Crystalline Solids and Lattice Energy
Crystalline Solids
Solids are rigid substances with definite shape and volume whose constituent atoms, ions, or molecules are closely packed and held together by strong cohesive forces. Based on the regularity of their internal arrangement, solids are classified into crystalline solids and amorphous solids.
Crystalline Solids: Solids in which atoms, ions, or molecules are arranged in a definite, repeating three-dimensional geometric pattern that extends throughout the entire crystal.
Amorphous Solids: Solids whose constituent particles lack a regular orderly arrangement. Glass, plastics, rubber, and glue are examples. They may contain small ordered regions called crystallites but lack long-range regularity.
Formation of Amorphous Solids: Many crystalline solids can be converted to amorphous solids by melting and then rapidly cooling the molten mass, so the constituent particles do not find time to arrange themselves.
Crystalline solids display a set of distinctive properties that arise directly from their ordered internal arrangement. These properties are useful for identifying and classifying crystalline materials.
Geometrical Shape: All crystalline solids have a definite, distinctive shape due to the orderly three-dimensional arrangement of particles. The interfacial angles at which crystal surfaces intersect are always the same, regardless of the crystal's overall size. Even when ground to powder, the characteristic angles persist.
Sharp Melting Points: Crystalline solids melt at a specific, well-defined temperature. Unlike amorphous solids, which soften over a temperature range and lack definite heats of fusion, crystalline solids transition sharply from solid to liquid.
Cleavage Planes: When crystalline solids are broken, they fracture along definite planes called cleavage planes. These planes are inclined to one another at characteristic angles that vary from one solid to another.
[Anisotropy]: Some crystalline solids show variation in physical properties depending on the direction of measurement. Properties like refractive index, thermal expansion, and electrical and thermal conductivities can differ along different crystal axes. For example, graphite conducts electricity far better parallel to its layers than perpendicular to them.
Symmetry: The repetition of faces, angles, or edges when a crystal is rotated through 360° along its axis. Symmetry elements include the center of symmetry, plane of symmetry, and axis of symmetry.
Crystal Habit: The shape in which a crystal usually grows. It depends on growth conditions — a cubic NaCl crystal becomes needle-like when 10% urea is present as an impurity.
Allotropy — Elements in Multiple Crystalline Forms
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Sulphur (S) — rhombic and monoclinic forms
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Carbon (C) — cubic (diamond) and hexagonal (graphite)
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Tin (Sn) — grey tin (cubic) and white tin (tetragonal)
Transition Temperature Examples
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Grey tin (cubic) ⇌ White tin (tetragonal)
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Sulphur S₈ (rhombic) ⇌ Sulphur S₈ (monoclinic)
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KNO₃ (orthorhombic) ⇌ KNO₃ (rhombohedral)
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Na₂SO₄·10H₂O (hydrated) ⇌ Na₂SO₄ (anhydrous) + 10 H₂O
Crystal Lattice and Unit Cell
A crystal lattice (or space lattice) is the three-dimensional array of points in a crystal, where each point represents the position of an atom, ion, or molecule. These positions are called lattice points or lattice sites. Every lattice point in a crystal has an identical environment.
The unit cell is the smallest repeating block of the crystal lattice that contains all the characteristic structural features of the entire crystal. If you know the exact arrangement of particles in a unit cell, you know their arrangement throughout the whole crystal. The unit cell is specified by six parameters called crystallographic elements: three edge lengths and three inter-axial angles.
$$a,\; b,\; c \quad \alpha,\; \beta,\; \gamma$$
The six crystallographic elements that define the size and shape of a unit cell
$a,\; b,\; c$=Lengths of the three edges along the x, y, and z axes(pm or Å)
$\alpha$=Angle between edges $b$ and $c$(degrees (°))
$\beta$=Angle between edges $a$ and $c$(degrees (°))
$\gamma$=Angle between edges $a$ and $b$(degrees (°))
Corner Sharing: A particle at a corner is shared by 8 adjacent unit cells, contributing $\frac{1}{8}$ per unit cell.
Edge Sharing: A particle on an edge is shared by 4 unit cells, contributing $\frac{1}{4}$ per unit cell.
Face Sharing: A particle on a face is shared by 2 unit cells, contributing $\frac{1}{2}$ per unit cell.
Body Centre: A particle at the body centre belongs entirely to one unit cell, contributing 1.
Crystals are classified into seven crystal systems based on the dimensions and angles of their unit cells. Each system has a unique combination of edge lengths ($a$, $b$, $c$) and inter-axial angles ($\alpha$, $\beta$, $\gamma$).
