Coordination Number Calculator
Calculate coordination number with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
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The radius ratio rule predicts coordination number based on the ratio of cation to anion radii. Specific ratio ranges correspond to stable geometries: below 0.155 = linear (CN=2), 0.155-0.225 = trigonal planar (CN=3), 0.225-0.414 = tetrahedral (CN=4), 0.414-0.732 = octahedral (CN=6), 0.732-1.0 = cubic (CN=8), and above 1.0 = cuboctahedral (CN=12).
Last reviewed: December 2025
Worked Examples
Example 1: NaCl Crystal Structure
Example 2: CsCl Crystal Structure
Background & Theory
The Coordination Number Calculator applies the following established principles and formulas. Chemistry is the science of matter's composition, structure, properties, and transformations. At the heart of quantitative chemistry lies the mole concept. One mole of any substance contains exactly 6.022×10²³ entities (Avogadro's number, Nₐ), and the molar mass of an element or compound in grams per mole is numerically equal to its atomic or molecular mass in atomic mass units. This allows chemists to convert between measurable mass and the number of reacting particles. Stoichiometry uses balanced chemical equations to relate the amounts of reactants and products. A balanced equation conserves both mass and charge. Molarity, the most common concentration unit, is defined as M = n/V, where n is moles of solute and V is volume of solution in liters, giving units of mol/L. Acidity and basicity are quantified by the pH scale, defined as pH = −log₁₀[H⁺], where [H⁺] is the molar concentration of hydrogen ions. Pure water at 25°C has pH 7.00; acids have lower values and bases higher values. Each unit change represents a tenfold change in hydrogen ion concentration. Gas behavior is described by the ideal gas law PV = nRT, where P is pressure in pascals, V is volume in cubic meters, n is moles, R = 8.314 J/(mol·K), and T is temperature in kelvin. Special cases include Boyle's Law (P₁V₁ = P₂V₂ at constant temperature) and Charles's Law (V₁/T₁ = V₂/T₂ at constant pressure). Thermochemistry quantifies heat changes in reactions through enthalpy, H. Hess's Law states that the total enthalpy change for a reaction is the sum of enthalpy changes for any sequence of steps leading to the same overall reaction, making it possible to calculate enthalpies for reactions that cannot be measured directly. Electron configuration describes the distribution of electrons in atomic orbitals according to the Aufbau principle, Pauli exclusion principle, and Hund's rule. Periodic trends including atomic radius, ionization energy, and electronegativity arise systematically from electron configuration and nuclear charge, enabling chemists to predict and rationalize chemical behavior across the periodic table.
History
The history behind the Coordination Number Calculator traces back through the following developments. Chemistry's roots lie in alchemy, the medieval practice combining proto-scientific experimentation with mystical aims. Alchemists developed practical techniques including distillation, calcination, and the preparation of acids, building a body of empirical knowledge despite their theoretical misunderstandings. Modern chemistry is conventionally dated to Antoine Lavoisier (1743–1794), often called the father of modern chemistry. Lavoisier demonstrated the law of conservation of mass in 1789, showing that matter is neither created nor destroyed in chemical reactions. He identified oxygen's role in combustion, dismantling the phlogiston theory, and co-authored the first systematic chemical nomenclature, establishing the language still used today. John Dalton proposed the first modern atomic theory in 1803, asserting that all matter is composed of indivisible atoms, that atoms of the same element are identical in mass, and that compounds form from fixed ratios of different atoms. This provided a physical basis for Lavoisier's conservation law and Proust's law of definite proportions. Dmitri Mendeleev published his periodic table in 1869, arranging the 63 known elements by atomic mass and revealing repeating patterns of chemical behavior. He boldly left gaps for undiscovered elements and predicted their properties with remarkable accuracy, predictions confirmed by the subsequent discovery of gallium, scandium, and germanium. Ernest Rutherford's gold foil experiment in 1911 revealed the nuclear model of the atom: a tiny, dense, positively charged nucleus surrounded by electrons. Niels Bohr refined this in 1913 with a quantized model of electron orbits that explained the hydrogen emission spectrum. Quantum chemistry and molecular orbital theory, developed through the 1920s and 1930s, provided the full quantum mechanical description of chemical bonding. The latter 20th century saw the rise of computational chemistry, enabling molecular simulation at unprecedented scale. The green chemistry movement, articulated in the 12 Principles of Green Chemistry in 1998, reoriented the field toward sustainability, waste reduction, and benign chemical design, reflecting chemistry's growing awareness of its environmental responsibilities.
