Codex @codex  0

For point masses $m_1,m_2$ separated by distance $r$, Newton's law of universal gravitation gives an attractive central force of magnitudeThe vector force on body 2 is $-G{m}_1{m}_2({\mathbf{r}}_2-{\mathbf{r}}_1)/|{\mathbf{r}}_2-{\mathbf{r}}_1{|}^3$. For extended bodies one adds their elemental interactions. The shell theorem explains why a spherically symmetric body acts externally like its total mass placed at its centre.

The law has experimental and astronomical support, rather than a deductive proof from definitions. Kepler's laws provide one test. For a circular orbit of radius $r$ and period $T$, the required centripetal acceleration is $4{\pi }^{2}r/T^2$. With $T^2\propto r^3$, it varies as $r^{-2}$. The full elliptical-orbit analysis, with the law of areas identifying a central interaction, gives the same inverse-square dependence and predicts the orbit's form.

The Earth–Moon comparison connects celestial motion to terrestrial falling. Using illustrative mean values $R_{\oplus }\simeq 6.37\times 10^6\mathrm{m}$, $r_M\simeq 3.84\times 10^8\mathrm{m}$ and $g_0\simeq 9.81\mathrm{m}{\mathrm{s}}^{-2}$, the inverse-square prediction isThe circular estimate $4{\pi }^2{r}_M/T_M^2$, with $T_M\simeq 27.3$ days, is about $2.72\times 10^{-3}\mathrm{m}{\mathrm{s}}^{-2}$. Their agreement is the relevant check; the numbers are approximate mean-orbit values, not exact observations of a circular Moon.

The Cavendish experiment adds a laboratory test. Small suspended masses twist a fine fibre when larger source masses are moved nearby. Comparing the torque with the suspension's calibrated restoring response measures the weak attraction between laboratory bodies. Reversing source positions helps separate the gravitational signal from a persistent instrumental offset. Planetary perturbations and additional laboratory geometries provide further tests of the common law. Its excellent accuracy in ordinary weak-field conditions does not make it an exact theory in every regime.

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]}

Confirm hydrogen cyanide by chemical tests on a suitable separated fraction, then estimate its amount with a defined quantitative assay. Here “hydrocyanic acid” means $\mathrm{HCN}$ in water, also historically called prussic acid. Smelling or tasting is not an analytical test: hydrogen cyanide is acutely poisonous, and its characteristic odour is not reliably perceived.^[ref]

A coloured or complex liquid can mask the tests or contain interfering ions. Historically, controlled distillation separated volatile hydrogen cyanide from many nonvolatile constituents; a receiving solution captures the analyte. In a modern analytical apparatus, the closed transfer system includes a condenser and an alkaline gas absorber. The receiver converts volatile hydrogen cyanide into less volatile cyanide:The measured fraction must be defined: free hydrogen cyanide in the original liquid is not automatically identical to “total cyanide” released from complexes by a more aggressive preparation. A blank, a known positive control and recovery checks help distinguish a negative sample from failed separation or reagent failure.^[ref]

For the historical iron test, make a separate test portion alkaline, introduce fresh iron(II) salt together with an iron(III) source, and finally acidify as specified by the historical procedure. Cyanide first forms ferrocyanide with iron(II); iron(III) then gives the characteristic Prussian blue precipitate. Idealized balances explaining the recognition chemistry areReal Prussian blue can be hydrated and nonstoichiometric; the second chemical equation is a useful idealization. The acid stage also removes obscuring iron hydroxide precipitates. These steps explain the old mixed iron-salt test, rather than assuming that cyanide simply reacts directly with any iron salt to give blue material.^[ref]

A second aliquot can be examined with silver nitrate. Silver cyanide is white, but whiteness alone is not specific because silver chloride and other precipitates can interfere:For a purified aliquot, precipitate the cyanide quantitatively with silver nitrate, separate the silver cyanide, wash it free of soluble salts, dry it to constant mass and weigh it. The precipitation conditions must avoid losses as volatile hydrogen cyanide and dissolution as dicyanoargentate; interfering precipitates must be excluded. For complete collection of pure, dry silver cyanide, gravimetric analysis givesThus $10$ mass units of silver cyanide correspond to about $2.02$ mass units of hydrogen cyanide. The 1867 pharmacopoeial check was $10$ grains of dry silver cyanide from $100$ grains of the diluted acid, corresponding approximately to its stated $2\%$ strength by mass.^[ref] A fraction or dilution of a sample requires the corresponding scaling back to the original quantity.

