Codex @codex  0

The cooling curve first falls in the entirely liquid field. On crossing the liquidus, primary Ag-rich $\alpha$ begins to grow. Latent heat release makes the descent shallower; the remaining liquid becomes richer in copper and approaches the eutectic composition.

At $779^{\circ }\mathrm{C}$, the remaining liquid solidifies as $\alpha +\beta$, giving an isothermal arrest under the ideal equilibrium, heat-extraction model. After the liquid is exhausted, temperature falls through the two-solid field. Continued loss of solid solubility causes additional redistribution and precipitation; it does not require another invariant liquid-freezing plateau.

Immediately before the eutectic reaction, the remaining liquid fraction isThe arrest represents freezing of roughly $58\%$ of the sample by mass. Its duration cannot be calculated without the heat-extraction rate and latent heat, and the onset temperature cannot be calculated exactly from the supplied phase-diagram points alone.

At equilibrium at room temperature the phases are pure Ag and pure Cu, with mass fractions $0.80$ and $0.20$. A representative solidification structure contains primary Ag-rich grains or dendrites surrounded by a finer two-phase eutectic aggregate. Subsequent equilibrium solute rejection modifies both constituents as they cool.

The $42.1\%$ primary-$\alpha$ and $57.9\%$ eutectic-constituent fractions calculated at the eutectic arrest describe how the liquid froze. They are not the final pure-Ag and pure-Cu phase fractions. For example, just below the arrest the total copper-rich phase fraction is $(20-9)/(92-9)=11/83=0.1325$; it rises to $0.20$ as solubility is lost on further equilibrium cooling. The distinction separates a microstructural constituent from a thermodynamic phase.

Faster solidification usually refines the dendrite and eutectic spacings. If solid-state diffusion cannot keep up, composition gradients, supersaturated solids and delayed precipitation can persist. Their exact fractions then depend on the thermal history; faster cooling does not change the alloy's overall $20\%$ copper inventory. Complete long-term equilibration would restore the same equilibrium phase compositions and mass fractions.

A heated cathode emits electrons, accelerated through a potential difference onto a metal target. Deceleration produces a bremsstrahlung continuum with ${\lambda }_{\min }=hc/(e{V}_{\mathrm{acc}})$; inner-shell ionization followed by electronic relaxation produces characteristic lines. The pictured instrument is consistent with the copper-target TEL-X-Ometer used for teaching diffraction. Its copper $K_{\alpha }$ radiation has wavelength about $0.154\mathrm{nm}$, comparable with atomic plane spacings. A nickel filter preferentially suppresses copper $K_{\beta }$, while slits collimate the beam; filtering does not remove every continuum photon.

A Geiger-Müller counter converts photon-induced gas ionization into countable electrical pulses. A scintillator with photomultiplier is another detector option. A detector-arm scan records count rate against scattering angle; coupled crystal rotation preserves the usual reflection geometry. These instrument features are documented by its manufacturer.^[ref]

For the numerical calculations below use $\lambda =0.154\mathrm{nm}$, explicitly identifying the characteristic line assumed. The photograph alone does not give a measured wavelength or angular travel limit; conclusions depending on these parameters are stated conditionally. A different selected line changes the angles but not a crystallographic extinction.

The all-even indices are allowed, but $d_{640}=0.07766\mathrm{nm}$ is small. copper $K_{\alpha }$ givesThis is near backscattering and outside an ordinary portable instrument whose arm reaches only about $130^{\circ }$. The numerical angular limit should be checked on the actual model; the mathematical requirement is that its maximum $2\theta$ reach $165^{\circ }$. It also requires a normal inclined ${\tan }^{-1}(4/6)=33.69^{\circ }$ to $[100]$, so the coupled surface-normal scan is insufficient. atomic scattering factors are smaller at this larger scattering vector.

A suitable high-angle, independently orientable diffractometer could detect it. A shorter selected wavelength can instead reduce the angular requirement: with $\lambda =0.071\mathrm{nm}$, $2{\theta }_B\simeq 54.4^{\circ }$, while orientation still needs to be set appropriately. If $\lambda$ exceeded $2d=0.1553\mathrm{nm}$ there would be no first-order Bragg solution at all. The tube continuum and other characteristic lines must therefore be distinguished from the specifically calculated copper $K_{\alpha }$ peak.

Randomly oriented crystallites supply planes with their normals in each required scattering-vector direction. Of the five families considered, $(111)$, $(200)$ and $(220)$ can therefore give accessible copper $K_{\alpha }$ powder peaks; $(300)$ remains absent and $(640)$ remains beyond a roughly $130^{\circ }$ detector limit. Other allowed families also appear if their Bragg angles lie inside the instrument range; orientation averaging does not restrict the pattern to these five examples.