Frequently Asked Questions
Formula
Radius Ratio = r(cation) / r(anion)
The radius ratio rule predicts coordination number based on the ratio of cation to anion radii. Specific ratio ranges correspond to stable geometries: below 0.155 = linear (CN=2), 0.155-0.225 = trigonal planar (CN=3), 0.225-0.414 = tetrahedral (CN=4), 0.414-0.732 = octahedral (CN=6), 0.732-1.0 = cubic (CN=8), and above 1.0 = cuboctahedral (CN=12).
Worked Examples
Example 1: NaCl Crystal Structure
Problem: Determine the coordination number of Na+ (radius 102 pm) with Cl- (radius 181 pm).
Solution: Radius ratio = r(Na+) / r(Cl-) = 102 / 181 = 0.5636\nRatio falls in range 0.414 - 0.732\nPredicted CN = 6 (octahedral)\nBond length = 102 + 181 = 283 pm\nThis matches the known NaCl rock salt structure
Result: CN = 6 | Geometry = Octahedral | Bond length = 283 pm
Example 2: CsCl Crystal Structure
Problem: Find the coordination number of Cs+ (radius 170 pm) with Cl- (radius 181 pm).
Solution: Radius ratio = r(Cs+) / r(Cl-) = 170 / 181 = 0.9392\nRatio falls in range 0.732 - 1.0\nPredicted CN = 8 (cubic)\nBond length = 170 + 181 = 351 pm\nCsCl adopts the body-centered cubic structure
Result: CN = 8 | Geometry = Cubic | Bond length = 351 pm
Frequently Asked Questions
What is the coordination number in chemistry?
The coordination number is the number of atoms, ions, or molecules (ligands) directly bonded to a central atom in a complex or crystal structure. It describes the immediate environment around a metal center and determines the geometry of the complex. For example, in the complex ion [Co(NH3)6]3+, cobalt has a coordination number of 6 because six ammonia molecules surround it. Coordination numbers typically range from 2 to 12, with 4 and 6 being the most common in transition metal chemistry. The coordination number affects properties like color, magnetism, and reactivity of the compound.
How does the radius ratio predict coordination number?
The radius ratio rule uses the ratio of the cation radius to the anion radius (r+/r-) to predict the coordination number and geometry in ionic crystals. When the cation is very small relative to the anion (ratio below 0.155), only 2 anions can fit around it in a linear arrangement. As the ratio increases, more anions can pack around the cation: 0.155-0.225 gives CN=3 (trigonal planar), 0.225-0.414 gives CN=4 (tetrahedral), 0.414-0.732 gives CN=6 (octahedral), and 0.732-1.0 gives CN=8 (cubic). While this rule provides good approximations, real crystal structures can deviate due to covalent bonding character and orbital effects.
What factors affect coordination number besides size?
Several factors beyond ionic radii influence coordination number. Electronic configuration plays a major role because certain d-electron counts favor specific geometries due to crystal field stabilization energy. Ligand properties matter too, as bulky ligands like triphenylphosphine reduce coordination numbers through steric effects, while small ligands like CO allow higher numbers. Charge and electronegativity affect the degree of covalent versus ionic bonding, shifting preferred geometries. Temperature and pressure can also change coordination numbers, as seen in the high-pressure transformation of NaCl from 6-coordinate to 8-coordinate CsCl-type structure.
What is the relationship between coordination number and hybridization?
Coordination number directly determines the hybridization of the central atom according to valence bond theory. A coordination number of 2 corresponds to sp hybridization with linear geometry, 3 to sp2 with trigonal planar geometry, 4 to sp3 (tetrahedral) or dsp2 (square planar), 5 to sp3d (trigonal bipyramidal) or dsp3, and 6 to sp3d2 (octahedral). For transition metals, d orbitals participate in hybridization, enabling higher coordination numbers. This relationship is essential for predicting molecular shapes, bond angles, and understanding reaction mechanisms in coordination chemistry and organometallic compounds.
How do you experimentally determine coordination numbers?
Coordination numbers are determined experimentally through several techniques. X-ray crystallography is the gold standard, directly revealing the three-dimensional arrangement of atoms around the central metal with sub-angstrom precision. Extended X-ray absorption fine structure (EXAFS) measures coordination numbers in both solid and solution states by analyzing oscillations in X-ray absorption spectra. Nuclear magnetic resonance (NMR) spectroscopy can reveal coordination numbers through chemical shifts and coupling patterns. Other methods include neutron diffraction for light atoms, pair distribution function analysis for amorphous materials, and computational methods like density functional theory to predict and validate experimental findings.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy