Radioactive Decay Calculator

Calculate remaining nuclei, activity, or decay constant using N = N₀·e^(−λt) and A = λN.

⚛️ Nuclear📐 N=N₀e^(-λt)☢️ Activity
Number of nuclei (N)
Decay constant λ (s⁻¹)
⚠️ Enter valid positive numbers.

What Is Radioactive Decay?

Radioactive decay follows an exponential law: N(t) = N₀·e^(−λt), where N₀ is the initial number of radioactive nuclei, λ is the decay constant (s⁻¹), and t is time. The relationship between λ and half-life: λ = ln(2)/t½ = 0.693/t½. Activity A = λN (decays per second, measured in Becquerels).

The exponential form arises because each nucleus decays independently with constant probability λ per unit time. The differential equation dN/dt = −λN gives the exponential solution. This is a first-order process — the same mathematics governs capacitor discharge (Q = Q₀e^(−t/RC)), drug elimination, and Newton's law of cooling.

Mean lifetime (τ = 1/λ = t½/ln2) is the average time a nucleus survives before decaying. After one mean lifetime, N = N₀/e ≈ 36.8% remains (compared to 50% after one half-life). Radioactivity A = λN = (ln2/t½)N means a sample with more atoms but same t½ has proportionally higher activity.

Units: 1 Becquerel (Bq) = 1 decay per second. 1 Curie (Ci) = 3.7×10¹⁰ Bq. Absorbed dose: 1 Gray (Gy) = 1 J/kg. Effective dose: 1 Sievert (Sv) = dose in Gy × quality factor (1 for gamma, 20 for alpha). Annual background radiation ≈ 2–3 mSv.

Formula Reference Table

Solve ForFormulaNotes
Decay lawN(t) = N₀·e^(−λt)λ = decay constant (s⁻¹)
ActivityA = λ·NBq = decays/s
Half-life relationλ = ln(2)/t½ = 0.693/t½t½ in same units as t
Mean lifetimeτ = 1/λ = t½/ln(2)τ = 1.443 × t½
Activity at time tA(t) = A₀·e^(−λt)Activity also decays exponentially
Becquerel1 Bq = 1 decay/s1 Ci = 3.7×10¹⁰ Bq

3 Worked Examples

Example 1
C-14 Activity in Sample

1 g of ancient wood has N = 5.97×10¹⁰ atoms of C-14. t½ = 5,730 yr = 1.808×10¹¹ s. Find activity.

  • λ = 0.693/(1.808×10¹¹) = 3.83×10⁻¹² s⁻¹
  • A = λN = 3.83×10⁻¹² × 5.97×10¹⁰ = 0.229 Bq
  • Modern C-14 activity in 1 g wood ≈ 0.226 Bq — confirms this sample is approximately 'fresh' (current)
✓ Activity = 0.229 Bq per gram (≈modern value)
Example 2
I-131 Medical Dose

Patient receives 370 MBq of I-131 (t½ = 8.02 days). Activity after 24 days?

  • λ = 0.693/(8.02×86400) = 1.0×10⁻⁶ s⁻¹
  • t = 24 days = 2.074×10⁶ s
  • A = 370×10⁶ × e^(−1.0×10⁻⁶ × 2.074×10⁶) = 370×10⁶ × e^(−2.074)
  • A = 370 × 0.126 = 46.6 MBq
✓ Activity after 24 days = 46.6 MBq (≈3 half-lives)
Example 3
Convert t½ to λ

Carbon-14: t½ = 5,730 years. Find λ in s⁻¹.

  • t½ = 5730 × 365.25 × 24 × 3600 = 1.807×10¹¹ s
  • λ = ln(2)/t½ = 0.6931/1.807×10¹¹ = 3.836×10⁻¹² s⁻¹
  • Mean lifetime τ = 1/λ = 2.608×10¹¹ s = 8,267 years
✓ λ = 3.836×10⁻¹² s⁻¹; mean lifetime = 8,267 years

Real-World Applications

🏥
Radiation Therapy
I-131 therapy for thyroid cancer: the thyroid preferentially absorbs iodine. A = λN decays over time, delivering therapeutic dose precisely. t½ = 8 days means most radiation delivered within 2–3 weeks.
☢️
Radiation Safety
Monitoring contamination uses A = λN. A spill of 100 MBq must be tracked until activity drops to safe levels. Short-lived isotopes (hours) decay quickly; long-lived (decades) require sustained containment.
🌍
Environmental Monitoring
Cs-137 (t½=30 yr) from nuclear accidents is tracked using γ-spectroscopy. Activity maps show contamination levels. Decay law predicts future activity for land use planning.
⚛️
Nuclear Medicine Production
Tc-99m is produced in generators: Mo-99 (t½=66h) decays to Tc-99m (t½=6h). Hospitals 'milk' the generator daily to extract fresh Tc-99m. The generator lasts about a week before Mo-99 activity drops too low.
🔬
Detector Calibration
Radioactivity standards use certified sources (often Cs-137 or Co-60) with known A₀ and t½. Labs apply A = A₀e^(−λt) to correct for decay since calibration date.

