The concept of coordination number is fundamental in understanding the structure of ionic solids, particularly in the context of the Collegeboard AP Chemistry curriculum. It defines the number of nearest neighbor ions surrounding a central ion in a crystal lattice, influencing the geometry, stability, and properties of the compound. Mastery of coordination numbers is essential for students to comprehend complex ionic structures and their applications in various chemical processes.

Key Concepts

Definition of Coordination Number

The coordination number (CN) refers to the number of atoms, ions, or molecules directly bonded to a central atom or ion in a coordination complex or crystal lattice. In ionic solids, it specifically denotes the number of oppositely charged ions surrounding a central ion. For instance, in sodium chloride (NaCl), each sodium ion (Na⁺) is surrounded by six chloride ions (Cl⁻), resulting in a coordination number of 6.

Determining Coordination Number

The coordination number can be determined by analyzing the geometry of the ionic lattice and the relative sizes of the ions involved. It is influenced by several factors:

Ionic Radii: Smaller ions can accommodate more neighbors due to their compact size. For example, the coordination number of Al³⁺ in aluminum chloride (AlCl₃) is 6, whereas that of larger ions like K⁺ in potassium chloride (KCl) is 6.
Charge Density: Ions with higher charges attract more oppositely charged ions, increasing the coordination number. For example, Mg²⁺ has a higher coordination number in magnesium oxide (MgO) compared to Na⁺ in sodium oxide (Na₂O).
Crystal Packing: The arrangement of ions in the crystal lattice affects the coordination number. Close-packed structures like face-centered cubic (FCC) and hexagonal close-packed (HCP) typically have higher coordination numbers.

Common Coordination Numbers and Geometries

Coordination numbers are associated with specific geometric arrangements around the central ion:

Coordination Number 2: Linear Geometry - Example: BeCl₂ in which beryllium chloride adopts a linear structure.
Coordination Number 4: Tetrahedral or Square Planar Geometry - Example: TiCl₄ adopts a tetrahedral geometry.
Coordination Number 6: Octahedral Geometry - Example: FeCl₃ in which iron chloride forms an octahedral structure.
Coordination Number 8: Square Antiprismatic or Dodecahedral Geometry - Example: Mo(CN)₈⁴⁻ has a coordination number of 8.

Coordination Number in Different Ionic Compounds

The coordination number varies among different ionic compounds based on their structural requirements and the factors mentioned earlier:

Sodium Chloride (NaCl): Each Na⁺ ion is surrounded by 6 Cl⁻ ions, and each Cl⁻ is surrounded by 6 Na⁺ ions, resulting in a coordination number of 6.
Copper(II) Sulfate (CuSO₄): In this compound, the Cu²⁺ ion typically has a coordination number of 4, adopting a square planar geometry.
Cesium Chloride (CsCl): Each Cs⁺ ion is surrounded by 8 Cl⁻ ions and vice versa, giving a coordination number of 8.
Beryllium Fluoride (BeF₂): Beryllium in BeF₂ has a coordination number of 4, forming a tetrahedral geometry.

Impact of Coordination Number on Physical Properties

The coordination number significantly influences the physical properties of ionic solids:

Melting and Boiling Points: Higher coordination numbers typically lead to stronger bonding and higher melting and boiling points due to increased electrostatic attractions.
Hardness and Brittleness: Compounds with higher coordination numbers often exhibit greater hardness and brittleness because of the tightly packed and rigid lattice structures.
Solubility: The solubility of an ionic compound can be affected by its coordination number. Higher coordination numbers may lead to lower solubility as more energy is required to break the ionic lattice.

Coordination Number in Complex Ions

Beyond simple ionic solids, coordination numbers play a crucial role in the formation of complex ions:

Transition Metal Complexes: Transition metals often form complex ions with coordination numbers ranging from 4 to 8. For example, [Fe(CN)₆]⁴⁻ has a coordination number of 6, adopting an octahedral geometry.
Ligand Bonding: The nature and number of ligands attached to the central ion determine the coordination number. Polydentate ligands can bind to multiple sites, affecting the overall coordination environment.
Stability of Complexes: Coordination number influences the stability of complex ions. Higher coordination numbers can enhance stability through more extensive bonding interactions.

