Vacancy Concentration Calculator
Copper at 1000 °C contains a vacancy fraction of 0.000273741, about one empty site in every 3650.
Copper 0.9 eV, aluminium 0.67 eV, iron about 1.4–1.6 eV.
0.000273741199917975
Full answer, then cut to fewer decimals (not rounded):
- 0.000273741199917975
- 0.00027374119991797
- 0.0002737411999179
- 0.000273741199917
- 0.00027374119991
- 0.0002737411999
- 0.000273741199
- 0.00027374119
- 0.0002737411
- 0.000273741
- 0.00027374
- 0.0002737
- 0.000273
- 0.00027
- 0.0002
- Atomic sites N
- 8.491231739 × 10²⁸ per m³
- Vacancies n_v
- 2.324399965 × 10²⁵ per m³
- One vacancy per
- 3653.08547 atoms
Step-by-step solution
- n_v/N = exp(−Q_v/kT) = exp(−0.9 ÷ (8.617 × 10⁻⁵ × 1273.15))
- = exp(−8.203327424) = 0.0002737411999
Vacancy Concentration Formula
n_v / N = exp(−Q_v / kT)
k = 8.617 × 10⁻⁵ eV/K; N = Nₐρ/A atoms per m³
Worked Example
Copper (Q_v = 0.9 eV) at 1000 °C:
- n_v/N = exp(−Q_v/kT) = exp(−0.9 ÷ (8.617 × 10⁻⁵ × 1273.15))
- = exp(−8.203327424) = 0.0002737411999
Vacancies n_v: 2.324399965 × 10²⁵ per m³ · One vacancy per: 3653.08547 atoms
GATE tip
Keep units consistent: with Q in J/mol use R; with Q in eV per atom use Boltzmann’s constant 8.617 × 10⁻⁵ eV/K.
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Example: α-iron, body-centred cubic with a = 2.8665 Å: Theoretical density ρ = 7.87422 g/cm³.
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nλ = 2d sin θ, d = a/√(h² + k² + l²)
Cu Kα₁ λ = 1.5406 Å; cubic crystals
Example: The (110) peak of α-iron at 2θ = 44.67° with Cu Kα radiation: Interplanar spacing d = 2.02699 Å.
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ASTM Grain Size Calculator
N = 2^(G − 1) grains/in² at 100×
G = −6.6439 log₁₀ ℓ − 3.288 (ℓ in mm); grains per mm² = 2^(G + 2.954)
Example: A micrograph showing 64 grains per square inch at 100×: ASTM grain size number G = 7.
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r* = 2γ/ΔG_v, ΔG* = 16πγ³/(3ΔG_v²)
ΔG_v = ΔH_f ΔT / T_m; heterogeneous S(θ) = (2 + cos θ)(1 − cos θ)²/4
Example: Copper solidifying with 200 K of undercooling: Critical radius r* = 1.27854 nm.
Frequently asked questions
Why are vacancies unavoidable?
Each vacancy raises the energy but also the configurational entropy; the free energy is lowest at a finite vacancy concentration that rises with temperature.
Why do vacancies matter?
Substitutional diffusion needs a neighbouring vacancy, so diffusion, creep and sintering all scale with the vacancy concentration.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Vacancy Concentration Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/vacancy-concentration-calculator/
IEEE
Scientific Converters, "Vacancy Concentration Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/vacancy-concentration-calculator/