Inorganic Chemistry I: Structures and Energetics of Metallic and Ionic Solids - Part II

Structures and Energetics of Metallic and Ionic Solids - Part II

Close-Packed Arrangements and Space Filling

• Two Different Close-Packed Arrangements:

  • Hexagonal Close-Packing (hcp): Involves two repeating layers (e.g., ABAB…).
  • Cubic Close-Packing (ccp): Also known as face-centered cubic (fcc), involves three repeating layers (e.g., ABCABC…). All fcc structures are ccp.

• Most Frequent Packings of Metals:

  • Hexagonal Close-Packing (hcp): Achieves $74\%$ space filling.
  • Cubic Close-Packing (ccp or fcc): Achieves $74\%$ space filling.
  • Body-Centered Cubic (bcc): Achieves $68\%$ space filling.

• Packing-of-Spheres Model Application:

  • This model is particularly applicable to Group 18 elements (Ne, Ar, Kr, Xe), which typically form ccp structures.
  • Similar behavior is observed for $H<em>2$ (at $14.0 \text{ K}$) and $F</em>2$ (at $45 \text{ K}$).
  • The model is valid for these elements because they contain freely rotating molecules.

Polymorphism

• Definition: Polymorphism means "many forms" and refers to the ability of a solid material to exist in more than one crystal structure or solid form.

• Relation to Allotropy: Polymorphism is related to allotropy, but allotropy specifically refers only to chemical elements (e.g., carbon).

• Example (Carbon): Carbon exhibits polymorphism in forms like Diamond, Graphite, and Fullerene.

• Significance: Different polymorphic forms can have significantly different structures and physical and chemical properties.

• Types of Polymorphism:

  1. Packing Polymorphism: Occurs when the packing and bonding arrangements of a structure are significantly different in its various polymorphic forms.
     • Example: $\alpha$-glycine (monoclinic symmetry) vs. $\beta$-glycine (hexagonal symmetry).
  2. Conformational Polymorphism: Involves the existence of different conformers of the same molecule in different polymorphic modifications.
     • Characterized by low energy differences between various conformations.
     • Example: Spiperone existing in Form I and Form II due to different molecular conformations.
  3. Pseudopolymorphism: (Pseudo means "false") Refers to a new structure formed when solvent molecules are incorporated into the crystal lattice. Also known as a solvate.
     • Example: N-acetyl-L-phenylalanine methyl ester forming a solvate with $0.5$ molecules of water.

• McCrone's Law: "Every compound has different polymorphic forms, and that, in general, the number of forms known for a given compound is proportional to the time and money spent in research on that compound."

  • Phase Transition: The process of transformation of one polymorph into another.

• Properties of Polymorphs: Polymorphs can differ in a wide range of properties:

  • Packing Properties: Molar volume, density, refractive index, conductivity, hygroscopicity.
  • Thermodynamic Properties: Melting and sublimation temperature, structural energy, enthalpy, heat capacity, entropy, free energy and chemical potential, thermodynamic activity, vapor pressure, solubility.
  • Kinetic Properties: Dissolution rates, rates of solid-state reactions, stability.
  • Spectroscopic Properties.
  • Surface Properties: Surface free energy, interfacial tension, morphology.
  • Mechanical Properties: Hardness, tensile strength, compactability, handling, flow.
  • Bioavailability.

• Characterization Methods: Various techniques are used to study polymorphs:

  • Crystallography: X-Ray Diffraction (Single Crystal X-Ray Diffraction, X-Ray Powder Diffraction).
  • Morphology: Microscopy (Polarizing Optical Microscopy, Thermal Microscopy).
  • Phase Transitions: Thermal Methods of Analysis (Thermogravimetry (TGA), Differential Thermal Analysis (DTA), Differential Scanning Calorimetry (DSC)).
  • Molecular Motion: Vibrational Spectroscopy (Infrared Absorption Spectroscopy, Raman Spectroscopy).
  • Chemical Environment: Nuclear Magnetic Resonance Spectrometry (NMR).

• Thermodynamic and Kinetic Stability:

  • Thermodynamic Theory: Crystallization must result in an overall decrease in the free energy of the system. Crystal structures with greater (more negative) lattice (free) energies are preferred.
  • Kinetic Tendency: Systems also tend to crystallize as quickly as possible to relieve imposed supersaturation.
  • Example (Graphite/Diamond): Graphite is the thermodynamically preferred crystalline form of carbon. However, kinetic factors, specifically a high activation barrier, make the rate of transformation from diamond to graphite infinitely slow, allowing diamond to persist.

• Energetic Types of Polymorphism:

  • Enantiotropic System: One form is stable over a certain pressure and temperature range, while the other polymorph is stable over a different pressure and temperature range.
    • Characterized by a reversible transition at a definite transition temperature ($T_t$), where the free energy curves cross below the melting point.
    • Form A might be stable below $T<em>t$ (lower $G</em>A$), and Form B stable above $T<em>t$ (lower $G</em>B$).
    • Transition is endothermic.
    • Lower melting point form is thermodynamically stable below $T<em>t$, higher melting point form stable above $T</em>t$.
    • Lower melting point form has lower enthalpy of fusion ($H_f$).
  • Monotropic System: Only one polymorph is stable at all temperatures below the melting point, with all other polymorphs being unstable.
    • Characterized by an irreversible transition because the free energy curves do not cross below the melting point.
    • Form A always has a lower free energy ($G_A$) than Form B throughout the range below the melting point.
    • Transition is exothermic.
    • Higher melting point form is always the thermodynamically stable form.
    • Higher melting point form has higher $H_f$.