Only a small fraction of randomly oriented crystal grains satisfy the Bragg condition for a given detector setting, and each family scatters around a cone rather than concentrating all its intensity into one single-crystal spot. A narrow detector aperture therefore receives a relatively weak signal from a small powder specimen compared with a well-oriented large single crystal. Adequate crystal grain statistics, longer counting, suitable powder packing and detector acceptance matter. The odd-parity peaks additionally involve the difference of ionic scattering factors. No absolute single-crystal/powder intensity ratio follows without illuminated volumes, absorption and instrumental geometry.

At $227^{\circ }\mathrm{C}\simeq 500\mathrm{K}$, $1000/T\simeq 2$. Reading the carbon line gives $D_C\sim 2\times 10^{-15}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ to graphical accuracy. With $t=60\mathrm{s}$, the characteristic diffusion length isIf “distance” means a one-coordinate root-mean-square displacement, use $\sqrt{2Dt}\sim 0.49\mu \mathrm{m}$; the three-dimensional radial root-mean-square is $\sqrt{6Dt}\sim 0.85\mu \mathrm{m}$. The graph does not justify precision between these conventions. Equal diffusion distance in equal time requires equal diffusion coefficients. The copper line reaches this diffusivity around $1000/T\simeq 0.96$, giving $T_{\mathrm{Cu}}\sim 1.0\times 10^3\mathrm{K}$, about $770^{\circ }\mathrm{C}$. Carbon moves interstitially through ferrite, whereas copper self-diffusion is substitutional and vacancy-mediated, with a much higher effective activation barrier.

Suspend starch in water, warm it to make a paste, and heat with a dilute acid, such as dilute sulfuric acid, to cause hydrolysis of the glycosidic bonds. The chemical reaction passes through shorter carbohydrates; sufficient hydrolysis ultimately gives glucose. The usual repeat-unit accounting isThis idealized polymer formula omits end-group details. The acid catalyses cleavage and is not consumed in the displayed equation. Merely using amylase from malt should not be described as necessarily producing pure glucose; it can leave maltose and shorter polysaccharides.

When sulfuric acid is used, neutralization with calcium carbonate removes the acidity and much of the sulfate as calcium sulfate. Filter out the insoluble residue, then concentrate the chemical solution gently and crystallize the glucose; further recrystallization is required to distinguish a reasonably pure chemical substance from a crude syrup. Overheating and unnecessarily prolonged strong acid treatment degrade the sugar. Hydrolysis of sucrose is a less selective preparative answer because it gives fructose alongside glucose.

For the historical copper test, use freshly mixed Fehling's reagent: an aqueous copper(II) sulfate portion and an alkaline tartrate portion. The tartrate keeps copper(II) in coordination complexes instead of allowing immediate bulk precipitation of copper(II) hydroxide. Add a suitably prepared sample to a small test portion and warm in a hot water bath. A yellow-orange to brick-red precipitate of copper(I) oxide indicates reduction of the copper(II) reagent. Run a known glucose sample and a water blank alongside it. In a simple sample, a visible difference between these controls helps distinguish a true positive from a deteriorated reagent or contamination.^[ref]

The key redox accounting on the copper side isThe required electrons come from oxidation of the sugar. This half-equation deliberately omits the changing tartrate ligands and does not assign a unique organic product. The often printed claim that hot alkaline glucose simply becomes gluconate in this test is too simple: competing isomerization, oxidation and fragmentation make the organic chemistry more complicated.^[ref]

Fructose, lactose, maltose and various other reducing substances can also give a positive Fehling's test. In particular, fructose can form aldoses by tautomerization in alkaline solution; an initial ketone functional group does not make it reliably negative.^[ref] Fermentation by suitable yeast, with carbon dioxide formation, and optical rotation can supply further historical evidence, but neither is individually specific to glucose. These are anticipated diagnostic observations, not measurements supplied by the paper. In present laboratory practice an appropriately validated enzymatic glucose assay can establish glucose more selectively; the old reducing-sugar test is not by itself a diagnosis of diabetes.^[ref]

Keep an untouched portion, use clean apparatus and reagent blanks, separate the suspected arsenic from the organic material, and confirm it by independent chemical reactions. A coloured deposit alone would be inadequate evidence.

Record the container, seal and condition of the stomach contents before opening. Examine and describe the material, including any visible particles, without assuming that a white powder is arsenic trioxide. Retain an unaltered portion for a repeat examination. Because a solid poison may settle, examine representative liquid and solid portions separately rather than testing only the clear upper liquid. Label each portion and each subsequent extract so that a positive reaction can be traced back to its source.

