Arrow Lab

Química Geral · Liquids, solids and solutions

Sólidos e células unitárias

Uma lição resolvida gratuita e 40 problemas de prática em que você mesmo desenha cada seta curva.

Travou em sólidos e células unitárias? Escaneie o problema.

Tire uma foto da reação da sua lista ou do livro e desenhe o mecanismo com cada seta corrigida. 1 scan grátis agora, sem cadastro; mais 1 com uma conta gratuita.

Escaneie sua lição de graça

O que você vai aprender

  • Classify solids as ionic, molecular, covalent network or metallic, and as crystalline or amorphous, from their particles and properties.
  • Count the atoms or formula units in simple cubic, body-centred cubic and face-centred cubic unit cells and give their coordination numbers and packing efficiencies.
  • Relate edge length, atomic radius and density for cubic cells, working in either direction.
  • Describe the NaCl and CsCl lattices and use ionic radii to find their edge lengths.
  • Use the band picture to explain why metals conduct, why insulators do not, and how semiconductors respond to temperature and doping.

Lição gratuita

O texto da lição resolvida. No Arrow Lab, cada etapa vem com as estruturas e as setas desenhadas.

Ler a aula gratuita (7 partes)

Four kinds of crystalline solid

Classify a solid by the particles at its lattice points and the forces between them. Ionic solids (NaCl) are cations and anions held by electrostatic attraction: hard, brittle, high-melting, and conducting only when melted or dissolved, because only then can the ions move. Molecular solids (iodine, drawn; ice; dry ice) are molecules held by dispersion forces, dipole forces or hydrogen bonds: soft and low-melting, and they do not conduct. Covalent network solids (diamond, graphite, quartz SiO₂, silicon carbide) are one giant molecule of covalent bonds: very high-melting and insoluble, and very hard when the network extends in three dimensions (graphite, whose covalent sheets are stacked by weak forces, is soft). Metallic solids are cations in a sea of delocalized electrons: they conduct as solids, and because the bonding is not directional, layers can slide, so metals are malleable. Composition alone can mislead: ammonium chloride contains only nonmetals but is ionic, and SiO₂ is a network solid even though CO₂ is molecular. A crystalline solid has long-range order and melts sharply at one temperature; an amorphous solid such as glass has only short-range order and softens over a range of temperature.

Counting atoms in a cubic unit cell

A unit cell is the smallest repeating box that rebuilds the crystal by stacking. An atom on a corner is shared by 8 cells, on an edge by 4, on a face by 2, and one in the body belongs to the cell alone. Simple cubic (sc) has atoms only at the corners: 8 × 1/8 = 1 atom per cell, and each atom touches 6 neighbours (coordination number 6). Body-centred cubic (bcc) adds one atom at the centre: 8 × 1/8 + 1 = 2 atoms, coordination number 8. Face-centred cubic (fcc, also called cubic closest packed) adds an atom to each face: 8 × 1/8 + 6 × 1/2 = 4 atoms, coordination number 12. The same counting gives the formula of a compound: if A sits at the corners and B at the body centre, the cell holds 1 A and 1 B, so the formula is AB.

Where the atoms touch: radius and packing efficiency

Model atoms as hard spheres of radius r in a cube of edge a. In sc the atoms touch along an edge, so a = 2r. In bcc they touch along the body diagonal, √3·a = 4r. In fcc they touch along the face diagonal, √2·a = 4r, so a = 2√2·r. Packing efficiency is the fraction of the cell volume filled by spheres: (atoms per cell × 4/3 πr³) / a³. For fcc: 4 × 4/3 πr³ / (2√2·r)³ = 16.755r³ / 22.627r³ = 0.740, or 74.0 %. The same steps give 52.4 % for sc and 68.0 % for bcc. Simple cubic wastes almost half its volume, which is why α-polonium is the only element that adopts it under ordinary conditions.