• Energy-Temperature Diagrams:

  • The relative stability of two polymorphs depends on their free energies ($G$).
  • The more stable polymorph has a lower free energy, as described by the Gibbs free energy equation: $G = H - TS$ (where $H$ is enthalpy, $T$ is temperature, and $S$ is entropy).

• **Example: **$[Cu(tzc)(dpp)]<em>n \cdot 2H</em>2O$

  • This complex, involving tetrazolate-5-carboxylate (tzc) and 1,3-Di(4-pyridyl)propane (dpp), exhibits multiple polymorphic forms ($4, 5, 6<em>I, 6</em>{II}, 6_{III}$) with different structural and magnetic properties.
  • Thermal analysis (TGA and DSC) reveals weight loss events corresponding to dehydration (loss of $H_2O$) and decomposition (loss of tzc, dpp).
  • DSC shows endothermic and exothermic transitions at specific temperatures ($87 \text{ °C}$, $129 \text{ °C}$, $188/190 \text{ °C}$), indicating phase changes.
  • Structural studies show transformations between different triclinic, orthorhombic, and monoclinic forms upon heating, cooling, and hydration/dehydration.
  • Energy-temperature diagrams for this system demonstrate enantiotropic relationships, where polymorphs like $6<em>I$ and $6</em>{II}$ or $6<em>{II}$ and $6</em>{III}$ are stable at different temperature ranges, with defined transition temperatures ($T_t$).

Polymorphism in Metals

• Structures of metals are generally considered at $298 \text{ K}$ and standard pressure ($1 \text{ bar}$).

• Changes in temperature and/or pressure can induce changes in crystal structure.

• Phases formed at high temperatures can sometimes be quenched to lower temperatures, retaining their high-temperature structure.

• Pressure-Temperature Phase Diagram for Iron: Each line on such a diagram represents a phase boundary, and crossing it requires changes in temperature and/or pressure. For example, at $298 \text{ K}$ and $1 \text{ bar}$, iron has a bcc structure.

• Example: Tin Pest:

  • Historical examples like Robert Scott's Antarctic expedition (1910) and Napoleon's invasion of Russia (1812) are linked to tin pest.
  • Below $13.2 \text{ °C}$, pure tin transforms from its silvery, ductile metallic allotrope ($\beta$-form white tin) to a brittle, nonmetallic, $\alpha$-form grey tin.
  • This phase change is associated with a significant volume increase (about $27\%$).
  • The $\alpha$-form decomposes into a powder, hence the name "tin pest." The transformation can be observed over time, for instance, a complete transformation in about $20 \text{ h}$ at $-40 \text{ °C}$.

Definition of Different Radii

• Covalent Radius: Refers to the size of an atom that forms part of a covalent bond. The sum of two covalent radii should approximate the covalent bond length.

• Metallic Radius: Half of the distance between nearest-neighbor atoms in a solid-state metal lattice. It is dependent upon the coordination number of the metal atom.

• Van der Waals Radius: Equal to one half the distance between two unbonded atoms when the electrostatic forces between them are balanced. It represents the effective radius of an atom when it is not chemically bonded.

• Ionic Radius: The measure of an atom's ion in a crystal lattice. Ionic radii are typically measured indirectly by comparing internuclear distances in salts where the positive ions vary in size.

• Trends and Data: Periodic tables often provide extensive data for Van der Waals, Covalent, and Ionic radii, which vary significantly based on the element, its charge, and its bonding environment.

  • For example, covalent radii generally decrease across a period and increase down a group.
  • Ionic radii differ from covalent radii due to electron gain or loss; cations are smaller and anions are larger than their neutral parent atoms.

Why Ion Formation? (Born-Haber Cycle)

• Simple Calculation Example (Li and Cl):

  • The ionization of lithium: $Li<em>{(g)} \rightarrow Li^+</em>{(g)} + e^-_{(g)} \quad (+520 \text{ kJ/mol})$
  • The electron affinity of chlorine: $Cl<em>{(g)} + e^-</em>{(g)} \rightarrow Cl^-_{(g)} \quad (-356 \text{ kJ/mol})$
  • Summing these, the formation of gaseous ions is an endothermic process: $Li<em>{(g)} + Cl</em>{(g)} \rightarrow Li^+<em>{(g)} + Cl^-</em>{(g)} \quad (+164 \text{ kJ/mol})$
  • This calculation demonstrates that simply gaining a 'noble gas configuration' is not the driving force for ion formation in the gas phase; it actually costs energy.

• Born-Haber Cycle: An approach used to analyze reaction energies, particularly for the formation of ionic compounds like LiF or LiCl from their elements.