Determining Coordination Number Using the Radius Ratio Rule

The radius ratio rule is a method used to predict the coordination number based on the ratio of the radii of the cation (r₊) to the anion (r₋): $$ \text{Radius Ratio} = \frac{r_+}{r_-} $$

Radius Ratio < 0.155: Coordination Number 2 (Linear)
0.155 ≤ Radius Ratio < 0.225: Coordination Number 3 (Triangular Planar)
0.225 ≤ Radius Ratio < 0.414: Coordination Number 4 (Tetrahedral)
0.414 ≤ Radius Ratio < 0.732: Coordination Number 6 (Octahedral)
0.732 ≤ Radius Ratio: Coordination Number 8 or 12 (Square Antiprismatic or Cubic)

For example, consider the coordination number in magnesium oxide (MgO). The radius of Mg²⁺ is approximately 0.72 Å, and the radius of O²⁻ is about 1.40 Å. The radius ratio is: $$ \frac{0.72}{1.40} = 0.514 $$ According to the radius ratio rule, a ratio of 0.414 to 0.732 suggests a coordination number of 6, indicating an octahedral geometry in MgO.

Applications of Coordination Number in Chemistry

Understanding coordination numbers is vital in various chemical applications:

Crystal Engineering: Designing materials with specific coordination numbers can tailor their physical properties for industrial applications.
Biochemistry: Coordination numbers are crucial in the structure of metalloproteins and enzymes, affecting their biological functions.
Catalysis: Transition metal catalysts rely on specific coordination environments to facilitate chemical reactions efficiently.
Pharmaceuticals: Coordination complexes are used in drug design and delivery systems, leveraging their structural properties to enhance efficacy.

Challenges in Determining Coordination Number

While the concept is straightforward, several challenges can arise:

Ambiguity in Coordination Environments: Some ions can adopt multiple coordination numbers depending on the surrounding environment, making it difficult to assign a single number.
Polyhedral Distortions: Real structures often deviate from ideal geometries, complicating the determination of coordination numbers based solely on geometric considerations.
Computational Limitations: Predicting coordination numbers in complex systems requires advanced computational methods, which may not always provide clear answers.
Mixed Valence States: Ions with varying oxidation states can exhibit different coordination numbers within the same compound, adding to the complexity.

Historical Perspective and Development

The concept of coordination number has evolved significantly since its inception:

Auger and Pauling's Contributions: Early theories by Auger and later Pauling laid the groundwork for understanding coordination in ionic compounds, introducing concepts like crystal field theory.
Advancements in Crystallography: The development of X-ray crystallography enabled precise determination of coordination numbers in various crystals, enhancing the accuracy of structural models.
Modern Computational Chemistry: Contemporary computational techniques allow for the prediction and visualization of coordination environments in complex systems, expanding the applicability of coordination number concepts.

Coordination Number and Electronic Configuration

The electronic configuration of the central ion influences its preferred coordination number:

High-Spin vs. Low-Spin Complexes: Transition metals can exhibit different coordination numbers based on their electronic configurations and ligand field strength.
Electron Counting: The number of electrons available for bonding affects the stability and geometry of the coordination complex, thereby influencing the coordination number.
Crystal Field Stabilization Energy: Coordination numbers are optimized to maximize the crystal field stabilization energy, contributing to the overall stability of the complex.

Examples of Coordination Number Determination

Let's explore specific examples to illustrate how coordination numbers are determined:

Ammonium Chloride (NH₄Cl): In the ammonium ion (NH₄⁺), the nitrogen atom is surrounded by four hydrogen atoms, resulting in a coordination number of 4.
Copper(II) Sulfate Pentahydrate (CuSO₄.5H₂O): The copper ion (Cu²⁺) is coordinated by four water molecules in a square planar arrangement, giving a coordination number of 4.
Iron(III) Chloride (FeCl₃): Iron in FeCl₃ typically exhibits an octahedral geometry with a coordination number of 6, surrounded by six chloride ions.
Nickel(II) Hexaammine Complex ([Ni(NH₃)₆]²⁺): The nickel ion is coordinated by six ammonia molecules, resulting in a coordination number of 6.

Coordination Number vs. Coordination Geometry

While coordination number refers to the number of ligands attached to the central ion, coordination geometry describes the spatial arrangement of these ligands:

Correlation: A higher coordination number typically allows for more complex geometries. For example, a coordination number of 6 often leads to an octahedral geometry.
Variability: The same coordination number can result in different geometries based on factors like ligand type and electronic configurations. For instance, a coordination number of 4 can correspond to either a tetrahedral or square planar geometry.
Impact on Properties: Coordination geometry affects the physical and chemical properties of the compound, including solubility, reactivity, and color.

