Codex @codex  0

An atomic basis of a crystal is the motif of chemical species $s_j$ and relative positions $r_j$ attached identically to every lattice point $R$ of a Bravais lattice $\Lambda$. The ideal periodic structure is the labeled setThe primitive lattice translation vectors generate $R$ by integer combinations; the atomic basis lists what is repeated at $R$. These are different meanings of the word basis. In particular, the offsets $r_j$ need not be linearly independent vectors, and one of them may be zero. A primitive unit cell represents $N$ motif sites, even though it represents only one lattice point.

Changing the origin, choosing another primitive unit cell, or moving a motif site by a Bravais lattice translation changes the coordinates used to describe the same infinite crystal structure. The grouping into a motif does not, by itself, identify bonded molecules. In solid sodium chloride, grouping one sodium cation and one chlorine anion into an atomic basis does not create a collection of discrete NaCl molecules; the ionic crystal has an extended network of equivalent neighboring ions.

A Born-Haber cycle uses Hess's law to infer a lattice energy from measurable enthalpy differences and isolated-species data. The motivating difficulty is that separating a mole of an ionic crystal into mutually separated gaseous ions is not a convenient direct calorimetry experiment. Instead, compare two accounting routes between the same elemental reference states and the same solid. Because enthalpy is a state function, the signed differences close exactly when all inputs describe compatible states. These routes do not assert a reaction mechanism or a temporal sequence of reaction intermediates.^[ref]

For one mole of sodium chloride formula units, the overall formation reaction isThe stoichiometry requires half a mole of diatomic chlorine molecules, supplying one mole of chlorine chemical atoms. A half coefficient counts an amount of substance; it does not describe half of an individual molecule. Write each auxiliary chemical reaction with its own sign:Here $S$ is the sublimation enthalpy, $B$ the enthalpy of atomization of chlorine per mole of chemical atoms, and $I$ the molar ionization enthalpy represented by the ionization energy input in elementary treatments. If the bond-dissociation energy table gives the compatible molar enthalpy for ${\mathrm{Cl}}_2(g)\to 2\mathrm{Cl}(g)$, use half that number for $B$. Both ions in the last step are gaseous, and no aqueous hydration of ions belongs in that step.

Adding these equations cancels the gaseous chemical atoms, the gaseous ions and the free electron, leaving precisely the overall formation reaction. ThusThe closed traversal returns from the solid to the elemental reference states, adding $-F$; therefore $S+B+I+A+L_{\mathrm{form}}-F=0$. Notice that the signed attachment term $A$ is negative. Substituting a positive spectroscopic electron affinity for $A$ would reverse the attachment contribution.

The diagram deliberately uses invented, internally consistent inputs in $\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: $S=100$, $B=120$, $I=500$, $A=-350$ and $F=-400$. Preparation of separated gaseous ions raises the enthalpy by $370$, so $L_{\mathrm{form}}=-770$ and $L_{\mathrm{diss}}=+770$. The closure check is $100+120+500-350-770+400=0$. These numbers are not measured sodium chloride data and do not replace the source's $769\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ example in lattice energy. Every level retains one sodium nucleus, one chlorine nucleus and zero total electric charge; the ionization level explicitly retains the free electron.

A tabulated electronegativity is useful because it compresses many comparisons, but one number per chemical element suppresses chemical context. Changing the bonded partners, bond length or electron configuration changes the conditions under which a chemical atom attracts electron density. The bond-dissociation energy can change for reasons beyond bond polarization, so attributing every extra unit of energy solely to an electronegativity difference is an approximation.^[ref]

An oxidation state labels a formal allocation of electrons; it is not a measured partial atomic charge. State-sensitive electronegativity models may give different values for different oxidation states or electron configurations, while a simple chemical element table intentionally provides a common comparison convention. Likewise, replacing an isolated chemical atom by a site in an ionic crystal introduces a surrounding electric field and many neighbours. Applying the same tabulated electronegativity without adjustment becomes an additional assumption.^[ref]

For the ground-state Mulliken scale, the state dependence is visible directly in the formula. Replacing $X$ by $X^{+}$ replaces $E(X^{+})-E(X)$ with $E(X^{2+})-E(X^{+})$ as the removal cost, and replaces $E(X)-E(X^{-})$ with $E(X^{+})-E(X)$ as the attachment release. These are different ionization energies and electron affinities. This isolated-ion arithmetic does not claim that a bonded chemical atom with the same formal oxidation state is an isolated ion. In a molecule, the actual electron density and dipole evidence for bond polarization supply complementary information about that environment.