  • It's an application of Hess's Law, breaking down the overall enthalpy of formation ($\Delta H_f^o$) into a series of hypothetical steps:
    1. Enthalpy of Atomization/Sublimation: Converting the solid metal to gaseous atoms (e.g., $Li<em>{(s)} \rightarrow Li</em>{(g)}$).
    2. Bond Dissociation Enthalpy: Converting non-metal molecules to gaseous atoms (e.g., $1/2 F<em>{2(g)} \rightarrow F</em>{(g)}$).
    3. Ionization Energy: Removing electrons from gaseous metal atoms to form cations (e.g., $Li<em>{(g)} \rightarrow Li^+</em>{(g)} + e^-_{(g)}$).
    4. Electron Affinity: Adding electrons to gaseous non-metal atoms to form anions (e.g., $F<em>{(g)} + e^-</em>{(g)} \rightarrow F^-_{(g)}$).
    5. Lattice Enthalpy: The energy released when gaseous ions combine to form the solid ionic lattice (e.g., $Li^+<em>{(g)} + F^-</em>{(g)} \rightarrow LiF_{(s)}$). This highly exothermic process is typically the main driving force for ionic compound formation.

Deformation in Solids

• Deformation of Salts (Ionic Solids): Salts are typically brittle.

  • When forces are applied, layers of similarly charged ions can be brought into proximity.
  • The strong electrostatic repulsion between like-charged ions causes the crystal to cleave or break.

• Deformation of Metals (Metallic Solids): Metals are generally ductile and malleable.

  • The "electron sea" model explains this: metal atoms are held together by delocalized electrons.
  • When a force is applied, layers of metal atoms can slide past each other without significant repulsion because the delocalized electrons can simply redistribute to maintain the metallic bonding, preventing strong electrostatic interactions that would cause fracture.

Ionic Lattices (Common Structure Types)

• Rock Salt/Halite Structure Type (e.g., NaCl):

  • Each ion (both $Na^+$ and $Cl^-$) is $6$-coordinate, meaning each ion is surrounded by six ions of opposite charge.
  • The structure consists of two interpenetrating face-centered cubic (fcc) lattices (one for cations, one for anions).
  • For salts of formula MX, the coordination numbers of M and X must be equal.
  • Many halides, oxides, and sulfides crystallize with the NaCl type structure.

• CsCl Structure Type (e.g., CsCl):

  • Each ion (both $Cs^+$ and $Cl^-$) is $8$-coordinate, meaning each ion is surrounded by eight ions of opposite charge.
  • The structure consists of two interpenetrating simple cubic lattices (often described as bcc packing of one ion type with the other ion type occupying the body center).
  • Examples include CsBr, TlCl, and $NH_4Cl$ (below $457 \text{ K}$, above which it adopts the NaCl structure).

• Fluorite ($CaF_2$) Structure Type:

  • Each cation ($Ca^{2+}$) is $8$-coordinate.
  • Each anion ($F^-$, for instance) is $4$-coordinate.
  • The calcium ions form an fcc lattice, and the fluoride ions occupy all tetrahedral sites.
  • For salts of formula $MX_2$, the coordination number of X must be half that of M.

• Zinc Blende ($ZnS$) Structure Type:

  • Based on a diamond-type network structure.
  • Cation:anion ratio is $1:1$. Both $Zn^{2+}$ and $S^{2-}$ are $4$-coordinate (tetrahedral).
  • Examples include Si, Ge, and $\alpha$-Sn, where atoms are covalently bonded in a similar network.

• $\beta$-Cristobalite ($SiO_2$) Structure Type:

  • Related to the diamond-type network.
  • The structure of $\beta$-cristobalite can be visualized as being derived from the Si (diamond) structure by placing an oxygen atom between adjacent silicon atoms, forming Si-O-Si linkages.

• Wurtzite ($ZnS$) Structure Type:

  • Exhibits hexagonal symmetry.
  • Both zinc and sulfur centers are tetrahedrally coordinated, similar to zinc blende but with a different stacking sequence.

• Rutile ($TiO_2$) Structure Type:

  • Titanium ($Ti^{4+}$) is $6$-coordinate (octahedral geometry).
  • Oxygen ($O^{2-}$) is $3$-coordinate (trigonal planar geometry).
  • Examples include $PbO<em>2$ and $SnO</em>2$.

Exam #1 Information

• Exam Duration: $1$ hour $15$ minutes.
• Memorization: Memorize the first three rows of the periodic table plus the first transition row.
• Preparation Materials: Use slides for preparation and textbook Chapter $1, 2, 3$ and $6$ (only sections covered in lecture).
• Review Quizzes: Be able to answer questions from Quizzes #$1$ and #$2$.
• Key Concepts to Understand:
  • Quantum numbers and associated rules.
  • Ionization energies (no specific values, but trends).
  • VSEPR theory.
  • Molecular symmetry and point groups.
  • Close-packing (hcp, ccp).
  • Molecular orbitals (MOs) and Atomic orbitals (AOs).
  • Polymorphism (definitions, types, stability).
  • Simple structure types (e.g., NaCl, CsCl, Fluorite, Zinc Blende).