{\mathrm{B}}_6{\mathrm{H}}_{11}]$ and $\mathrm{C}={\mathrm{B}}_6{\mathrm{H}}_{12}$ are arachno. The initial hydride removes an acidic bridging hydrogen as a proton, releasing ${\mathrm{H}}_2$ and retaining the potassium counterion. Addition of one $\mathrm{B}{\mathrm{H}}_3$ unit expands the boron framework; final protonation supplies the neutral hexaborane and KCl. These assignments conserve all the atoms in the displayed sequence, without inserting an unshown dehydrogenation step.

For ${\mathrm{B}}_{\mathrm{n}}{\mathrm{H}}_{\mathrm{m}}^{\mathrm{q}-}$, take one terminal B–H bond per vertex and remove its $2n$ electrons from the valence total. The skeletal electron count is $3n+m+q-2n=n+m+q$. ThusWade's rules assign $n+1$, $n+2$ and $n+3$ pairs to closo, nido and arachno frameworks respectively. A derives from a six-vertex closo octahedron by removal of one vertex, giving a square-pyramidal boron skeleton. B and C have six-vertex open frameworks formally derived from an eight-vertex closo polyhedron by removing two vertices. Their additional hydrogens are distributed over bridging and other open-face positions; counting every hydrogen as an independent terminal B–H vertex would give the wrong classification.

The arachno assignment of $[{\mathrm{B}}_6{\mathrm{H}}_{11}{]}^{-}$ has specific theoretical and NMR support. It should not be replaced by a permanently localized, unrearranged $[{\mathrm{B}}_5{\mathrm{H}}_8{]}^{-}\cdot \mathrm{B}{\mathrm{H}}_3$ picture: the corresponding symmetric adduct participates in fluxional rearrangement rather than being the lowest-energy structure.^[ref]

In the Hodgkin-Huxley model, $V$ is the membrane potential, the voltage inside the axon relative to outside. The space clamp assumption makes it spatially uniform. The gating variables lie between zero and one: $m$ describes sodium activation, $h$ describes availability against sodium inactivation, and $n$ describes potassium activation. Thus large $h$ means channels remain available, not that inactivation is strong.

The left side $C_m\dot{V}$ is capacitive electric current per unit membrane area. The imposed $I$ is inward-positive applied electric current. The other terms are outward-positive ionic electric currents: sodium with electrical conductance $g_{Na}{m}^{3}h$, potassium with conductance $g_K{n}^4$, and a passive leak with conductance $g_L$. Each current is its conductance times the difference between $V$ and the corresponding reversal potential $V_{Na}$, $V_K$ or $V_L$. The coefficients $g_{Na},g_K$ are maximal conductances; the gate factors represent independent permissive gates in an ion channel.

For each gate, $k_{\infty }(V)$ is the equilibrium opening/availability fraction at fixed voltage and ${\tau }_k(V)>0$ its exponential relaxation time. The equation follows from two-state ion-channel gating: if $\dot{k}={\alpha }_k(V)(1-k)-{\beta }_k(V)k$, thenVoltage feedback couples these otherwise elementary gating equations to the membrane potential and produces action potentials.

Write $M(v)=m^3(v)$. In the physically relevant strip $-v_K<v<1$, $0<n<H$, the zero-current nullcline satisfiesThe left side increases strictly from zero to infinity, so there is exactly one nullcline height $n=N(v)$ for each interior voltage. If $M$ is positive at the endpoints, its curve joins $(-v_K,H)$ to $(1,0)$; when low-voltage activation is idealized as exactly zero, the limiting left branch can lie on $v=-v_K$.

The signs of the derivatives explain its folds:At low voltage, potassium's approximately linear contribution dominates and $g_v>0$. During rapid sodium activation the last negative term can dominate, giving $g_v<0$. Once activation saturates, $M^{\prime }$ becomes small and $g_v>0$ again. This folded conductance-model nullcline has the familiar shape: the curve has descending outer branches and an ascending middle branch in the $(v,n)$ plane. Equivalently, an intermediate fixed $n$ can give three zeros of $g$ as voltage rises, with positive, negative and positive $g_v$ at those crossings.