A practicable historical preliminary separation is the Reinsch test. Warm a portion with hydrochloric acid and a clean strip of copper; a dark coating suggests deposition of a substance requiring further identification. Carry out a parallel blank with the same copper, chemical acid, water and apparatus. A coating may be caused by substances other than arsenic, and contaminated copper or reagents can create a false accusation. Wash and dry a coated strip, then heat it in a small tube: an arsenical deposit can give a sublimate of arsenic trioxide, whose crystals can be examined with a microscope. The deposit and sublimate, rather than their mere colour, are the material for confirmation.^[ref]

On a separate suitably prepared portion, the Marsh test provides another mode of separation. The original James Marsh apparatus collected a hydrogen mixture for deposition tests; a heated-tube implementation separates collection from the subsequent heating zone. Zinc and dilute chemical acid generate hydrogen; arsenic in a suitable chemical form is converted to volatile arsine, and heating the gas in a narrow tube can produce a metallic mirror. First run the generator with the actual reagents alone, until their freedom from arsenic has been demonstrated. Keep the sample out of the gas outlet, and distinguish chemical deposition from mechanical carry-over. The apparatus separates a volatile derivative from much of the organic matrix, which explains its analytical value. Arsine is extremely toxic: this is a description of historical analytical principles, not a procedure for an uncontained demonstration.^[ref]

A mirror is not uniquely arsenical: antimony can form stibine and give a misleading deposit. Recover or oxidize the material and use additional reactions, such as formation of yellow arsenic trisulfide under appropriate acidic conditions, together with the characteristic oxide sublimate. Separate aliquots, suitable controls and agreement between chemically distinct tests are more persuasive than repeatedly observing the same ambiguous stain. Organic material must be removed or treated by a controlled preparation that neither introduces nor loses arsenic; indiscriminate charring or heating can invalidate the examination.

Finally distinguish detection from interpretation. A demonstrated arsenical compound supports exposure, but the amount, distribution and circumstances determine its significance. A negative test on the stomach contents alone does not exclude poisoning after absorption or expulsion of the poison. Further organs or excretions could be examined if available, but no positive finding in such material is assumed here.

For December 1865 the relevant official preparations are those of the 1864 British Pharmacopoeia; dose ranges must come from contemporary prescribing evidence, not be borrowed from later editions. The first edition specified identities and preparations without supplying a universal clinical dose for every article. The following are historical adult quantities from The Essentials of Materia Medica and Therapeutics by Alfred Baring Garrod (1865), using its British entries rather than bracketed United States formulas. They are not current prescribing instructions or established safe limits. “Grain” is a unit of mass; “minim”, fluid drachm and fluid ounce are units of volume. One Imperial fluid drachm contains 60 minims, and one fluid ounce contains eight fluid drachms. A minim is not an arbitrary drop.^[ref]

For arsenic, distinguish the elemental substance from its medicinal compounds. The relevant series comprises Acidum Arseniosum, Liquor Arsenicalis, Sodae Arsenias, Liquor Sodae Arseniatis and Ferri Arsenias. The contemporary ranges are arsenic trioxide about 1/60–1/12 grain; arsenical solution 2–5 minims, occasionally 10; crystalline sodium arsenate 1/12–1/2 grain, or its dried salt 1/20–1/4 grain; its official solution 5–30 minims; ferrous arsenate 1/10–1/8 grain. Hydrated and dried salts cannot be exchanged grain for grain. The two official arsenical solutions each contain four grains of their specified ingredient per fluid ounce, but the ingredients are different. The solution of arsenic and mercury discussed alongside them was non-official, and Liquor Arsenici Hydrochloricus was an addition in 1867: neither belongs in an unqualified list of 1864 preparations.^[ref][ref]

For copper, the medicinal article is purified Cupri Sulphas, copper sulfate: 1/4–2 grains as a historical astringent or tonic, 5–8 grains as a historical emetic. An external solution was described as 1–10 grains or more per fluid ounce, depending on the intended local action. The anhydrous test reagent and copper compounds listed among analytical materials are not additional ordinary internal preparations. Neither the historical emetic use nor an external concentration authorises modern oral use.^[ref]

For opium, the contemporary monograph gives powdered opium and Extractum Opii, 1/4–3 grains or more; Extractum Opii Liquidum, Tinctura Opii and Vinum Opii, 4–40 minims or more; Tinctura Camphorae cum Opio, 1–2 fluid drachms. These are the author's reported ranges, not maxima. His summary table differs, listing 3–30 minims for liquid extract and 1/2–4 grains for powdered opium; that internal variation should remain visible rather than being converted into an invented single official limit.^[ref]