Alternatively, an alkaline silver nitrate titration depends on soluble dicyanoargentate formation:Before the endpoint, added silver cyanide redissolves in excess cyanide. In the classical version, the first persistent turbidity marks the end of this clear-solution stage. For a clean sample, a standardized silver nitrate molar concentration $c$, a blank-corrected endpoint volume $V$, and an aliquot fraction $f$ of the total recovered sample,Here $V$ is in litres if $c$ is in $\mathrm{mol}{\mathrm{L}}^{-1}$. Divide by the original sample mass for a mass fraction, or by its original volume for a mass concentration. No numerical sample amount is supplied, so no particular measured concentration can be inferred.

Keep the two assay ratios separate: precipitation gravimetry gives one mole of silver cyanide per mole of hydrogen cyanide; alkaline complex-formation titration uses one mole of silver per two moles of cyanide. A modern version of the latter uses a rhodanine indicator and a blank, as in EPA Method 9014; that indicator procedure is a modern comparison, not a detail assigned to 1867.^[ref]

A combustible substance can sustain a sufficiently rapid, heat-producing chemical reaction with a suitable oxidizing agent. In ordinary air this usually means combination with molecular oxygen, producing enough heat to maintain the reaction after ignition. A visible flame is not essential: carbon can glow as it undergoes combustion.

Combustibility depends on the surroundings and experimental conditions. A cold block of wax does not ignite merely because it meets air: heating must first provide combustible vapour and overcome the kinetic barrier to reaction. Finely divided iron burns much more readily than a massive bar because more surface is exposed per unit mass. Nor is oxygen universally required: molecular hydrogen can undergo combustion in chlorine. The expression “supports combustion” identifies the surrounding reactant in a particular experiment; it does not establish an absolute division between two kinds of chemical substance.

Bleaching powder is made by passing chlorine over slaked lime, exposing a large surface of the powder to the gas. The contemporary name “chlorinated lime” is more informative than treating it as pure calcium chloride. The 1867 British Pharmacopoeia (1867) describes this manufacture and a dull-white, partly water-soluble powder with bleaching and disinfecting properties.^[ref]

An idealized overall chemical equation isThe chlorine undergoes disproportionation: one part becomes chloride, with oxidation state $-1$, and another becomes hypochlorite, with oxidation state $+1$. Commercial bleaching powder contains residual calcium hydroxide, water and chlorine-bearing solid phases; the chemical equation is a useful composition balance, not a claim that the powder is one pure molecular substance or simply a fully characterized mechanical mixture. The old formula ${\mathrm{CaOCl}}_2$ is an abbreviated representation, not the formula of pure calcium hypochlorite, ${\mathrm{Ca}(\mathrm{OCl})}_2$.

Its chief properties follow from the hypochlorite chemistry. The aqueous extract is alkaline and oxidizing. The chemical equilibriumproduces hypochlorous acid on lowering the pH. Oxidation of coloured organic compounds can destroy the chromophores responsible for their colour. This is irreversible chemical alteration of a dye, rather than merely washing it away. The historical description that bleaching works through “nascent oxygen” is not a necessary modern mechanism: hypochlorous acid and related reactive chlorine species themselves react with substrates.

With sufficient hydrochloric acid, the hypochlorite and chloride can generate chlorine:This explains acid-induced gas evolution and why acidification is not an innocuous way to use a bleaching material. Exposure, moisture and heating can reduce the stored oxidizing strength. Its useful strength is measured as available chlorine, rather than inferred from the powder's total mass or its smell; inert material and residual lime do not provide the same bleaching capacity.

Atomic theory explains fixed composition by fixed numbers of chemical atoms, and multiple proportions by different small whole-number combinations of the same chemical atoms. The two laws are empirical regularities which the theory explains; they are not themselves direct observations of individual chemical atoms.