Density from the unit cell

One unit cell has mass n·M/NA, where n is the number of atoms per cell, M the molar mass and NA = 6.022 × 10²³ mol⁻¹, and volume a³. So ρ = n·M / (NA·a³). Convert the edge to centimetres first: 1 pm = 10⁻¹⁰ cm. Example, aluminium (fcc, a = 404.9 pm, M = 26.98 g/mol): a = 4.049 × 10⁻⁸ cm, a³ = 6.638 × 10⁻²³ cm³, and ρ = 4 × 26.98 / (6.022 × 10²³ × 6.638 × 10⁻²³) = 107.92 / 39.97 = 2.70 g/cm³. The equation can be run backwards: from ρ and a, n = ρ·NA·a³/M reveals the cell type, and from ρ, a and n, M identifies the element. To get a radius from a density, find a³ = n·M/(ρ·NA), take the cube root, then apply the touching condition of that cell.

Ionic lattices: NaCl and CsCl

In rock salt, NaCl (drawn), the Cl⁻ ions form an fcc array and Na⁺ ions fill the holes between them: one at the body centre and one at each edge centre. The cell holds 8 × 1/8 + 6 × 1/2 = 4 Cl⁻ and 12 × 1/4 + 1 = 4 Na⁺, so 4 NaCl formula units. Each ion has 6 nearest neighbours of opposite charge (6:6 coordination), and cation and anion touch along the edge, so a = 2(r₊ + r₋). In CsCl the Cl⁻ ions sit at the corners of a cube and one Cs⁺ ion at its centre: 1 formula unit per cell and 8:8 coordination. The centre ion differs from the corner ions, so CsCl is not body-centred cubic; it is a simple cubic lattice with a Cs⁺–Cl⁻ pair at each lattice point. Here the ions touch along the body diagonal: √3·a = 2(r₊ + r₋).

Bands: metals, insulators and semiconductors

When very many atoms combine, their orbitals merge into bands of closely spaced levels. In a metal the highest occupied band is only partly filled (or overlaps an empty band), so electrons need almost no energy to move into empty levels and carry current. In an insulator a filled valence band is separated from an empty conduction band by a large band gap, about 5.5 eV in diamond, far more than the thermal energy available (kT ≈ 0.026 eV at 25 °C). A semiconductor has a small gap (silicon 1.12 eV, germanium 0.67 eV), so a few electrons are promoted thermally. Raising the temperature promotes more electrons, so a semiconductor conducts better when hot, while a metal conducts worse because stronger lattice vibrations scatter its electrons. Doping silicon with a group 15 atom such as phosphorus adds electrons just below the conduction band (n-type); a group 13 atom such as boron leaves holes in the valence band (p-type).

What to carry forward

Identify the particles and forces to classify a solid. For cubic cells, count shared atoms (corner 1/8, edge 1/4, face 1/2, body 1): sc 1, bcc 2, fcc 4 atoms with coordination numbers 6, 8 and 12 and packing efficiencies 52.4 %, 68.0 % and 74.0 %. Use the touching condition (a = 2r, √3·a = 4r, √2·a = 4r) with ρ = n·M/(NA·a³), always converting pm to cm. NaCl has 4 formula units and 6:6 coordination; CsCl has 1 and 8:8. Band gaps separate metals, semiconductors and insulators.

Problemas de prática (40)

Praticar é grátis com uma conta.

Preveja o produto 1

  • Rank solids by melting pointPredict the melting-point order of four solids from the forces that must be overcome to melt each.

Classificar 5

Ionic lattices and band theory

  • Classify a solid from its band gapClassify a solid as a metal, semiconductor or insulator from its band structure.
  • Coordination number in cesium chlorideGive the coordination number of the cation in the cesium chloride structure.

Types of solids

  • What kind of solid is iodine?Classify solid iodine and name the force that holds its particles together.
  • Which classification holds up?Judge classifications of solids that a misleading property or composition can make look wrong.
  • Pick the covalent network solidIdentify a covalent network solid among ionic, molecular and metallic examples.

Comparar 3

Types of solids

  • Glass versus quartz on heatingDistinguish an amorphous solid from a crystalline one by its melting behaviour and order.

Ionic lattices and band theory

  • Band gaps down group 14Rank the band gaps of diamond, silicon and germanium from the trend in bond strength down group 14.

Unit cells and packing

  • Rank cubic cells by packing efficiencyOrder the three cubic unit cells by the fraction of space their atoms fill.