Coordination Number in Biological Systems

Coordination numbers are pivotal in biological systems, particularly in metalloproteins and enzymes:

Hemoglobin: The iron ion in hemoglobin has a coordination number of 6, bonded to four nitrogen atoms from the heme group, one nitrogen from a histidine residue, and one oxygen molecule.
Vitamin B12: The cobalt ion in Vitamin B12 exhibits a coordination number of 6, bonded to four nitrogen atoms from the corrin ring, one nitrogen from a nucleotide, and one methyl group.
Enzyme Active Sites: Transition metals in enzyme active sites often have specific coordination numbers that are crucial for catalytic activity.

Comparison Table

Aspect	Coordination Number	Coordination Geometry
Definition	Number of nearest neighbor ions around a central ion	Spatial arrangement of the surrounding ions
Typical Values	2, 4, 6, 8, 12	Linear, Tetrahedral, Octahedral, Square Antiprismatic, Cubic
Influencing Factors	Ionic radii, charge density, crystal packing	Ionic radii, electronic configuration, ligand type
Examples	NaCl (6), CsCl (8)	BeCl₂ (linear), FeCl₃ (octahedral)
Impact on Properties	Determines lattice structure, melting/boiling points	Affects geometry, stability, and reactivity

Summary and Key Takeaways

Coordination number is the count of nearest neighbors around a central ion in a crystal lattice.
It is influenced by ionic radii, charge, and crystal packing.
Common coordination numbers include 2 (linear), 4 (tetrahedral/square planar), and 6 (octahedral).
The radius ratio rule helps predict coordination numbers based on ion sizes.
Coordination numbers impact the physical and chemical properties of ionic compounds.
Understanding coordination numbers is essential for studying complex ions and biological systems.

Examiner Tip

Tips

Use the mnemonic "CRISP" to remember factors affecting coordination numbers: Crystal packing, Radius ratio, Ionic radii, Structure, and charge Presence. Additionally, always double-check your predicted geometry by visualizing the spatial arrangement to ensure accuracy for AP exam questions.

Did You Know

Coordination numbers play a crucial role in mineral formation and can influence the color of gemstones. For example, the vibrant blue of sapphire is due to the specific coordination of titanium and iron ions within the crystal lattice. Additionally, some metal-organic frameworks (MOFs) utilize high coordination numbers to create porous structures used in gas storage and separation technologies.

Common Mistakes

Students often confuse coordination number with oxidation state. For instance, in FeCl₃, the oxidation state of iron is +3, but its coordination number is 6. Another common error is misapplying the radius ratio rule, leading to incorrect predictions of coordination numbers. It's essential to consider all influencing factors, not just ion sizes, when determining the coordination number.

FAQ

What is the coordination number in water (H₂O) molecules?

Water molecules themselves are not coordination complexes, but when acting as ligands, each H₂O molecule can coordinate to a central metal ion, influencing the metal's coordination number based on how many water molecules attach.

How does the coordination number affect the geometry of a complex?

The coordination number determines the spatial arrangement of ligands around the central ion, leading to specific geometries such as linear, tetrahedral, or octahedral structures.

Can coordination numbers be fractional?

No, coordination numbers are always whole numbers as they represent the count of neighboring ions or ligands around a central ion.

Why does aluminum typically have a coordination number of 6?

Aluminum ions (Al³⁺) have a high charge density and a relatively small ionic radius, favoring an octahedral arrangement with six chloride ions in compounds like aluminum chloride (AlCl₃).

How do ligands influence the coordination number?

The size, charge, and binding mode of ligands can dictate how many can simultaneously coordinate to a central ion, thereby influencing the overall coordination number of the complex.

1. Kinetics

1.1 Elementary Reactions

1.1.1 Molecularity

1.1.2 Mechanisms of Reactions

1.1.3 Identifying Intermediates

1.2 Collision Model

1.2.1 Activation Energy

1.2.2 Orientation of Collisions

1.2.3 Factors Affecting Collision Frequency

1.3 Catalysis

1.3.1 Homogeneous and Heterogeneous Catalysts

1.3.2 Enzymatic Catalysis