The Seven Crystal Systems
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Cubic — $a=b=c$, $\alpha=\beta=\gamma=90°$ — Fe, Cu, Ag, Au, NaCl, NaBr, Diamond
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Tetragonal — $a=b \neq c$, $\alpha=\beta=\gamma=90°$ — SnO₂, MnO₂
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Orthorhombic — $a \neq b \neq c$, $\alpha=\beta=\gamma=90°$ — Rhombic sulphur, BaSO₄, K₂SO₄
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Monoclinic — $a \neq b \neq c$, $\alpha=\gamma=90°$, $\beta \neq 90°$ — Borax, Na₂SO₄·10H₂O
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Hexagonal — $a=b \neq c$, $\alpha=\beta=90°$, $\gamma=120°$ — Graphite, ZnO, CdS, Ice, Zn, Cd
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Rhombohedral (Trigonal) — $a=b=c$, $\alpha=\beta=\gamma \neq 90°$ (<120°) — Bi, Al₂O₃, NaNO₃, KNO₃
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Triclinic — $a \neq b \neq c$, $\alpha \neq \beta \neq \gamma \neq 90°$ — H₃BO₃, CuSO₄·5H₂O
Ionic Crystals
Ionic solids are crystalline solids in which positively charged cations and negatively charged anions are held together by strong electrostatic forces of attraction called ionic bonds. Examples include NaCl, KBr, MgO, and CsF. Ionic crystals do not exist as individual neutral molecules — the entire crystal is one continuous lattice of alternating cations and anions.
Hardness and Stability: Very high energy is required to separate cations from anions against the electrostatic forces, making ionic crystals very hard and stable.
High Melting and Boiling Points: The strong non-directional ionic bonds result in high melting and boiling points and low volatility.
Non-Directional Bonding: Since ionic bonds are non-directional, ions pack closely in a systematic alternating arrangement that minimises space.
Formula Mass, Not Molecular Mass: Because ionic solids do not exist as discrete molecules, we refer to their formula mass rather than molecular mass.
The structure and geometry of an ionic solid depend on the radius ratio of the cation to the anion. This ratio determines how ions can pack together and fixes the coordination number — the number of nearest neighbours of opposite charge surrounding each ion.
$$\text{Radius ratio} = \frac{r^+}{r^-}$$
The ratio of the cation radius to the anion radius, which determines the geometry and coordination number of the ionic crystal
$r^+$=Radius of the cation(pm or Å)
$r^-$=Radius of the anion(pm or Å)
$0.414 < \frac{r^+}{r^-} < 0.732$
→Octahedral geometry with coordination number 6 (e.g., NaCl)
$\frac{r^+}{r^-} \geq 0.732$
→Cubic geometry with coordination number 8 (e.g., CsCl)
Shape Determination: The geometry of an ionic crystal is independent of the chemical nature of the atoms and depends only on the number of ions and their radius ratio. NaCl and CsF have the same geometry because their radius ratios are the same.
Coordination Number in NaCl: Each Na⁺ ion is surrounded by six Cl⁻ ions at the corners of a regular octahedron, so the coordination number is 6. Each Cl⁻ is similarly surrounded by six Na⁺ ions.
Ionic Distance in NaCl: The distance between two nearest Cl⁻ ions is 5.63 Å, so the distance between adjacent Na⁺ and Cl⁻ ions is $\frac{5.63}{2} = 2.815$ Å.
Sodium chloride (NaCl) has a face-centred cubic structure. The unit cell contains exactly 4 NaCl formula units. Each Na⁺ is bonded equally to all six surrounding Cl⁻ ions — there are no discrete NaCl molecules in the solid.
Chloride Ion Count: 8 Cl⁻ at corners (each shared by 8 cells: $\frac{1}{8}$ each) plus 6 Cl⁻ at face centres (each shared by 2 cells: $\frac{1}{2}$ each): $\frac{8}{8} + \frac{6}{2} = 1 + 3 = 4$ Cl⁻ per unit cell.
Sodium Ion Count: 12 Na⁺ at edge centres (each shared by 4 cells: $\frac{1}{4}$ each) plus 1 Na⁺ at the body centre (fully owned): $\frac{12}{4} + 1 = 3 + 1 = 4$ Na⁺ per unit cell.
No Discrete Molecules: All six Cl⁻ ions around each Na⁺ are at the same distance. NaCl exists as independent molecules only in the vapour phase; in the solid it has formula units of NaCl.
Ionic solids have several characteristic physical properties that follow directly from their structure and bonding.
Electrical Conductivity: Ionic crystals do not conduct electricity in the solid state because ions remain tightly held in fixed positions. They conduct only when molten or dissolved, where ions become free to move.
Brittleness: Ionic crystals are highly brittle. They consist of parallel layers of alternating cations and anions. When external force causes one layer to slide, like charges align face-to-face and repel, causing the crystal to fracture.
