Decay Constant Calculators

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The decay constant (λ, lambda) is the probability per unit time that a single radioactive nucleus will decay. It characterizes the rate of radioactive decay and is inversely related to the half-life: λ = ln(2) / t₁/₂ = 0.693 / t₁/₂. The decay constant appears in the exponential decay law: N(t) = N₀ × e^(−λt), where N(t) is the number of undecayed nuclei at time t and N₀ is the initial number. Activity (A = λN) gives the number of decays per second (Becquerels or Curie). The decay constant is also used in first-order chemical kinetics and in pharmacokinetics (elimination rate constant).

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Decay Constant and Half-Life

λ = ln(2) / t₁/₂ = 0.693 / t₁/₂

t₁/₂ = 0.693 / λ

Example: ¹⁴C has t₁/₂ = 5,730 years: λ = 0.693/5730 = 1.21 × 10⁻⁴ yr⁻¹ = 3.84 × 10⁻¹² s⁻¹.

Exponential Decay Law

N(t) = N₀ × e^(−λt)

N₀ = initial number of nuclei; N(t) = number remaining at time t; λ = decay constant. Equivalent: N(t) = N₀ × (1/2)^(t/t₁/₂).

Example: start with 1.0 × 10⁸ nuclei of ³²P (t₁/₂ = 14.3 days). After 30 days: N = 1.0 × 10⁸ × e^(−0.693/14.3 × 30) = 1.0 × 10⁸ × e^(−1.454) = 2.34 × 10⁷ nuclei (23.4% remaining).

Activity

A = λ × N (decays per second = Becquerels, Bq)

1 Ci (Curie) = 3.7 × 10¹⁰ Bq. Activity also decays exponentially: A(t) = A₀ × e^(−λt).

Applications

  • Radiocarbon dating: t = (1/λ) × ln(N₀/N) — requires knowing λ for ¹⁴C and measuring remaining ¹⁴C
  • Medical imaging: ⁹⁹ᵐTc (t₁/₂ = 6.02 h, λ = 0.115 h⁻¹) — short t₁/₂ = low patient dose
  • Drug pharmacokinetics: elimination rate constant k_e ≡ λ; t₁/₂ = 0.693/k_e

Glossary

Decay Constant (λ)
The probability per unit time that a nucleus decays: λ = ln(2)/t₁/₂ = 0.693/t₁/₂; units s⁻¹ or year⁻¹; appears in N(t) = N₀ × e^(−λt); larger λ = faster decay.
Activity (A)
Rate of radioactive decay: A = λN; units Becquerel (Bq = 1 decay/s) or Curie (Ci = 3.7×10¹⁰ Bq); decreases exponentially with same half-life as the parent nuclei.
Half-Life (t₁/₂)
Time for half the nuclei to decay: t₁/₂ = ln(2)/λ = 0.693/λ; after n half-lives, fraction remaining = (1/2)^n; ranges from nanoseconds (unstable isotopes) to billions of years.

Frequently Asked Questions

The decay constant λ is the probability per unit time that a nucleus decays. Relationship to half-life: λ = ln(2)/t₁/₂ = 0.693/t₁/₂. Example: ¹³¹I has t₁/₂ = 8.02 days: λ = 0.693/8.02 = 0.0864 day⁻¹ = 1.00 × 10⁻⁶ s⁻¹. Interpretation: each second, each ¹³¹I nucleus has a 1.00 × 10⁻⁶ probability of decaying. The decay constant is the inverse of the mean lifetime τ = 1/λ. Larger λ = shorter half-life = more rapid decay.

N(t) = N₀ × e^(−λt) or equivalently N(t) = N₀ × (1/2)^(t/t₁/₂). Example: 10 μg of ³²P (t₁/₂ = 14.3 days) after 28.6 days (= 2 half-lives): N = 10 × (1/2)² = 10/4 = 2.5 μg remaining. Or using e: λ = 0.693/14.3 = 0.0485 day⁻¹; N = 10 × e^(−0.0485×28.6) = 10 × e^(−1.386) = 10 × 0.25 = 2.5 μg ✓. Each half-life halves the quantity — after n half-lives, fraction remaining = (1/2)^n.

Activity A = λ × N = rate of decay (disintegrations per second). Units: Becquerel (Bq) = 1 disintegration/second; Curie (Ci) = 3.7 × 10¹⁰ Bq (activity of 1 gram of radium-226). Example: 1 μg of ¹⁴C (MW = 14 g/mol; t₁/₂ = 5730 yr = 1.806 × 10¹¹ s). N = 1×10⁻⁶/14 × 6.022×10²³ = 4.30×10¹⁶ nuclei. λ = 0.693/(1.806×10¹¹ s) = 3.84×10⁻¹² s⁻¹. A = λN = 3.84×10⁻¹² × 4.30×10¹⁶ = 165,120 Bq = 165 kBq.

In pharmacokinetics, the elimination rate constant k_e is mathematically equivalent to the decay constant λ. Drug elimination follows first-order kinetics: C(t) = C₀ × e^(−k_e × t). k_e = ln(2)/t₁/₂ = 0.693/t₁/₂. Half-life of drug = 0.693/k_e. Volume of distribution and clearance relate: k_e = CL/Vd. Example: drug with t₁/₂ = 4 hours: k_e = 0.693/4 = 0.173 h⁻¹. After 12 hours (3 half-lives): plasma concentration = C₀ × (1/2)³ = C₀/8. Clinical application: after 4–5 half-lives, a drug is essentially eliminated (>96% cleared) — used to determine dosing intervals and washout periods.