Half-Life Calculator – Radioactive Decay, Carbon Dating, and Medication Metabolism

  • Adam
  • November 4, 2025

Free online half-life calculator for radioactive decay, exponential decay, carbon dating, medication dosing, and decay constant calculations. Calculate remaining amount, half-life period, time elapsed, and initial amount with step-by-step solutions.

Half-Life Calculator

Advanced Tool for Radioactive Decay, Medication Metabolism, and Exponential Decay Analysis

Quick Navigation

Calculate Remaining Amount

Find how much remains after decay

Calculate Half-Life

Determine the half-life period

Calculate Time Elapsed

Find how much time has passed

Calculate Initial Amount

Find the starting amount

Calculate Decay Constant

Find the decay constant from half-life

What is Half-Life?

Half-life is the time required for a substance, quantity, or population to reduce to half its initial value. This concept applies to radioactive decay, medication metabolism, pollutant degradation, and any exponential decay process. It's a fundamental concept in nuclear physics, chemistry, pharmacology, and environmental science.

The key insight about half-life is that it's a constant property of any given substance. Carbon-14 always has a half-life of 5,730 years, regardless of how much carbon-14 you have. This constancy makes half-life incredibly useful for predicting how long substances persist and for applications like radiocarbon dating.

Half-life follows exponential decay patterns. After one half-life, 50% remains. After two half-lives, 25% remains. After three, 12.5% remains. This rapid decrease is why long-lived radioactive materials can still be hazardous for millennia, while short-lived medical isotopes are relatively safe within days.

Key Concept: Half-life is independent of the initial amount. Whether you start with 1 gram or 1000 grams of a radioactive isotope, the time to reach half that amount is the same.

Key Features & Capabilities

This comprehensive half-life calculator provides multiple calculation modes and detailed analysis:

📉 Calculate Remaining Find how much substance remains after a given time period using exponential decay formula
⏱️ Calculate Half-Life Determine the half-life period from initial amount, remaining amount, and elapsed time
⏰ Calculate Time Elapsed Find how much time has passed based on decay amounts and half-life period
📊 Calculate Initial Amount Determine the starting amount from current amount, time, and half-life
⚛️ Calculate Decay Constant Compute decay constant (λ) from the half-life period for advanced decay modeling
📋 Step-by-Step Solutions See detailed calculation breakdown showing exactly how results were obtained
🔢 Multiple Calculations Calculate half-lives passed, percentage remaining, and other derived values
📊 Statistical Analysis View related values like decay constant, percentage remaining, number of half-lives
📋 Copy to Clipboard One-click copy functionality to transfer results to other applications
🎓 Educational Content Comprehensive guides, examples, and explanations of half-life concepts
⚡ Real-Time Calculation Instant results with no delays or external dependencies
📱 Fully Responsive Works seamlessly on desktop, tablet, and mobile devices

How to Use This Calculator

Step-by-Step Guide

  1. Identify What You're Calculating: Choose the appropriate tab: Remaining (find what's left), Half-Life (find decay period), Time Elapsed (find how long), Initial Amount (find starting quantity), or Decay Constant (find λ parameter).
  2. Gather Your Known Values: Collect the values you know. For "Calculate Remaining," you need initial amount, half-life, and time elapsed. For other modes, different inputs are required.
  3. Enter Values with Correct Units: Input your numbers. Ensure time units are consistent (all years, all days, etc.). The calculator performs all computations in whatever units you use.
  4. Click Calculate: Press the Calculate button to perform the computation using the exponential decay formula: N(t) = N₀ × (1/2)^(t/t_half)
  5. Review Results: The main result displays prominently. For remaining amount, you see the final quantity in the same units as your initial amount.
  6. Study the Breakdown: Below the result, see step-by-step showing: how many half-lives passed, percentage remaining, and the calculation method used.
  7. Analyze Statistics: See additional derived values like decay constant, percentage remaining, and number of half-lives that have occurred.
  8. Copy or Clear: Use Copy to transfer results to documents. Use Clear to reset for a new calculation.