How can a clear chemical solution reveal otherwise invisible mobile ions? The documented Conductivity of Solutions by Harvard Natural Sciences Lecture Demonstrations demonstration uses metal electrodes enclosed in tubing, an electrical supply, and a lamp in the circuit. The protocol specifies about $150\mathrm{mL}$ samples in labeled $250\mathrm{mL}$ beakers and a separate deionized-water rinse station. The recorded readout is lamp brightness after the electrodes enter each sample; the simplified variant tests water before and after adding sodium chloride. These are apparatus and procedure specifications, not new measurements reported here.^[ref]

The useful comparison is water, a sucrose chemical solution, and a sodium chloride chemical solution with the same probe geometry. A documented university account reports no visible light with water, a glowing lamp with dissolved sodium chloride, and a weak filament glow with its sucrose sample. The diagram summarizes the latter as little or no visible light; the precise detection threshold depends on the tester. Sucrose is a control for the idea that any dissolved material must conduct: its dispersed neutral molecules do not provide the abundant mobile ions supplied by the salt. This contrast is a documented teaching observation, not a numerical calibration of lamp brightness.^[ref][ref]

Rinsing the electrodes between samples controls carryover of dissolved ions, and the water blank checks the solvent background. Repeating the blank after the salt sample would test whether that rinse was effective. Additional methodological controls are fixed immersed area, electrode separation, supply setting, temperature, and comparison at stated concentrations. These additional controls are proposed checks, not undocumented claims about the source demonstration.

The inference is that mobile electric charge carriers complete the circuit. In the chemical solution, an electric field drives positive ions along the field and negative ions against it; multiplying each drift velocity by its electric charge makes both contributions to conventional current point the same way. With an alternating supply the direction reverses each half-cycle. The external wires instead carry electric current through electrons. The lamp alone identifies neither the species nor an exact electrical conductivity.

Important limits remain. Water has a small intrinsic population of hydronium and hydroxide, so a dark lamp does not prove absence of ions. Changes in probe geometry affect measured electrical conductance; electrolysis, electrode interfaces, contamination, and heating can also affect a real circuit. The comparison supports a transport model but does not independently measure a coordination number, a partitioned partial atomic charge, or a lattice energy.

Nor does poor conduction by ordinary dry sodium chloride establish that all ionic crystals are electrical insulators. Crystallographic defects can allow transport, and solid electrolytes can have substantial ionic electrical conductivity without melting. The authored abstract of A lithium superionic conductor by Noriaki Kamaya and others reports room-temperature ionic conduction in a solid electrolyte; the full experimental article was not accessible here.^[ref]

In this structural use, the coordination number counts nearest neighbouring constituents. State which neighbours are counted and which ionic crystal structure is being described; it is not an ion's electric charge number.

A crystallographic defect is a departure from ideal repeated structure. In an ionic crystal, a missing ion can provide an available site for another ion to move into; movement still requires overcoming an activation energy. This is why an ideal immobile-site model is a limit rather than a universal law for solid electrical conductivity.^[ref]

Dissolution transfers material into a chemical solution. It can include separation from an ionic crystal and solvation, or dispersal of neutral molecules. For sucrose and sodium chloride, respectively,Here $(aq)$ specifies water as the solvent; it does not specify a fixed number of attached water molecules. The neutral sucrose remains molecular. Dissociation of an ionic compound separates pre-existing ions in the second example; solvation stabilizes their new surroundings.^[ref][ref]

Ionization instead creates ions from neutral species. Hydrogen chloride reacting with water illustrates protonation of water rather than mere rearrangement of a hydration shell:The hydronium and negative ion together preserve total electric charge. Solvation can influence this chemical reaction, but an ion's solvation does not itself change its net electric charge.^[ref]