Common Mistakes to Avoid

⚠️
Confusing λ (s⁻¹) units

λ must have units of inverse time (matching t). If t½ is in years, λ = ln2/t½ is in yr⁻¹. For A = λN in Bq (s⁻¹), convert λ to s⁻¹ and ensure N is in atoms (not grams).

⚠️
N vs A (nuclei vs activity)

N = number of radioactive atoms (not Bq). A = λN = decay rate in Bq. Doubling λ doubles A even with same N. Doubling N doubles A for same λ.

⚠️
Half-life not the same as mean lifetime

t½ = 0.693/λ; τ = 1/λ = 1.443×t½. After one mean lifetime: 36.8% remains. After one half-life: 50% remains.

⚠️
Wrong exponential form

N = N₀e^(−λt), not N₀(e^λ)^t or N₀e^(+λt). The exponent must be negative for decay.

⚠️
Not converting half-life to s⁻¹ for Bq

To get activity in Bq: λ must be in s⁻¹. C-14 t½ = 5,730 yr = 1.807×10¹¹ s; λ = 3.84×10⁻¹² s⁻¹.

Frequently Asked Questions

What is the decay constant λ? ▾
λ (s⁻¹) is the probability per second that any given nucleus will decay. High λ = fast decay, short t½. Low λ = slow decay, long t½. The fundamental nuclear property: λ = ln(2)/t½. Activity A = λN measures total decays per second.
What is a Becquerel? ▾
1 Bq = 1 nuclear decay per second. 1 Ci = 3.7×10¹⁰ Bq = radioactivity of 1 g Ra-226. Medical doses: 400 MBq Tc-99m for bone scan. Natural K-40 in the human body: ≈4,000 Bq. Brazil nuts contain elevated Ra-226: ≈60 Bq/kg.
What is specific activity? ▾
A_sp = A/m = λN_A/M (Bq/g) where N_A = Avogadro's number, M = molar mass. For Co-60 (t½=5.27 yr, M=60): A_sp = 4.18×10¹³ Bq/g = 1,131 Ci/g. Short-lived isotopes have enormous specific activity; long-lived ones (U-238: 12,400 Bq/g) have very low specific activity.
What is secular equilibrium? ▾
In a decay chain, when parent t½ >> daughter t½, after several daughter half-lives, A_daughter = A_parent. This is secular equilibrium. Example: Ra-226 (t½=1,600 yr) → Rn-222 (t½=3.8 d): after a few weeks, A_Rn = A_Ra. Used in calibration of Rn measurement devices.
How does radiation damage tissue? ▾
Ionizing radiation (alpha, beta, gamma, neutrons) disrupts chemical bonds in cells. DNA double-strand breaks are most damaging. The effective dose (Sv) = absorbed dose (Gy) × radiation quality factor × tissue weighting factor. 5 Gy whole-body dose is approximately LD50 (50% mortality in 30 days without treatment).
What is the difference between absorbed dose and effective dose? ▾
Absorbed dose: D (Gray, Gy) = energy deposited per kg of tissue (J/kg). Same for all radiation types. Effective dose: H (Sievert, Sv) = D × quality factor (w_R). Quality factors: gamma = 1, beta = 1, alpha = 20, neutrons = 5–20. 1 mSv/yr is typical medical dose; 20 mSv/yr is radiation worker limit.
What is transmutation? ▾
Nuclear decay changes the element. Alpha decay: Z−2, A−4. Beta minus: Z+1, same A (neutron → proton + electron + antineutrino). Beta plus: Z−1 (proton → neutron + positron + neutrino). Electron capture: Z−1. Gamma decay: same Z and A (only energy released). The periodic table position of a nucleus changes each time it decays.
How do Geiger counters measure radioactivity? ▾
A Geiger-Müller tube contains gas at low pressure. An ionizing particle entering ionizes the gas, triggering a cascading discharge detected as an electrical pulse. Count rate (pulses/s) is proportional to A but depends on detector geometry and efficiency. Calibrated against known sources for quantitative A measurement.

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