Calcular 16

Unit cells and packing

  • Atoms in a body-centred cubic cellCount the net number of atoms in a body-centred cubic unit cell from corner and body-centre shares.
  • Packing efficiency of body-centred cubicDerive the packing efficiency of a body-centred cubic cell from its touching condition.
  • Density change from bcc to fccCompare the densities of bcc and fcc forms of the same element from their packing efficiencies.
  • Coordination number in face-centred cubicGive the number of nearest neighbours of each atom in a face-centred cubic metal.

Density, edge length and radius

  • Atomic radius of chromium from its densityCalculate the atomic radius of a body-centred cubic metal from its density and molar mass.
  • Density of copper from its unit cellCalculate the density of a face-centred cubic metal from its edge length and molar mass.
  • Identify a metal from its cell and densityCalculate the molar mass of a face-centred cubic metal from its density and edge length to identify it.
  • Edge length of iron from its atomic radiusCalculate the edge length of a body-centred cubic cell from the atomic radius.
  • Density of nickel from its atomic radiusCalculate the density of a face-centred cubic metal starting from its atomic radius.
  • Atomic radius of platinumCalculate the atomic radius of a face-centred cubic metal from its edge length.
  • Density of simple cubic poloniumCalculate the density of a simple cubic element from its edge length.
  • Unit cells in a gram of copperConvert a mass of a face-centred cubic metal into a number of unit cells.

Ionic lattices and band theory

  • Edge of the CsCl cell from ionic radiiEstimate the edge length of the cesium chloride cell from ionic radii using the body-diagonal contact.
  • Density of sodium chlorideCalculate the density of an ionic solid from its edge length and the number of formula units per cell.
  • Edge of the NaCl cell from ionic radiiEstimate the rock-salt cell edge from the cation and anion radii.
  • Formula units in the rock-salt cellCount the formula units in the NaCl unit cell from the positions of its ions.

Encontre o erro 4

Ionic lattices and band theory

  • Is cesium chloride body-centred cubic?Diagnose the error in calling the CsCl structure body-centred cubic and in its count of formula units.

Density, edge length and radius

  • Find the slip in a density calculationDiagnose which step of a unit-cell density calculation produced a result exactly one quarter of the true value.

Unit cells and packing

  • Find the error in an fcc edge lengthDiagnose the use of the wrong touching condition when finding the edge length of a face-centred cubic cell.

Types of solids

  • Is SiO₂ just like CO₂?Find the error in assuming that SiO₂ is a molecular solid because its formula resembles CO₂.

Interprete as evidências 2

Unit cells and packing

  • Which cubic cell fits the data?Identify a metal's cubic cell type by comparing its edge-to-radius ratio with the touching conditions.

Types of solids

  • Identify a solid from its propertiesUse hardness, melting point, solubility and conductivity data to identify an unknown solid as a covalent network solid.

Trabalhe de trás para frente 3

Unit cells and packing

  • Formula of an ordered copper–gold alloyDeduce the formula of an ordered alloy from which sites of an fcc cell each metal occupies.
  • Formula of perovskite from its cellWork back from atom positions in a cubic unit cell to the formula of a ternary compound.

Density, edge length and radius

  • Cell type of tungsten from its densityWork back from density, edge length and molar mass to the number of atoms per cubic cell and hence the cell type.

Planeje uma síntese 1

  • Choose a dopant for p-type siliconChoose a dopant that makes silicon a p-type semiconductor from its valence-electron count.

Explique por quê 5

Ionic lattices and band theory

  • Heating a metal and a semiconductorExplain with the band model why heating lowers the conductivity of a metal but raises that of a semiconductor.

Types of solids

  • Why graphite conducts and diamond does notExplain the conductivity of graphite and diamond from the bonding of carbon in each network.
  • Hammering a metal and a saltExplain why metals are malleable while ionic crystals shatter, from the nature of their bonding.
  • Why molten salt conductsExplain why an ionic solid conducts electricity only when melted or dissolved.

Unit cells and packing

  • Why so few metals are simple cubicExplain why metals rarely adopt a simple cubic structure in terms of coordination number and packing efficiency.

Travou em sólidos e células unitárias? Escaneie o problema.

Tire uma foto da reação da sua lista ou do livro e desenhe o mecanismo com cada seta corrigida. 1 scan grátis agora, sem cadastro; mais 1 com uma conta gratuita.

Escaneie sua lição de graça