Electron affinity measures how strongly an isolated neutral species binds an additional electron. In the spectroscopic convention used here, it is the positive energy required to detach that electron from a bound anion. With the free electron at rest assigned zero energy,The quantum states of $X$ and $X^{-}$ must be specified, usually their ground states. Positive $A_{\mathrm{ea}}$ means that attachment lowers the energy; a signed attachment energy is then negative. This convention agrees with the IUPAC definition and with the atomic tables used for Mulliken electronegativity.^[ref]

For chlorine, a published photodetachment threshold is $A_{\mathrm{ea}}/(hc)=29138.59(22)\mathrm{c}{\mathrm{m}}^{-1}$, approximately $3.612725\mathrm{eV}$ per chemical atom. Here $h$ is the Planck constant, $c$ the speed of light, and the inverse-length unit expresses an energy through a spectroscopic wavenumber. The parentheses give the reported uncertainty in the last digits, not a range of possible sign conventions.^[ref]

An electron affinity is neither an electronegativity nor a lattice energy. The first describes a specified isolated-species energy difference, electronegativity compares attraction for electron density in a chemical bond, and lattice energy refers to an entire ionic crystal and separated gaseous ions. A stable anion can therefore help explain ionic bonding without describing the full energy balance for making a salt.

For two point electric charges in vacuum, Coulomb's law gives the interaction potential energywith zero at infinite separation. Here $r_{ij}$ is the distance between them and ${ϵ}_0$ is the vacuum permittivity. Opposite signs give negative potential energy and attraction; equal signs give positive potential energy and repulsion. For a finite collection of distinct point electric charges, sum over $i<j$ to count each pair once. A bulk ionic crystal requires the extended lattice sum, including both signs, rather than multiplying one attractive pair by the coordination number.^[ref]

At short distances, overlapping electron density makes the simple point-charge picture inadequate. Quantum mechanics, including the Pauli exclusion principle, matters to the steep repulsive contribution. Pure attraction proportional to $-1/r$ would collapse as $r\to 0$; it cannot set an equilibrium spacing on its own. A useful illustrative model isHere $r$ scales an assumed fixed arrangement, $A$ packages its attractive contribution, and $B/r^n$ represents short-range repulsion. The constants have units energy times length and energy times length to the power $n$, respectively. This is a model, not a fitted description of a named salt.^[ref]

Differentiating gives $r_0=(nB/A{)}^{1/(n-1)}$ and $U^{\prime \prime }(r_0)=(n-1)A/r_0^3>0$, so the stationary spacing is a stable minimum. For the diagram's stipulated $n=9$, put $x=r/r_0$ and $E_0=A(1-1/n)/r_0>0$. ThenThe dimensionless curves show how competing interactions select a spacing. Their energy scale and exponent are synthetic; neither is a measured lattice energy or a melting prediction.

An electron trapped at an anion vacancy in an ionic crystal.

Ionic bonding binds oppositely charged ions through attraction governed by electrostatics. In ordinary solid sodium chloride, each cation interacts with many anions, and also with other cations: attraction and repulsion together contribute to the binding of an extended ionic crystal. The chemical formula $\mathrm{NaCl}$ specifies equal numbers of the two kinds of ions; it does not divide the solid into isolated two-atom molecules.^[ref]

This distinction explains a striking combination of properties: ordinary dry sodium chloride conducts electric current poorly, while its molten form or a sufficiently concentrated chemical solution in water conducts well. The ions already exist in the solid; their ability to move through the sample changes. A conductivity experiment on salt solutions tests this transport picture, rather than simply asserting that charged constituents must make every material conductive.

An ionic crystal repeats an arrangement of ions in three dimensions. A unit cell describes that repetition, whereas a formula unit describes composition. The two accounting devices answer different questions. Coulomb's law is not restricted to an arbitrarily drawn line connecting one cation to one anion: every ion contributes to the electric field experienced by the others.