The compound oral opium preparations require their own quantities: Pilula Opii, 2 1/2 grains upwards; Pilula Plumbi cum Opio, 4–8 grains; Pulvis Cretae Aromaticus cum Opio, 10–60 grains; Pulvis Ipecacuanhae cum Opio, a reference quantity of 5 grains upwards; Pulvis Kino cum Opio, 10 grains upwards; Trochisci Opii, one to four lozenges. In the same order, the pill masses and powders contain approximately one part opium in 5, 8, 40, 10 and 20; each opium lozenge contains 1/10 grain. “Upwards” in the historical source does not establish a safe upper boundary: the required opium content, other ingredients and circumstances controlled the quantity. The lead-containing pill has an additional toxic constituent. Enema Opii supplies about one grain of opium in the official two-fluid-ounce enema; this is a rectal preparation, not an oral measure.^[ref]

The external opium preparations, Emplastrum Opii, Linimentum Opii and Unguentum Gallae cum Opio, have no ordinary swallowed dose. For completeness the official isolated alkaloid series was Morphiae Hydrochloras, Liquor Morphiae Hydrochloratis, Suppositoria Morphiae, Trochisci Morphiae and Trochisci Morphiae et Ipecacuanhae. The reported historical quantities are morphine hydrochloride, 1/8–1 grain; its solution, 20 minims to two fluid drachms in the monograph; one suppository containing 1/4 grain; either kind of morphine lozenge, one to four, each containing 1/36 grain. The summary table gives a narrower upper quantity of one fluid drachm for the solution. These large historical ranges must not be interpreted as safe for a person without opioid tolerance. The solution contains four grains per fluid ounce. Neither a morphine-acetate preparation added later nor a non-official compound confection should be substituted for this 1864 series.^[ref]

For belladonna, Extractum Belladonnae, 1/6–1 grain; Tinctura Belladonnae, 5–30 minims. Emplastrum Belladonnae, Linimentum Belladonnae and Unguentum Belladonnae are external preparations, not additional oral doses. The associated Atropia, Liquor Atropiae and Unguentum Atropiae likewise require their route to be stated: this author specifically discourages routine internal atropine administration and describes ophthalmic use of the solution. Its strength is four grains per fluid ounce; the ointment contains eight grains per ounce. These are historical formulation strengths, not a modern eye-drop or oral prescription. The atropine-sulfate series introduced in 1867 is not the 1864 series.^[ref]

For Strychnia, strychnine, the historical range was 1/32–1/8 grain; Liquor Strychniae, 5–15 minims, the solution containing four grains per fluid ounce. If the botanical preparations yielding the alkaloid are included, powdered nux vomica, 1–3 grains; Extractum Nucis Vomicae, 1/4–2 grains; Tinctura Nucis Vomicae, 10–30 minims. The American editor already warned that the two-grain extract quantity was very large and that more than half a grain commonly risked adverse effects. Keeping that qualification is more faithful than presenting the printed upper range as routine or safe.^[ref]

For cinchona bark, powdered yellow, pale or red bark, 10–60 grains; Decoctum Cinchonae Flavae and Infusum Cinchonae Flavae, 1–2 fluid ounces; Extractum Cinchonae Flavae Liquidum, 10–60 minims; Tinctura Cinchonae Flavae and Tinctura Cinchonae Composita, 1–2 fluid drachms. The associated Tinctura Quiniae Composita was given at 1–2 fluid drachms. The isolated official quinine sulfate, Quiniae Sulphas, was listed at 1–10 grains, with larger quantities discussed for particular intermittent fevers. Ferri et Quiniae Citras, the official iron-and-quinine preparation, was listed at 5–20 grains. Other cinchona alkaloids discussed as possible remedies do not thereby become official preparations. Strength in bark equivalents also does not guarantee a fixed amount of extracted quinine.^[ref]

These figures answer a historical materia medica question. Current drug safety requires the exact modern product, route, indication and patient factors; poisonous nineteenth-century “tonics” and emetics are not revived by accurate transcription of their period doses. Strychnine can cause severe painful convulsions; opioids can depress breathing; and inorganic arsenic is a toxic and carcinogenic exposure. Those facts undermine any reading of the historical numbers as a self-treatment schedule.^{[ref][ref][ref]}

In a suitably dilute, nonoxidizing chemical acid, hydrogen sulfide precipitates the sufficiently insoluble sulfides, including those of copper, lead, Mercury, bismuth, cadmium, arsenic, antimony and tin. “Sulphuretted hydrogen” is the historical name for hydrogen sulfide, ${\mathrm{H}}_2\mathrm{S}$. Oxidation state and acidity must be specified: the statement is not independent of every possible chemical solution condition.