The rapid-activation description permits this folded shape; exact fold positions depend on $M,{\gamma }_K,H,v_K$. The figure supplies an explicit admissible sigmoid example. Its right panel shows $g=0$ as well as the recovery nullcline and an oscillating trajectory.

In the Hodgkin-Huxley model, $V$ is the intracellular-minus-extracellular membrane potential, $C_m$ is the membrane capacitance per unit area, and $I_{\mathrm{app}}$ is the applied inward electric current per unit area. A space clamp makes $V$ spatially uniform, leaving temporal ordinary differential equations. With outward ionic electric current positive, the capacitance balance is $C_m\dot{V}=I_{\mathrm{app}}-I_i$.

The variables $m,n,h$ lie between zero and one. They describe sodium activation, potassium activation and availability against sodium inactivation, respectively. Thus larger $h$ means less inactivation. Their voltage-dependent limiting values are $m_{\infty },n_{\infty },h_{\infty }$, and ${\tau }_m,{\tau }_n,{\tau }_h$ are their positive relaxation time constants.

Each ionic term has the Ohm's law form conductance times driving potential. The sodium term has instantaneous electrical conductance $g_{\mathrm{Na}}{m}^{3}h$, the potassium term has instantaneous electrical conductance $g_{\mathrm{K}}{n}^4$, and the leak term has constant electrical conductance $g_{\mathrm{L}}$. Here the $g$ constants are maximal electrical conductances per area, and $V_{\mathrm{Na}},V_{\mathrm{K}},V_{\mathrm{L}}$ are the corresponding reversal potentials. The factors $m^{3}h$ and $n^4$ arise from independent permissive gates in an ion channel. A conducting channel carries zero net ionic electric current at its reversal potential.

Cow's milk contains water, dispersed fat, casein, soluble milk proteins, lactose and mineral constituents, with smaller amounts of other substances. These constituents occupy different physical states, which makes their separation possible: the fat is present as an emulsion, much of the casein as colloidal casein micelles, and the lactose and many salts in chemical solution. Vitamins, enzymes and small nitrogenous constituents accompany these main fractions. The proportions vary between samples and are not fixed by the name cow's milk.^[ref]

First let fresh unhomogenized milk stand so that the less dense fat globules rise as cream; skimming gives a fat-enriched layer and skimmed milk beneath. This separates fractions rather than pure chemical substances: cream still contains water and other constituents. Churning cream coalesces the fat globules into butter; extraction of a separately prepared dried portion with a suitable organic solvent provides another way to isolate fat. A modern centrifugal separator accelerates the same density-based separation, but it is not needed to explain an 1864 bench separation.

Next add dilute acetic acid carefully to skimmed milk. Lowering its pH toward the isoelectric point of casein, about 4.6, produces a curd that can be collected by filtration and washed. Excess acid, retained fat, and shifts of calcium phosphate between curd and liquid limit purity. Rennet offers another route to a curd, but its enzyme-catalysed action and the mineral content of the resulting curd differ from simple acid precipitation; the two should not be treated as identical mechanisms.^[ref]

The liquid after removal of the curd is whey. Heat this fraction under suitable acidity to induce protein denaturation and aggregate much of its residual whey protein; collect the coagulum by filtration. Older descriptions commonly grouped this fraction under “albumin”, but modern milk chemistry distinguishes several proteins. Heating alone does not make every minor constituent fall into a single pure fraction.^[ref]

Concentrate the deproteinized whey and cool it to crystallize lactose. Filter and wash the crystals, then purify them by recrystallization; dissolved salts and residual organic matter remain substantially in the mother liquor. This provides a separation from most protein, not an automatic complete separation from all minerals.^[ref]

Finally use separate measured portions for water and mineral determinations. Controlled drying gives an operational total solids measurement; subtraction from the initial mass estimates the water lost. Ignition of a dried portion destroys most organic matter and leaves ash for mineral analysis. Ash is a transformed residue: it does not preserve all original salts unchanged, and some constituents can volatilize. Calcium, phosphorus, sodium, potassium and chloride are among the important mineral constituents. Skimming, curd formation, heat coagulation, lactose crystallization, drying and ashing separate or determine the principal fractions; none alone constitutes a complete pure-component analysis.