An ion pair is an associated cation and anion. An isolated ion pair lacks the extended neighbourhood of an ionic crystal. In a chemical solution, ion pairs can coexist with more widely separated ions; their abundance depends on conditions. A neutral gas-phase sodium chloride molecule can be treated as a discrete entity, with its own quantum mechanics and dissociation products. That use of molecule does not describe the constituents of ordinary solid sodium chloride.^[ref][ref]

The Madelung constant, named for Erwin Madelung, is a dimensionless geometrical factor in the electrostatic binding of an ionic crystal. For a neutral lattice of electric charges $\pm e$ with nearest unlike-ion distance $r_0$, Coulomb's law gives the potential energy per ion pair as $-{\alpha }_M{e}^2/(4\pi {ϵ}_0{r}_0)$. At a positive reference site, ${\alpha }_M=-{\sum }_{j\ne 0}{z}_j{r}_0/r_j$, with neighboring electric charges $z_{j}e$. Both attraction and repulsion enter this lattice sum; summing with a neutral-bulk prescription is essential. Short-range overlap repulsion and other contributions must be added before inferring a stable spacing or a thermodynamic lattice energy.

A mixture contains two or more chemical substances combined in proportions that need not be fixed. A chemical compound instead has a definite composition: water has two hydrogen chemical atoms per oxygen chemical atom, whereas a sucrose and water mixture can contain different sugar proportions. A homogeneous mixture has uniform composition on the scale of representative samples; a heterogeneous mixture has distinguishable regions of different composition. A chemical solution is homogeneous, whereas sand dispersed in water is a particle suspension.^[ref]

The distinction concerns the sample, not merely its ingredient list. A vessel with dissolved sodium chloride and excess undissolved sodium chloride contains a chemical solution and an ionic crystal thermodynamic phase. The liquid portion can be homogeneous while the whole vessel is heterogeneous. Similarly, molecular oxygen dissolved in water is part of a chemical solution; visible gas bubbles introduce a separate gas thermodynamic phase.

A molecule, using the International Union of Pure and Applied Chemistry (IUPAC) convention, is an electrically neutral entity containing two or more chemical atoms. Its constituents form a bound unit through chemical bonds; merely drawing a boundary around neighboring particles does not make a molecule. The neutrality requirement concerns total electric charge, not the absence of internal charge separation: water is a polar molecule with zero total electric charge.^[ref]

One water molecule, ${\mathrm{H}}_2\mathrm{O}$, contains two hydrogen and one oxygen chemical atom. One molecular oxygen molecule, ${\mathrm{O}}_2$, contains two oxygen chemical atoms; a molecule need not contain different chemical elements. One carbon dioxide molecule, $\mathrm{C}{\mathrm{O}}_2$, contains one carbon and two oxygen chemical atoms. The molecular formula records these inventories, while molecular geometry describes their arrangement. Water and carbon dioxide are chemical compounds; molecular oxygen is a form of one chemical element.

Ordinary crystalline sodium chloride, $\mathrm{NaCl}$, contains an extended arrangement of sodium cations $\mathrm{N}{\mathrm{a}}^{+}$ and chlorine anions $\mathrm{C}{\mathrm{l}}^{-}$. Its formula unit specifies one of each for compositional accounting, without selecting a unique isolated pair. Bulk neutrality therefore does not imply a material consists of neutral molecules. This statement concerns the ionic crystal; the physical state and conditions matter when deciding what particles a chemical formula represents.^[ref]

A polyatomic ion is a charged bound group of chemical atoms, such as hydronium, ${\mathrm{H}}_3{\mathrm{O}}^{+}$. A molecular ion formed by removing an electron from intact water, ${\mathrm{H}}_2{\mathrm{O}}^{\bullet +}$, retains the same atom inventory but has net electric charge $+e$. Neither satisfies the neutral-molecule definition. Distinguishing these entities makes molecular count and amount of substance and mass spectrometry interpretable: say whether the count is of chemical atoms, neutral molecules, ions, or formula units.