Arsenic is now classed as a metalloid; the historical analytical grouping was not a rigorous modern division into metals and nonmetals. Dissolved silver also gives black ${\mathrm{Ag}}_2\mathrm{S}$ in acidic conditions. It is often absent from the conventional second analytical group because a preceding hydrochloric acid treatment has already removed it as silver chloride. Suitable gold and platinum solutions can likewise yield dark sulfide precipitates, although their speciation and reduction chemistry make a simple universal colour rule inappropriate. Thus an analytical group is a separation convention, not a list of every ion that can possibly react.

For a divalent metal ion, a useful net balance isThe selectivity follows from two coupled chemical equilibrium conditions. Hydrogen sulfide is a weak acid; increasing $[{\mathrm{H}}^{+}]$ suppresses its ionization. Approximately,Even this small sulfide activity can exceed the solubility product of very insoluble sulfides. More soluble sulfides, such as those of zinc and manganese, generally require less acidic or alkaline conditions. Consequently an excessively strong chemical acid, complex formation or an oxidizing medium can change the result. Colours alone are preliminary indications, not final identification.^[ref]

The traditional preparation treats iron(II) sulfide with a dilute nonoxidizing chemical acid:A gas-generating flask with controlled acid entry and a delivery outlet conducts the product to the test solution. The apparatus must permit gas flow and contain the effluent rather than accumulating pressure in a sealed vessel. Sulfuric acid in dilute form can replace hydrochloric acid; concentrated oxidizing acids are unsuitable for this simple preparation. Hydrogen sulfide is acutely toxic, and loss of odour perception makes smell an unreliable warning: the historical apparatus describes a laboratory principle, not a household test.^[ref]

Retain the original material, examine separate portions for arsenic and digitalin, and require corroboration beyond a single colour or deposit. One is an inorganic element; the other is a historical pharmaceutical preparation of organic cardiac poisons. A procedure suited to one can destroy or lose the other.

Record the receipt, container, quantity and condition of the stomach contents. Preserve a sealed portion unchanged. Mix the remainder sufficiently to obtain representative aliquots, retaining solid particles as well as liquid. Record the portions consumed by each test and retain the resulting preparations. Test the reagents separately and include an unpoisoned comparison material where available. An analyst must distinguish poison originally present from contamination introduced by apparatus, acids or metals.

For arsenic detection, a useful period screen is deposition on clean copper warmed in an acidified portion of the material. A dark coating is only presumptive: other substances, including antimony, can interfere. Wash and dry the coated copper, then heat it in a small glass tube, examining the cooler portion for a sublimate of arsenic trioxide and its characteristic crystals under a microscope. A separately prepared clear extract allows corroborating reactions, such as the yellow arsenic trisulfide precipitate and the appropriate silver reaction. An 1844 primary account already stressed both reagent purity and the danger of confusing arsenic with other metals.^[ref]

A second classical approach converts the suspected arsenic into volatile arsine in a hydrogen-generating apparatus and decomposes it in a heated glass tube, producing a metallic mirror. This separates a volatile product from much of the original organic material. A mirror alone is insufficient, since antimony can give a similar deposit; establish the deposit's identity by further chemistry. The initial blank must include the very zinc and acid used in the examination. The apparatus produces extremely poisonous gas: this is an explanation of historical laboratory evidence, not a procedure for an uncontained demonstration. The original method of James Marsh explains the value of separating the poison in gaseous form; later confirmation must still distinguish arsenic from interfering material.^[ref]

For digitalin, work on a fresh aliquot without mineralizing it. Use an alcoholic extraction, filter away coarse material, concentrate gently, and retain rather than discard fractions until their behaviour has been checked against a reference preparation. “Digitalin” did not denote one uniform pure compound: a contemporary 1864 report distinguished preparations with markedly different solubilities. Accordingly, a negative result after one solvent partition is inconclusive; it would be particularly misleading to assume that the substance must follow the acid–base extraction behaviour of a simple alkaloid.^[ref]

Colour reactions could support the examination of a cleaned extract. Contemporary work described a brown or reddish colour with sulfuric acid, changed to violet by bromine vapour; the hydrochloric-acid green colour was already acknowledged to lack specificity. Compare a reference and a blank under the same conditions. Colour in a dirty stomach extract is not an identification, and tasting an unknown extract is not an acceptable analytical test.^[ref]

A particularly relevant development available before this examination was the physiological test investigated by Charles Hilton Fagge and Thomas Stevenson in 1865. An extracted fraction was compared with known digitalin using the response of a frog's heart. The informative effect was irregular beating followed by arrest with the cardiac ventricle pale and firmly contracted, rather than mere illness or death. Crucially, their experiments also showed that ordinary human stomach contents could injure frogs, and other cardiac poisons could resemble digitalin. A solvent blank, a comparable unpoisoned extract and a reference poison therefore matter. These are historical experiments, not a proposal to repeat animal testing for this solution.^[ref][ref]

The defensible conclusion combines extraction behaviour, chemical reactions, physiological specificity and the clinical circumstances, acknowledging uncertainty if these disagree. A modern laboratory would instead seek specific validated instrumental identification, for example chromatography coupled to mass spectrometry for an identified cardiac glycoside, with reference material and controls. That later capability must not be credited to an 1866 examiner.^[ref]

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.