Potassium iodide is prepared by converting iodine into iodide, then obtaining a pure crystalline potassium salt. One chemical route uses potassium hydroxide; an alternative first forms iron(II) iodide. The account here describes the chemical operations and their balances, without assigning a later pharmacopoeial formula to 1866.

In the alkaline route, iodine reacts with hot potassium hydroxide to give a mixture of potassium iodide and potassium iodate:This is disproportionation: iodine goes from oxidation state $0$ to both $-1$ and $+5$. Merely evaporating the solution would leave the unwanted iodate with the iodide. Heating the dried salts with charcoal reduces the iodate to iodide. A limiting balance with carbon monoxide as the carbon oxide product isThe carbon oxides can depend on the reaction conditions; the indispensable chemical point is removal of the iodate's oxygen without losing its iodine. Extraction with water, filtration of insoluble residue, concentration and crystallization then recover potassium iodide. A modern USDA technical evaluation explicitly describes this same reaction sequence; it is corroboration of the chemistry, not evidence that its modern document was the 1866 official recipe.^[ref]

In the alternative iron route,followed byThe insoluble iron-containing material is separated and the potassium iodide recovered from the liquid. Oxidation of the ferrous precipitate during working can alter its final form, so the displayed chemical equation is the ideal metathesis balance rather than a guarantee that every collected particle remains pure iron(II) carbonate.

Pure potassium iodide is a colourless or white crystalline ionic solid, commonly in cubic crystals, readily soluble in water and less soluble in ethanol. It has no intrinsic brown iodine colour. Its aqueous solution contains potassium ions and iodide ions. Free iodine dissolves in the presence of iodide through triiodide formation,With silver nitrate, iodide gives yellow silver iodide, sparingly soluble in water and resistant to ordinary aqueous Ammonia:A little chlorine liberates iodine, which gives the characteristic colour with starch:By contrast, iodate contamination can liberate iodine simply on acidification in the presence of iodide:That distinction explains why removal of the iodate is part of preparing a well-defined product, rather than an optional cosmetic step. These are chemical properties and recognition principles, not advice to administer the salt.

The main distinction is between metallic iron, iron(II) preparations, iron(III) preparations, and mixed iron oxides. The old prefixes “proto-” and “per-” ordinarily distinguish the lower and higher oxidation states here. “Peroxide of iron” means the historical iron(III) oxide preparation, not a modern peroxide containing an oxygen–oxygen bond.

The principal medicinal preparations can be grouped by the chemical form supplied:

These identifications follow the historical monographs, rather than the names of present commercial supplements.^[ref] A green iron(II) sulfate solution gives the characteristic blue reaction with potassium ferricyanide, whereas the higher iron solutions give blue with potassium ferrocyanide. Those diagnostic reactions explain why the official tests distinguish the two oxidation states; the historical colour test alone does not establish every detail of a complex preparation's composition.

The chief physiological routes are abrupt failure of blood circulation, failure of pulmonary ventilation or gas exchange, or catastrophic failure of the central control of those functions. Historical clinical accounts used labels such as syncope, asphyxia and medical coma in explaining deaths; these terms describe broad physiological failures rather than identifying the underlying diseases.^[ref] A brief faint or ordinary medical coma is not itself necessarily fatal.