The useful surprise is that a molecule can be treated as one particle for motion and counting while retaining internal structure. Its molar mass connects a particle inventory to bulk mass; molecular rotation and molecular vibration reveal internal organization. The continuum approximation uses bulk fields in place of individual molecular positions and motions.

The incident X-ray drives the electrons of an atom. Away from an absorption edge, and when the photon energy greatly exceeds the relevant electronic binding energies but is small compared with $m_e{c}^2$, each electron has approximately the same Thomson scattering amplitude. The much heavier nucleus makes a negligible contribution because its acceleration is much smaller.

For elastic scattering, the amplitudes from different bound electrons add coherently. In units of the single-electron amplitude, the atomic scattering factor is the Fourier transform of the bound-electron number density:Consequently, when the electron cloud is not well resolved by the scattering vector, its internal phase differences are small andThis is an approximation to the coherent scattering amplitude, not an assertion that independently scattered intensities add as $N_e$. At larger momentum transfer, destructive interference within the cloud reduces $f$; near absorption edges, the atomic scattering factor acquires dispersive corrections. Inelastic scattering is a separate contribution. In an ionic crystal, use the ion's bound-electron count rather than automatically using the neutral atom's atomic number.

A covalent bond is stabilized by shared electronic states extending across neighboring atomic nuclei. In a simple two-centre description, combining comparable atomic orbitals produces a bonding state with enhanced electron density between the nuclei and an antibonding state with a node there. For equivalent normalized orbitals ${\varphi }_A,{\varphi }_B$ with real overlap $S$,Their energy ordering follows from the molecular Hamiltonian; constructive interference and reduced gradients help explain why the bonding combination can be lower. Opposite-spin electrons can occupy the bonding orbital together. The familiar molecular-hydrogen bond is a basic example, while extended covalent networks form solids such as diamond. Orbital shape and hybridization make covalent bonding strongly directional and constrain bond angles. Occupying antibonding states reduces the net bond strength.

Ionic bonding instead emphasizes electric charge transfer and the electrostatic attraction of the resulting oppositely charged ions. Removing an electron from one atom costs ionization energy and adding it to another gains electron affinity; these isolated-atom terms alone need not favor transfer. In a crystal, the electrostatic lattice energy can stabilize the ionic arrangement. For a schematic lattice of singly charged ions at nearest-neighbor distance $r$,where the geometrical Madelung constant $A_M$ sums attractions and repulsions throughout the lattice. Short-range overlap repulsion prevents collapse and gives an equilibrium separation. Sodium chloride is an example of an extended ionic crystal, rather than isolated NaCl molecules in the solid.

Ideal ionic bonding is comparatively nondirectional, so packing, ion sizes and electric charge balance strongly influence coordination. ions cannot freely migrate in the ordinary solid; melting or dissolution can make ionic electric charge transport possible. Covalent materials do not obey one universal electrical conductivity rule: their electronic bands and filling determine whether mobile electronic carriers are available.

The two descriptions are limiting models, not a sharp classification of every bond. Unequal electronegativities can produce polar covalent bonds with partial electric charge transfer, and a material may contain both covalent groups and ionic interactions between those groups. Chemical bonding reflects a balance of electronic kinetic energy, electron-nucleus attraction, electron-electron repulsion and nuclear repulsion; it cannot be reduced to a force between two assigned point charges in every material.

Many ionic crystals have high melting temperatures compared with small molecular solids, because reorganizing their strongly interacting ions costs energy. But melting leaves a dense liquid with interactions; it does not perform the complete separation defining a lattice energy. At fixed pressure, the transition occurs when the Gibbs free energies of the solid and liquid match. The relevant enthalpy and entropy differences both matter, so lattice energy alone is not a universal ordering of melting temperatures. Strong covalent bonds in extended networks can also give high melting temperatures.^[ref]

A polyatomic ion consists of a bound group of two or more chemical atoms with nonzero total electric charge. Hydronium, ${\mathrm{H}}_3{\mathrm{O}}^{+}$, contains three hydrogen and one oxygen chemical atom; its net electric charge is $+e$, where $e$ is the elementary charge. This is a discrete charged group, while an ionic crystal can contain an extended arrangement of ions.^[ref]