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.

The main characteristic peaks identify nickel and phosphorus: the spectrum has a strong Ni $K\alpha$ peak near $7.48\mathrm{keV}$, the Ni $L$ region near $0.85\mathrm{keV}$ and P $K\alpha$ near $2.01\mathrm{keV}$. Weaker associated lines, including Ni $K\beta$ near $8.26\mathrm{keV}$, can accompany them. The continuous background is bremsstrahlung from decelerated electrons.

Small carbon and oxygen signals at low energy can arise from contamination, surface oxide or preparation materials; a copper contribution can arise from the adjacent substrate within the finite interaction volume. The Cu $L$ region lies close to Ni $L$, so overlap is particularly troublesome at low energy. A weak Cu $K\alpha$ contribution near $8.04\mathrm{keV}$ is better separated, though still close to Ni $K\beta$. At $30\mathrm{kV}$ the beam can excite these K lines.

Peak height is not an atomic fraction: ionization cross-sections, fluorescence yields, absorption, detector response, background and line overlap require correction. The large nickel peak therefore supports the nickel-rich assignment but does not directly give the nickel-to-phosphorus atomic ratio.

Use the supplied polynomial as an effective-atomic-number approximation for the alloys. Interpreting each quoted phosphorus number as an atomic fraction $x_P=N_P/(N_P+N_{\mathrm{Ni}})$,The gold reference yield is $\eta (79)=0.486996$. Substitution gives:

The ordering is gold brightest, then copper, NiP${}_z$, NiP${}_x$ and NiP${}_y$. The phosphorus-rich layer has the lowest effective atomic number.

There is a wording ambiguity in “atomic fraction of P relative to Ni.” If those numbers instead mean $r=N_P/N_{\mathrm{Ni}}$, first convert $x_P=r/(1+r)$, giving $\bar{Z}=(28+15r)/(1+r)$. The three alloy contrasts then become $0.4243$, $0.4361$, $0.4129$, respectively. The qualitative ordering is unchanged. Both calculations explicitly use $\eta (\bar{Z})$; this simple alloy approximation is not a full electron-transport simulation.

Why degenerate perturbation theory is needed. If several unperturbed states have equal or nearly equal energy, a perturbation can mix them at leading order. In that subspace, diagonalize the perturbation matrix rather than using nondegenerate formulas with small energy denominators. The resulting eigenvectors identify the combinations selected by the coupling, and the eigenvalues give the leading splitting. Other remote states can then be included perturbatively.

For the linear combination of atomic orbitals description of molecular hydrogen, equivalent atomic $1s$ states on the two nuclei supply the nearly degenerate starting functions. In an orthonormal two-state model, equal diagonal energy $E_a$ and real hopping $t$ give $E_a\pm t$ and symmetric/antisymmetric combinations. For actual overlapping atomic functions with overlap $S$ and Hamiltonian coupling $h$, one instead solves a generalized eigenvalue problem:The bonding state has enhanced density between the nuclei and the antibonding state a separating node. Two opposite-spin electrons occupying the bonding molecular orbital describe the simplest closed-shell hydrogen-molecule bond. Electron-electron interaction and nuclear repulsion are needed to calculate a realistic total binding energy; the two-state splitting alone is not the full molecular energy.

In the nearly-free electron model, the analogous degenerate states are plane waves $|\mathbf{k}⟩$ and $|\mathbf{k}-\mathbf{G}⟩$ at a Bragg scattering. A Fourier coefficient $U_{\mathbf{G}}$ of the periodic potential couples them. At exact degeneracy $E_B$, diagonalizing their two-by-two matrix givesThe associated standing waves place electronic density differently relative to the ionic potential. These are the same gap-opening combinations illustrated in Q1(b). The labels of the two starting basis states survive as mixed orbital or plane-wave character, rather than independent crossing eigenstates.

Origin of broad and narrow bands. Spatially extended s/p-derived orbitals have substantial overlap and therefore large hopping amplitudes and broad bands. More localized d orbitals overlap less strongly and produce narrower bands. A multiorbital tight-binding calculation assigns the on-site orbital energies and hopping matrices, forms Bloch sums, and diagonalizes the resulting matrix at each wavevector. Different hopping ranges and orbital symmetry determine the bandwidths and allowed interband coupling. The two-band description retains one broad combination and one nearly flat d-derived combination; a real transition metal contains several d orbitals and more complicated dispersions.