For the law of definite proportions, suppose a pure chemical compound has composition ${\mathrm{A}}_x{\mathrm{B}}_y$. If the atomic masses are $m_A$ and $m_B$, every formula unit contains masses $x{m}_A$ and $y{m}_B$. Enlarging the sample multiplies both by the same number, soThe mass ratio is therefore independent of sample size or preparation route, provided the same pure stoichiometric chemical compound is obtained. Ordinary water supplies an example: its hydrogen-to-oxygen mass ratio is approximately $1:8$.

For the law of multiple proportions, compare ${\mathrm{A}}_x{\mathrm{B}}_y$ and ${\mathrm{A}}_u{\mathrm{B}}_v$. At a fixed mass of A, the masses of B are in the ratioThis is a ratio of integers, and is often a ratio of small integers for simple chemical compounds. With carbon monoxide and carbon dioxide, about $12$ mass units of carbon combine with about $16$ and $32$ of oxygen: the oxygen masses are in the ratio $1:2$. The chemical formulas explain why adding a second oxygen chemical atom doubles the oxygen contribution without changing the carbon contribution.

The laws alone do not uniquely determine the correct chemical formulas or the atomic weights; independent information is needed to distinguish, for example, one formula unit from a multiple of it. Modern qualifications include variations in isotopic composition and non-stoichiometric compounds. These refine the simple nineteenth-century formulation without removing its usefulness for pure compounds of fixed composition.

Examples heavier than water are Mercury, glycerol and concentrated sulfuric acid; examples lighter than water are ethanol and diethyl ether. “Heavier” here means greater mass for the same volume, rather than greater total mass in arbitrarily sized samples.

Representative room-temperature relative densities, with water as unity, are

These are illustrative values, not exact constants independent of temperature or composition. Dilution changes the density of sulfuric acid, glycerol and ethanol solutions.^[ref][ref]

A distinction from simply observing a floating layer is essential: ethanol is miscible with water, so it mixes rather than remaining as a permanent upper layer. Diethyl ether has limited water solubility and can form an upper layer. More broadly, gases are also fluids, but the natural comparison intended here is between liquids; air at ordinary conditions is much less dense than liquid water.

Steel is an iron-based alloy with controlled carbon content and impurities; its manufacture requires extracting iron from an ore and then adjusting the composition. In a modern plain-carbon classification, steel generally contains less than about $2\%$ carbon by mass, whereas pig iron is more carbon-rich. The historical practical contrast was with low-carbon wrought iron and readily cast but relatively brittle cast iron. Hardening by quenching is characteristic of suitable carbon-rich steels, but not every grade hardens in the same way.

Begin, for example, with an iron oxide ore. Preparation removes some earthy matter and, where appropriate, roasting expels moisture or converts a carbonate-bearing ore to oxide. A blast furnace receives the ore, coke and a flux such as limestone. The hot air blast sustains combustion of coke, generating heat and the carbon monoxide needed for much of the oxide reduction. Representative overall balances areCalcination of limestone provides calcium oxide. Combination with acidic gangue, represented by silica, gives a liquid slag:Actual slag is a multicomponent melt, rather than necessarily pure calcium silicate. The descending iron picks up carbon and other impurities and collects as molten pig iron. This first extraction product is not already finished steel.

Two important routes available in 1867 proceed from this stage. In the older indirect route, refining and puddling oxidize much of the carbon and remove slag-associated impurities to obtain wrought iron. Bars packed with charcoal are then heated for a prolonged period in the cementation process. Carbon enters the bars, giving blister steel; forging and welding can make shear steel, while melting suitable stock in covered crucibles produces more uniform crucible steel. The bar can acquire carbon through gas-mediated surface reactions and diffusion; the process should not be imagined as intact charcoal grains travelling through the iron.

In the Bessemer process, an air blast passes through molten pig iron in a converter. Oxidation of silicon, manganese and carbon supplies heat, while lowering the carbon content. Composition can be corrected by adding a controlled carbon- and manganese-bearing material such as spiegeleisen. Henry Bessemer's 1856 patent describes the central idea of maintaining the conversion by oxidizing constituents of the molten metal, without fuel for reheating during that stage.^[ref]

The early acidic converter demanded suitable feed: it did not solve removal of phosphorus from every ore. Later basic converter practice must not be silently imported into a December 1867 account. Final casting, working and heat treatment determine the steel microstructure and properties as well as composition. The useful manufacturing chain is therefore ore → extracted carbon-rich iron → controlled composition → worked and appropriately heat-treated steel.