Tips for Accurate Use

  • Unit Consistency: Keep time units consistent throughout. Don't mix years and days. Convert everything to the same unit first.
  • Significant Figures: Report answers with appropriate precision. Decay calculations are meaningful to about 3-4 significant figures for most applications.
  • Fractional Half-Lives: You can use fractional half-life periods. For example, 2.5 half-lives means the elapsed time is 2.5 times the half-life period.
  • Very Long Periods: For very small remaining amounts, the calculation remains accurate even if amount approaches zero asymptotically.
  • Real-World Precision: Remember that real decay follows probability. These calculations give average behavior for large quantities.

Complete Formulas Guide

Basic Half-Life Formula

Exponential Decay with Half-Life
N(t) = N₀ × (1/2)^(t / t_half)

Where:
N(t) = amount remaining at time t
N₀ = initial amount
t = time elapsed
t_half = half-life period

Example: If you start with 100g and half-life is 5 years, after 10 years:
N(10) = 100 × (1/2)^(10/5) = 100 × (1/2)^2 = 100 × 0.25 = 25g

Number of Half-Lives

Calculating Half-Lives Passed
n = t / t_half

Where:
n = number of half-lives
t = time elapsed
t_half = half-life period

Then: N(t) = N₀ × (1/2)^n

Example: 100g sample, 5-year half-life, 15 years elapsed
n = 15 / 5 = 3 half-lives
Remaining = 100 × (1/2)^3 = 100 / 8 = 12.5g

Decay Constant

Relationship Between Half-Life and Decay Constant
λ = ln(2) / t_half ≈ 0.693147 / t_half

Alternative form: N(t) = N₀ × e^(-λt)

Where:
λ = decay constant (per unit time)
ln(2) = natural log of 2 ≈ 0.693147
t_half = half-life period

Example: C-14 half-life is 5730 years
λ = 0.693147 / 5730 ≈ 1.209 × 10^-4 per year

Solving for Half-Life

Calculate Half-Life from Decay Data
t_half = t × ln(2) / ln(N₀ / N(t))

Or: t_half = -t × ln(2) / ln(N(t) / N₀)

Example: 100g reduced to 25g over 10 years
t_half = 10 × ln(2) / ln(100/25)
t_half = 10 × 0.693147 / ln(4)
t_half = 10 × 0.693147 / 1.38629 ≈ 5 years

Percentage Remaining

Calculate Percentage After Decay
Percentage Remaining = [(1/2)^n] × 100%

Where n = number of half-lives

Quick Reference:
After 1 half-life: 50%
After 2 half-lives: 25%
After 3 half-lives: 12.5%
After 4 half-lives: 6.25%
After 5 half-lives: 3.125% (≈3%)

Real-World Applications

Radiocarbon Dating

Archaeology and geology use carbon-14 half-life (5,730 years) to date organic materials. By measuring remaining C-14, scientists determine how long ago the organism died. This technique works for materials up to about 50,000 years old.

Medical Isotopes

Medical imaging uses short-lived radioactive isotopes. Technetium-99m (6 hours) is used for bone scans. The short half-life means the patient receives minimal radiation exposure while getting diagnostic information.

Pharmacology and Medicine

Drug metabolism follows half-life principles. Aspirin has ~20 minute half-life (needs frequent dosing), while levothyroxine has ~7 day half-life (once daily dosing). Doctors use half-life to determine dosing schedules.

Nuclear Waste Management

Environmental safety requires understanding half-lives of radioactive waste. Iodine-131 (8 days) becomes safe relatively quickly, while Plutonium-239 (24,000 years) requires extremely long-term storage and containment.

Environmental Contamination

Pollution studies track how long contaminants persist. Pesticides, oil spills, and other pollutants have characteristic half-lives determining how quickly environments recover.

Biological Half-Life

Healthcare considers biological half-life (how long body takes to eliminate substance). This differs from radioactive half-life but follows similar exponential patterns.