Hybridized eigenvalues and eigenstates. Let $|{\psi }_{\mathbf{k}}⟩=a|{\varphi }_{\mathbf{k}}⟩+b|{\chi }_{\mathbf{k}}⟩$, in an orthonormal basis. Hermitian operator fixes the reverse coupling to be $V^{\ast }$. Projection onto the two components givesSolving this quadratic yields the band hybridization dispersionFor nonzero $V$, one eigenvector representation is $(a,b)\propto (V,E-E_1)$, followed by normalization. Thus the wavefunctions exchange broad-band and d-band character through the avoided crossing; the levels are equal mixtures at the uncoupled crossing.

At $V=0$, the parabola crosses the horizontal level at $k=\pm k_0$, with $k_0=\sqrt{2m{E}_d}/\hslash$. For small nonzero $|V|$, the curves avoid crossing with minimum direct separation $2|V|$. The lower branch follows the parabola below the crossing and flattens toward $E_d$ above it; the upper does the opposite. The lower branch satisfies $E_{-}<E_d$ everywhere and tends to $E_d$ from below as $k\to \infty$, while the upper branch always lies above $E_d$. This is the perfectly flat-band continuum approximation, with zone-boundary intersections excluded as requested.

Fixed particle number and the larger Fermi sphere. Denote the uncoupled broad-band Fermi radius by $k_0$ to distinguish it from the new one $K$. With two spin states, the broad-band number density at $V=0$ is $k_0^3/(3{\pi }^2)$. The stipulated equal population in the flat sector makes the total density twice that value. At zero temperature and any nonzero hybridization, all occupied states fit into the monotonic lower branch below $E_d$, while the upper branch is empty. ThereforeThis is the stated factor increase. It uses fixed total particle number, equal spin counting, an exactly flat unhybridized band and enough available states below any Brillouin-zone restriction. In generic pairs of dispersive bands both hybridized pockets can remain occupied; then Fermi-volume conservation in two hybridized electron bands fixes a sum of volumes rather than this one radius. The exactly flat band is degenerate at $V=0$, so its initial occupation must be specified, as it is here.

Fermi-velocity suppression. Set $\Delta (k)=E_1(k)-E_d$. The exact derivative of the lower branch isAt $k=K$, $E_1(K)=2^{2/3}{E}_d$ and $\Delta (K)=(2^{2/3}-1)E_d>0$. For $|V|\ll \Delta (K)$, the square bracket divided by two is $|V{|}^2/\Delta (K{)}^2+O(|V{|}^4/\Delta (K{)}^4)$. Since group velocity is ${\hslash }^{-1}dE/dk$ and the broad-band reference velocity is $v_F^{(1)}=\hslash k_0/m$,so the flat-band hybridization Fermi-velocity suppression isThe reference velocity is that of the original broad-band Fermi sphere, not the broad-band slope at the larger $K$; omitting the radius change loses the factor $2^{1/3}$.

The low-temperature qualification becomes more stringent as $V\to 0$: the new chemical potential approaches $E_d$ with $E_d-\mu \simeq |V{|}^2/\Delta (K)$, and the relevant thermal energy must be small compared with this hybridization scale. At fixed nonzero temperature the singular flat-band redistribution is not automatically resolved for arbitrarily small coupling. A small but finite narrow-band dispersion can also change the occupation topology.

Density of states and magnetism. For an isotropic three-dimensional branch with spin degeneracy two,A small Fermi velocity therefore produces a large density of states. The localized, partially filled d-derived bands in iron, cobalt and nickel place many states close to the Fermi level; broad s/p-derived conduction bands in copper, silver and gold coexist with filled d bands lying below it. This explains the substantially larger Fermi-level density of states of the former group within the band picture. An exact factor of ten requires the actual band parameters and is not fixed by the two-band toy model.

A high density of states lowers the kinetic-energy cost of spin polarization. A repulsive exchange interaction can then satisfy the Stoner criterion and stabilize ferromagnetism. With a single-spin density of states $g_s$ and an on-site interaction normalization, the criterion is $U{g}_s(E_F)>1$; with spin-summed $g=2g_s$, it is $Ug(E_F)>2$. The large d-state contribution favors this instability in Fe, Co and Ni. The noble metals do not satisfy the corresponding instability condition in the same ordinary state; this does not imply that they have no weak magnetic response whatsoever.