In primary circulatory failure, effective delivery of blood to the brain stops. Modern examples include ventricular fibrillation, pulseless ventricular tachycardia, severe bradyarrhythmia and asystole. They may arise in coronary artery disease, after myocardial infarction, in cardiomyopathy, or from inherited cardiac ion channel disorders. An electrical cause may leave little distinctive gross evidence at autopsy. These modern mechanisms refine the broad historical description of cardiac syncope.^[ref]

Mechanical interruption of blood circulation gives another group: massive haemorrhage, rupture of an aortic aneurysm, cardiac tamponade, and large pulmonary embolism. The immediate deficit differs in each. Haemorrhage removes circulating volume; cardiac tamponade prevents adequate filling; pulmonary embolism obstructs flow through the pulmonary circulation. All can end with critically low cardiac output even if some electrical activity persists.^[ref]

Respiratory failure includes obstruction of the airway by a foreign body, laryngeal oedema or aspiration; drowning; and depression of breathing by drugs or major brainstem injury. Oxygen delivery then fails and secondary cardiac arrest follows. Carbon monoxide poisoning and cyanide poisoning illustrate related failure of oxygen transport or use: not every hypoxic death is caused by physical obstruction of an airway.^{[ref][ref][ref]}

A major intracranial haemorrhage or rapidly developing brain herniation can destroy or compress the pathways maintaining consciousness and breathing. A severe disturbance of electrolytes, especially potassium, may instead act directly on the heart's electrical function. A fatal seizure can involve respiratory and circulatory mechanisms; sudden unexpected death in epilepsy remains a diagnosis requiring exclusion of other causes, not merely recognition of a history of epilepsy.^[ref]

The practical distinction is between mode, immediate mechanism and underlying cause. For example, rupture of an aneurysm can cause haemorrhage, then circulatory failure, then irreversible hypoxic-ischaemic brain injury. A complete explanation identifies that chain. Inspection, histology, examination of the airway and heart, and appropriately interpreted toxicology may establish it; an unexplained sudden death cannot be assigned a precise cause from its suddenness alone.

The expected broad contrast is a less mature, commonly thinner early juvenile continental crust, with more mafic material and abundant sodic tonalite-trondhjemite-granodiorite, rather than the more differentiated, potassium-rich granitoid component characteristic of much Phanerozoic continental crust. This is a secular tendency, not a single stratified column applicable throughout the Archean.

Archean terrains commonly expose tonalite-trondhjemite-granodiorite gneiss associated with greenstone belts containing basalt, komatiite and metasedimentary rocks. Many later granitoids are richer in potassium, including granite and granodiorite. The proportions changed during the Archean: a reconstruction from trace element ratios in sedimentary rocks infers a markedly mafic upper continental crust before about 3.0 Ga and a more felsic upper continental crust by 2.5 Ga. Thus “all Archean continental crust was mafic” would be wrong. A preserved tonalite-trondhjemite-granodiorite-dominated nucleus can be strongly felsic, including at depth; its composition is not identical to that of its original juvenile precursor.^[ref]

Thickness needs a stated comparison. An influential seismic compilation found preserved Archean crust about 27–40 km thick, commonly without the thick high-velocity basal layer of its 40–55 km thick Proterozoic comparison provinces. That supports a relatively thin endmember, but is not a measurement of every original Archean continental crust, nor a demonstration that every Archean region was thinner than today's 35–40 km reference column. Tectonic thickening, magmatic underplating, erosion, delamination and preservation bias all alter what survives. The very thick lithospheric mantle beneath a craton must also not be mistaken for thick continental crust.^[ref]

The transferable answer is therefore more mafic early juvenile precursors, widespread sodic tonalite-trondhjemite-granodiorite production, and variable thickness evolving through time; late Archean felsic nuclei can already resemble substantial modern continental crust.