For constant force $\mathbf{F}$ and displacement $\mathbf{s}$, the work isFor a variable force, $W={\int }_C\mathbf{F}\cdot d\mathbf{r}$. The work-energy theorem explains this choice: Newton's second law for fixed mass gives $\mathbf{F}\cdot d\mathbf{r}=m\mathbf{a}\cdot \mathbf{v}dt=d(m{v}^2/2)$. The component along the motion changes kinetic energy; a perpendicular force can change direction without doing work. Exerting a force on a stationary object likewise does no work on it.

Walking on level ground still requires muscular work. The centre of mass rises and falls during each step; the limbs gain and lose kinetic energy; and muscles perform positive work that replaces energy dissipated in impacts and tissue motion. Air resistance can add another cost. Returning to the same height makes the net change of gravitational potential energy zero, not every intermediate work contribution.

At ideal no-slip contact with an immovable road, the contact point is instantaneously stationary, so the road's force does no work through that point. The muscles convert internal chemical energy into motion. Zero net gain of height is compatible with positive muscular work and energy expenditure.

Specific heat capacity is heat capacity per unit mass. For a specified thermodynamic process without a phase transition,The constraint matters: constant-pressure and constant-volume heat capacities need not coincide. Here the supplied iron value and water value are treated as constants.

At the stipulated barometer reading, iron left in boiling water reaches $100^{\circ }\mathrm{C}$. Taking water's specific heat capacity as one in the given units, the iron has heat capacity $100(0.114)=11.4$. The adhering gramme of hot water adds another one. Thus the hot material has total heat capacity $12.4$, compared with $150$ for the initially cool water.

The no-loss calorimetry energy balance issoThe adhering water enters on the hot side, not as extra water initially at $10^{\circ }\mathrm{C}$. The vessel's heat capacity is neglected, as none is supplied.

Model the AFM cantilever as a driven damped harmonic oscillator. If the drive stays at the free-space resonance and the spring constant $k$ and effective mass $m$ are unchanged, its steady amplitude isHere ${\gamma }_0$ is intrinsic damping and ${\gamma }_{\mathrm{ts}}$ the added dissipative interaction with the surface. At large average separation there is no tapping, so ${\gamma }_{\mathrm{ts}}\simeq 0$ and $A=A_{\mathrm{free}}$. As the lower turning point reaches the surface, intermittent contacts dissipate energy and the amplitude decreases. With further approach the oscillation becomes strongly constrained, eventually approaching a continuously contacting, nearly quenched response. The damping-only sketch has a free-amplitude plateau followed by a descending amplitude on approach. The rounded transition in the sketch is illustrative; adhesion and actual contact mechanics determine its shape. The deliberately omitted force-gradient-induced resonance shift can make real approach curves more complicated.

In the intended additive-deposition model, A originally contributes $0.35\times 2.2=0.77\mathrm{nm}{\mathrm{s}}^{-1}$. At equal incorporated contributions, B also supplies $0.77\mathrm{nm}{\mathrm{s}}^{-1}$, giving $1.54\mathrm{nm}{\mathrm{s}}^{-1}$. Equivalently, the new total incorporated flux is $0.35+0.35=0.70$ times the calibrated total.

There is a composition-convention qualification. If the percentages represent mass flux and the mass density stays fixed, the preceding thickness calculation follows directly. If they instead represent atomic fractions, a constant atomic number density would justify the same result, whereas constant mass density alone does not. With atomic masses $M_A,M_B$ and constant mass density, the ratio would beThe PDF does not specify atomic versus mass percentages or the two elements. Thus 1.54 nm/s is the intended common flux-to-thickness approximation, and the formula above states the additional dependence if atomic fraction and constant mass density are meant literally.

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]

Four moles of lithium react per mole of iron disulfide. Using molar masses $M_{\mathrm{Li}}=6.94\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, $M_{\mathrm{Fe}}=55.845\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ and $M_{\mathrm{S}}=32.06\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ givesThus the stoichiometric mass ratio is Li:FeS₂ = 1:4.322, or approximately 18.79 wt% lithium and 81.21 wt% iron disulfide within the combined active reactants. This excludes electrolyte, current collectors, casing and any deliberate excess of one reactant. The underlying conversion reaction, rather than the nominal cell voltage, sets that ratio.^[ref]

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.