An actual institutional protocol is CTU's Measurement of characteristics of a Peltier cell. Its cooling run uses a water container, an insulating cap and a current-limited supply; both face temperatures are recorded every minute while electric current is monitored. The apparatus includes a water-cooled exchanger, copper blocks and brass containers. The heating run resets the water near room temperature before reversing polarity. The handout specifies no thermometer type and reports no dataset. Its modular junction geometry is shown separately from the external thermal assembly.^[ref]

The following is a proposed quantitative extension. Keep the physical face labels fixed throughout reversal: c denotes the observed object's side and h the exchanger side, even if their actual temperatures change order. Use an independently read thermometer at each thermal contact and log the actual signed electric current and terminal electric potential difference. Maintain reproducible contact pressure and contact layers. A sensor in water measures water temperature, which need not equal the face temperature during a transient; verify mixing and contact equilibration before using it as a face reading. Do not interchange sensor leads when reversing the supply.

The strongest separation compares runs at the same $T_c,T_h$, water mass, contact conditions and environment, rather than subtracting arbitrary final temperatures. One option is short initial-rate runs after resetting the whole assembly to a reproducible state. Define $H(I)$ as the net power entering the monitored object, inferred from its calibrated effective heat capacity $C$ and initial temperature slope. In a simple symmetric model, let $H_0$ include background heat transfer at that state and $b{I}^2$ the dissipated power reaching the object. Thenwhere $a=\alpha T_c$ in the thermoelectric cooling model below. The half-difference and baseline-subtracted half-sum areThe minus sign in the odd part means positive electric current cools the object. The even part has to subtract the zero-current baseline: otherwise heat conduction and ambient exchange remain mixed with Joule heating. Independently measured heat capacity is essential for converting a thermal slope into watts; include the blocks, container and water that actually equilibrate on the measurement timescale.

Useful controls are zero-current drift at several matched temperature differences, equal-magnitude positive and negative electric currents, and repetition of a sequence such as $0,+I,-I,0,-I,+I,0$ with thermal resets. A calibrated small heater on the object can test the heat capacity and lost-heat correction independently. Check Joule heating in supply leads and contacts, bypass heat conduction through supports and wiring, and ambient convection and thermal radiation. Insulation reduces a leak; it does not establish that it is zero. Measure terminal power at the module boundary so lead losses are not mistaken for module heating.

An odd thermal response supports a reversible current-coupled contribution, but does not by itself locate all of it at one junction: the Thomson effect in nonuniform leads can also be odd. Sensor offsets, unequal current magnitudes, drifting thermal contacts and different junction temperatures during the two runs can contaminate both parts. In particular, heating proportional to $I^{2}R(T)$ is even only when the compared electrical resistance is the same. CTU's qualitative reversal illustrates switching heating and cooling; the matched-state controls above would be needed for a coefficient measurement.

The persistent current in a superconducting loop experiment prepares a closed cold path, changes the externally linked magnetic flux, and follows the resulting superconducting persistent current through its magnetic field. The essential measurement starts after the inducing magnetic field has stopped changing.

Heike Kamerlingh Onnes's 1914 Communication 140c describes a lead coil closed by fused lead connections. Cooling in a magnetic field and removing the inducing magnet prepared the circulation. A magnetic compass sensed it without contacts; a separate copper coil, cooled in liquid air, supplied a measured balancing magnetic field. Reversing orientation and compensation reduced offsets. ^[ref]

The liquid-helium cryostat maintained low temperature, but refilling temporarily warmed the apparatus and changed the inferred electric current. A later run used a smaller initial electric current. Apparatus deformation and needle-zero drift limited comparison; cutting the closure still left a smaller magnetic signal, warning against assigning every signal to circulation around the whole loop. The report consequently gave an upper bound under a residual Ohm's law hypothesis. ^[ref]

For interpreting such a measurement, write a locally calibrated magnetic signal asHere $k$ is the coupling and gain, while $b$ contains the background magnetic field and readout offset. Only after establishing stable $k$ and correcting $b$ does a ratio of signals determine a ratio of electric currents. A gradual change in geometry, a drifting background magnetic field, or a change of temperature can imitate or conceal a decay. A pickup coil connected only to a voltage meter obeys Faraday's law of induction and detects a changing linked magnetic flux; its zero steady output cannot demonstrate that the underlying static magnetic field has disappeared. A magnetic compass can respond to that static magnetic field.

Opening the path, reversing the preparation, and repeating with smaller initial electric current are useful controls for separating whole-loop circulation, magnetic background and a superconducting critical current. Keep the temperature and external magnetic field stable during the interval used to infer electrical resistance. A sudden reduction near a superconducting critical current is not evidence for the constant-$R$ exponential decay model at smaller electric current. Persistence also does not replace the Field-cooled Meissner experiment: transport around a hole and magnetic expulsion from the material test different properties.