Potassium has a closed 18-electron core and one valence electron. In $3d{}^2{D}_{5/2}$ the orbital label specifies principal quantum number $n=3$ and single-electron orbital angular momentum $l=2$. The term letter D specifies total orbital angular momentum $L=2$, the superscript $2S+1=2$ specifies $S=1/2$, and the subscript is total electronic angular momentum $J=5/2$. The $5p{}^2{P}_{3/2}$ state instead has $n=5$, $l=L=1$, the same spin, and $J=3/2$. The closed electron shell that forms the core contributes no net $L$ or $S$ in this alkali-like approximation. The dash identifies a pair of levels rather than by itself specifying the direction of emission.

The leading electric dipole transition rules for one active electron areThe electronic parity must change, since the electric dipole operator is odd under spatial reflection. The electric-dipole selection rule imposes no general restriction on changing $n$ by more than one. The rule for magnetic sublevels depends on the radiation's polarization relative to the quantization axis: the $q=0$ component couples equal $m_J$, whereas $q=\pm 1$ supplies the other changes. In a many-electron description using LS coupling the total orbital rule is $\Delta L=0,\pm 1$, excluding $L=0\leftrightarrow 0$; for a single active electron the stronger $\Delta l=\pm 1$ applies. Thus the displayed D–P pair satisfies the leading rules. Relativistic mixing and higher-multipole transitions can permit weaker lines outside the approximation of electric dipole transitions.

Choose the following original phenomenological mathematical model solely to calculate a paired-pulse readout. It is not experimental data, a fit to the teaching experiment, or a simulation of the Hodgkin-Huxley model. The conditioning action potential defines time zero. Test pulses have fixed width $0.2\mathrm{ms}$; $I$ denotes their inward electric current amplitude. Impose an absolute neuronal refractory period $t_{\mathrm{abs}}=2\mathrm{ms}$. No finite test amplitude generates an event for $0<\Delta <t_{\mathrm{abs}}$. After that hold, stipulate a required amplitudewith $I_{\infty }=0.2\mathrm{nA}$, dimensionless $A=2$ and recovery time constant ${\tau }_r=3\mathrm{ms}$. Emit a test event exactly when $I\ge I_{\mathrm{req}}(\Delta )$, including equality. For this ideal event readout, assign events to pulse onset and assume the conditioning event is successfully evoked. There is no calculated membrane potential, spike waveform, response latency or stochastic success probability.

The exponential decay represents a combined recovery deficit; it is neither the Hodgkin-Huxley model's $h$ nor a measured potassium electrical conductance. One such curve cannot separately identify sodium recovery and persistent potassium electrical conductance. A constant pulse-width amplitude criterion also does not establish an equivalent rule for other pulse widths or voltage trajectories.

At the hold boundary, $I_{\mathrm{req}}(2\mathrm{ms})=0.2(1+2)=0.6\mathrm{nA}$. At $\Delta =4\mathrm{ms}$,at $\Delta =6\mathrm{ms}$,Thus a $0.35\mathrm{nA}$ test fails at $1\mathrm{ms}$ because of the imposed hold, fails at $4\mathrm{ms}$ because recovery is insufficient for that amplitude, and succeeds at $6\mathrm{ms}$. At $4\mathrm{ms}$, a $0.5\mathrm{nA}$ pulse would instead succeed: failure of the weaker pulse was relative refractoriness.

For $I>I_{\infty }$, solve $I\ge I_{\mathrm{req}}(\Delta )$ using the logarithm. The earliest allowed delay isFor $I=0.35\mathrm{nA}$ this is $2+3\ln (8/3)\approx 4.942488\mathrm{ms}$; for $I=0.5\mathrm{nA}$ it is $2+3\ln (4/3)\approx 2.863046\mathrm{ms}$. If $I\ge 0.6\mathrm{nA}$, the imposed $2\mathrm{ms}$ hold is the only delay restriction. If $I\le I_{\infty }$, no finite delay passes the rule: the required amplitude stays strictly above its asymptotic baseline. A finite experimental “fully recovered” time therefore also needs a tolerance and a readout definition.

A potassium salt whose final net thermal decomposition gives potassium chloride and oxygen.