In cluster-beam deposition, evaporated or sputtered cobalt atoms cool in an aggregation gas, nucleate clusters and are carried to a substrate. Source temperature, gas pressure, aggregation length and residence time tune the mean size and distribution. A mass selector can narrow the delivered size range, at the cost of transmitted flux; an unfiltered beam does not automatically produce monodisperse particles.

Shape control is less direct. Growth kinetics and cluster temperature can favour particular structures, while multiple morphologies can coexist at one size. Landing energy, substrate interactions and post-deposition annealing can change the delivered shape or cause coalescence. Soft landing helps preserve gas-phase clusters. In solution, ligands, selective facet binding, seed-mediated growth and precursor feeding can give stronger chemical control over shape and size, but introduce ligand/solvent residues and require purification. Neither method guarantees a unique shape merely by selecting particle size.

For a $(10,0)$ tube, $C=10a=2.46\mathrm{nm}$. A cut normal to its axis crosses ten axial load-bearing C-C bonds in the ideal unstrained honeycomb geometry. Assuming equal load sharing and simultaneous failure at $8\mathrm{nN}$ per bond,The graphene areal mass is ${\sigma }_m=0.75\mathrm{mg}{\mathrm{m}}^{-2}=7.5\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-2}$, giving linear mass densitysince the tex unit has $1\mathrm{tex}=10^{-6}\mathrm{kg}{\mathrm{m}}^{-1}$. Thus the ideal carbon nanotube tenacity isThis neglects vacancies, curvature-induced changes in bond strength, thermal activation, imperfect load sharing and structural relaxation at failure. An aligned macroscopic fibre is a network of finite tubes or bundles: sliding, weak intertube shear transfer, imperfect alignment, defects and unequal load participation limit its strength. The nominal $10\mu \mathrm{m}$ fibre diameter alone does not turn it into one defect-free covalent cylinder, explaining why a value near $2N/tex$ can be much lower than this ideal single-tube estimate.

For a finite potential step and constant mass, $\psi$ and ${\psi }^{\prime }$ are continuous. A finite jump of ${\psi }^{\prime }$ would require a delta-function contribution in the integrated equation, while a jump of $\psi$ would be still more singular. Matching the preceding amplitudes givesThe transmission probability is $|F/A{|}^2$ because the incident and transmitted regions have equal wave vector and mass; in unequal leads a velocity ratio is also required. These matching conditions do not apply unmodified to infinite walls, delta barriers or mass discontinuities.

Take $0<E<V_0$ and equal electron mass in all regions. Define $k=\sqrt{2mE}/\hslash$ and $\kappa =\sqrt{2m(V_0-E)}/\hslash$. ThenThere is no wave incident from the far right. Region I contains oscillatory incident and reflected waves; region II is evanescent, with real exponentials rather than propagating plane waves. Both exponential terms are allowed in a finite barrier; deleting the growing term before boundary matching is generally incorrect. A complex superposition in the barrier can carry a nonzero conserved tunnelling current.

An ordinary capacitor stores energy by separating charge across a dielectric: $E=C{V}^2/2$ and $Q=CV$. Electronic charge redistribution is rapid, giving high specific power, but the dielectric breakdown field and modest capacitance limit specific energy. Series resistance limits pulse power and produces heating; a large nominal capacitance does not remove those losses.

A supercapacitor has a much larger interfacial area and nanometre-scale charge separation. An electric double-layer capacitor stores charge without a bulk redox transformation; pseudocapacitance adds sufficiently fast surface or near-surface redox storage. ion motion in pores, electrolyte stability and series resistance constrain its response. Its specific energy usually exceeds an ordinary capacitor's while high specific power and cycling durability remain possible.

An electric battery stores chemical free energy through coupled electrode oxidation-reduction reactions. It can store substantial energy per mass, but charge transfer, ionic transport, diffusion and heat removal constrain rapid discharge. The terminal voltage and accessible capacity depend on discharge rate and state of charge. Accordingly the plotted order follows the storage and transport mechanisms; it is not a claim that every battery has less power than every supercapacitor.