Kapustinskii Equation: Lattice Energy Calculations
The Kapustinskii equation is a fundamental relationship in inorganic chemistry and solid-state energetics. In competitive postgraduate examinations such as CSIR-NET, GATE, and IIT-JAM, calculating lattice energy is a frequent numerical problem. Classical formulations like the Born-Lande and Born-Mayer equations require knowledge of the crystal structure and the exact Madelung constant. However, for newly synthesized ionic crystals or unstable salts where crystallographic structures are not yet known, students cannot look up a Madelung constant. Anatoli Kapustinskii resolved this by recognizing that the Madelung constant divided by the number of ions per formula unit is nearly constant across common ionic lattices. This led to a formula predicting lattice energy using only ionic charges and ionic radii.
The Kapustinskii Formula and Parameters
The Kapustinskii equation calculates the lattice dissociation energy of an ionic crystal. Lattice energy is defined as the energy required to dissociate one mole of a crystalline solid into gaseous ions at infinite distance. In this convention, lattice energy is positive (endothermic). When defined as lattice formation from gaseous ions, it has an equal magnitude but a negative sign.
UL = [ (K · ν · |z+| · |z−|) / (r+ + r−) ] × [ 1 − (d / (r+ + r−)) ]
Where:
• UL = Lattice energy in kilojoules per mole (kJ/mol)
• K = 1.202 × 105 kJ·pm/mol (or 120.2 kJ·nm/mol)
• ν = Total number of ions per formula unit (e.g. 2 for KCl, 3 for CaF2)
• |z+| = Absolute charge of the cation
• |z−| = Absolute charge of the anion
• r+ = Cation radius in picometres (pm)
• r− = Anion radius in picometres (pm)
• r0 = Interionic distance (r+ + r−) in picometres (pm)
• d = Born-Mayer constant = 34.5 pm (or 0.0345 nm)
The equation is also used in reverse to determine thermochemical radii for complex non-spherical ions such as sulfate or nitrate from Born-Haber cycle data.
Step-by-Step Worked Examples
Step 1: Identify the ion counts and charges.
KCl dissociates into one K+ and one Cl− ion.
Total ions, ν = 2. Charges: |z+| = 1 and |z−| = 1.
Charge product = |z+| × |z−| = 1 × 1 = 1.
Step 2: Find the sum of ionic radii.
r0 = r+ + r− = 138 pm + 181 pm = 319 pm.
Step 3: Calculate the repulsion term.
1 − (d / r0) = 1 − (34.5 / 319) = 1 − 0.10815 = 0.89185.
Step 4: Compute the electrostatic term.
Electrostatic term = (1.202 × 105 × 2 × 1) / 319 = 240400 / 319 = 753.605 kJ/mol.
Step 5: Multiply to get lattice energy.
UL = 753.605 × 0.89185 = 672.1 kJ/mol.
Step 6: Second route verification.
UL = [ 120200 × 2 × (319 − 34.5) ] / 3192 = 68393800 / 101761 = 672.1 kJ/mol.
Answer: 672.1 kJ/mol
Step 1: Identify the ions and charges.
CaF2 contains one Ca2+ and two F− ions per formula unit.
Total ions, ν = 1 + 2 = 3.
Charges: |z+| = 2, |z−| = 1.
Charge product = 2 × 1 = 2.
Step 2: Calculate the sum of ionic radii.
r0 = 100 pm + 133 pm = 233 pm.
Step 3: Calculate the repulsion factor.
1 − (34.5 / 233) = 1 − 0.14807 = 0.85193.
Step 4: Compute the electrostatic term.
Electrostatic term = (1.202 × 105 × 3 × 2) / 233 = 721200 / 233 = 3095.279 kJ/mol.
Step 5: Compute final lattice energy.
UL = 3095.279 × 0.85193 = 2637.0 kJ/mol.
Step 6: Second route verification.
UL = [ 721200 × (233 − 34.5) ] / 2332 = 143158200 / 54289 = 2637.0 kJ/mol.
Answer: 2637.0 kJ/mol
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: CSIR-NET Chemical Sciences, GATE Chemistry, IIT-JAM Chemistry, CUET-PG
- Topic: Kapustinskii Equation and Lattice Enthalpy
- Difficulty: Advanced
Common Mistakes in Kapustinskii Calculations
Students regularly lose precision in examinations due to these errors:
1. Incorrect ion count (ν): In salts like CaF2, Al2O3, or Na2SO4, students often substitute ν = 2. Always count every individual ion present in the chemical formula unit. For CaF2, ν = 3 (one Ca2+ and two F−). For Al2O3, ν = 5 (two Al3+ and three O2−).
2. Unit mismatch: The constant K = 1.202 × 105 requires radii in picometres (pm). If radii are given in angstroms (Å) or nanometres (nm), convert them before calculating. Note that 1 Å = 100 pm and 1 nm = 1000 pm.
3. Sign convention confusion: Lattice dissociation energy is positive (endothermic), whereas lattice formation enthalpy is negative (exothermic). Check which quantity the question specifies.
Exam Relevance and Format Breakdown
The Kapustinskii equation is tested across several higher chemistry examinations. The summary below illustrates how different exam papers approach the topic.
| Exam | Question Format | Key Focus Area |
|---|---|---|
| CSIR-NET Chemical Sciences | Multiple choice and numerical conceptual options | Thermochemical radii extraction and lattice stability comparisons |
| GATE Chemistry | Numerical Answer Type (NAT) | Accurate arithmetic substitution using given ionic radii without Madelung constants |
| IIT-JAM Chemistry | Multiple choice and numerical answer questions | Comparison between Kapustinskii estimates and Born-Haber cycle experimental values |
| CUET-PG Chemistry | Objective questions | Direct dependence of lattice energy on ionic charge products and interionic distance |
Explore real crystal structures in the Crystal Lab, then test yourself on previous-year questions.
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