Solubility: Ionic solids dissolve in polar solvents (e.g., water) where they dissociate into ions and undergo fast ionic reactions.
High Density: Due to close packing of ions, ionic solids generally have high density.
Molecular Crystals
Molecular solids are crystalline solids whose constituent particles are individual polar or non-polar molecules (or atoms, as in solidified noble gases). The particles are held together by weak intermolecular forces — dipole-dipole interactions, van der Waals (London dispersion) forces, or hydrogen bonding — which are much weaker than ionic, covalent, or metallic bonds.
Polar Molecular Solids: Crystals of polar molecules such as ice (H₂O) and sugar. They generally have higher melting and boiling points than non-polar molecular solids.
Non-Polar Molecular Solids: Crystals of non-polar molecules such as iodine (I₂), sulphur (S₈), phosphorus, and solid CO₂ (dry ice). They have the lowest melting and boiling points among all crystalline solid types.
Ionic vs Molecular Crystals — Key Differences
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Particles: Ions (ionic) vs molecules or atoms (molecular)
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Binding forces: Strong electrostatic (ionic) vs weak intermolecular (molecular)
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Hardness: Hard and rigid (ionic) vs soft and compressible (molecular)
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Melting point: High (ionic) vs low (molecular)
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Volatility: Low (ionic) vs high (molecular)
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Electrical conductivity: Conducts when molten or dissolved (ionic) vs non-conductor (molecular)
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Density: High (ionic) vs low (molecular)
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Examples: NaCl, MgO, KBr (ionic) vs I₂, CO₂, ice, sugar (molecular)
The weak intermolecular forces in molecular solids give them a distinct set of physical properties that contrast sharply with ionic and covalent crystals.
Softness and Compressibility: Molecular crystals are soft and easily compressible because the intermolecular forces holding molecules together are very weak.
Low Melting and Boiling Points: Minimal energy is needed to overcome the weak intermolecular attractions, so molecular solids are volatile and melt or boil at low temperatures.
Electrical Insulation: No free electrons or mobile ions are present, making molecular solids poor conductors of electricity.
Solubility: Polar molecular crystals dissolve in polar solvents (sugar in water); non-polar molecular crystals dissolve in non-polar solvents (iodine in CCl₄).
Transparency: Many molecular crystals are transparent to light.
Lattice Energy
Lattice energy is the energy released when one mole of an ionic crystal is formed from its constituent gaseous ions. Equivalently, it is the energy required to separate one mole of a solid ionic compound into isolated gaseous ions. It is expressed in kJ mol⁻¹.
$$Na^+(g) + Cl^-(g) \rightarrow NaCl(s) \quad \Delta H = -787 \; \text{kJ mol}^{-1}$$
The lattice energy of NaCl — energy released when one mole of solid NaCl forms from gaseous Na⁺ and Cl⁻ ions
$Na^+(g)$=Gaseous sodium ion(kJ mol⁻¹)
$Cl^-(g)$=Gaseous chloride ion(kJ mol⁻¹)
$\Delta H$=Enthalpy change (lattice energy)(kJ mol⁻¹)
Formation (gaseous ions → solid)
→$\Delta H$ is negative (exothermic, energy is released)
Dissociation (solid → gaseous ions)
→$\Delta H$ is positive (endothermic, energy must be absorbed)
The magnitude of lattice energy depends on the sizes and charges of the ions. Understanding these trends allows you to predict and compare lattice energies across different compounds.
Cation Size Effect: Keeping the anion constant, lattice energy decreases (becomes less negative) as the cation size increases. Larger cations reduce the tightness of packing, weakening electrostatic attraction.
Anion Size Effect: Keeping the cation constant, lattice energy decreases as the anion size increases. NaCl (−787) > NaBr (−728) > NaI (−690) because Cl⁻ < Br⁻ < I⁻ in ionic radius.
Charge Effect: Ions with higher charges produce much stronger electrostatic attraction. MgO (Mg²⁺, O²⁻) has a far greater lattice energy than NaCl (Na⁺, Cl⁻) because the charges are doubled.
Lattice Energies of Selected Ionic Compounds
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LiCl: −833 kJ mol⁻¹
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NaF: −895 kJ mol⁻¹
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NaCl: −787 kJ mol⁻¹
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NaBr: −728 kJ mol⁻¹
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NaI: −690 kJ mol⁻¹
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KCl: −690 kJ mol⁻¹
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KBr: −665 kJ mol⁻¹
A more negative lattice energy indicates stronger ionic bonding and a more stable crystal. Compounds with high lattice energy tend to have higher melting points, greater hardness, and lower solubility.