Worked Examples

Example 1: Carbon Dating

Problem: An ancient bone contains 25% of its original C-14. How old is the bone? (C-14 half-life: 5,730 years)

Solution:
Remaining: 25% = 0.25 = (1/2)^n
0.25 = (0.5)^n
n = 2 (because 0.5^2 = 0.25)
Elapsed time = 2 × 5,730 = 11,460 years

Verification: 100 × (1/2)^(11460/5730) = 100 × (1/2)^2 = 25 ✓

Example 2: Medical Isotope

Problem: A patient receives 10 mCi of Tc-99m (half-life: 6 hours). How much remains after 24 hours?

Solution:
Initial: 10 mCi
Half-life: 6 hours
Time elapsed: 24 hours

Number of half-lives = 24 / 6 = 4
Remaining = 10 × (1/2)^4 = 10 × 1/16 = 0.625 mCi

The activity drops by factor of 16 in 24 hours

Example 3: Drug Dosing

Problem: Aspirin has a 20-minute half-life. After 2 hours, what percentage of the drug remains?

Solution:
Half-life: 20 minutes
Time elapsed: 2 hours = 120 minutes

Number of half-lives = 120 / 20 = 6
Percentage remaining = (1/2)^6 × 100% = (1/64) × 100%
= 1.56%

After 2 hours, only 1.56% of the dose remains

Example 4: Radioactive Decay

Problem: Start with 80g of Iodine-131 (half-life: 8 days). How much remains after 32 days?

Solution:
Initial amount: 80g
Half-life: 8 days
Time elapsed: 32 days

Number of half-lives = 32 / 8 = 4
Remaining = 80 × (1/2)^4 = 80 × 1/16 = 5g

After 32 days: 80→40→20→10→5g ✓

Example 5: Finding Half-Life

Problem: A sample decreased from 500g to 125g over 30 years. What is the half-life?

Solution:
Initial: 500g
Final: 125g
Time: 30 years

125 / 500 = 0.25 = (1/2)^n
n = 2 (two half-lives occurred)

t_half = 30 years / 2 = 15 years

Verification: 500 × (1/2)^(30/15) = 500 × (1/2)^2 = 125 ✓

Frequently Asked Questions

❓ Why does half-life matter for radioactive materials?
Half-life determines how long a radioactive material remains hazardous. Short half-lives mean rapid decay and reduced danger (Tc-99m in medical imaging). Long half-lives mean materials stay dangerous for millennia (Plutonium-239 in nuclear waste). Understanding half-life is critical for safety planning.
❓ Is half-life different from decay constant?
Yes. Half-life (t_half) is how long to reach 50%. Decay constant (λ) is related by λ = ln(2)/t_half. They're different mathematical expressions of the same decay property. Half-life is more intuitive; decay constant is used in exponential decay formulas: N(t) = N₀ × e^(-λt).
❓ After 5 half-lives, how much is left?
After 5 half-lives, (1/2)^5 = 1/32 ≈ 3.125% remains. This is why pharmaceutical dosing often considers "5 half-lives to elimination"—after 5 half-lives, only about 3% remains, usually considered negligible.
❓ Can I have a negative time?
Mathematically yes, but practically no. Negative time represents the past—working backwards to find how much was present before. The calculator uses positive values for forward-in-time decay calculations.
❓ How accurate is the half-life formula?
The half-life formula N(t) = N₀ × (1/2)^(t/t_half) is extremely accurate for large quantities. For individual atoms, quantum mechanics introduces randomness, but averaged over populations, the formula is accurate to better than one part per million.
❓ What if I don't know the exact half-life?
You can measure it empirically. If you know the initial amount, current amount, and elapsed time, this calculator can determine half-life. You take measurements at two points in time and the calculator solves for the half-life period.
❓ Does biological half-life equal radioactive half-life?
No. Radioactive half-life is fixed (doesn't change). Biological half-life depends on body elimination rates and varies by individual. For medical calculations, effective half-life = 1/(1/radioactive_t_half + 1/biological_t_half).
❓ Can half-life be less than one minute?
Yes! Some highly unstable isotopes have microsecond or nanosecond half-lives. The formula works for any time scale. Thorium-212 has a half-life of 368 nanoseconds.
❓ Is half-life used in anything other than radioactivity?
Absolutely! Half-life concepts apply to: medication metabolism, pollutant degradation, population decline, data storage degradation, financial investments, and any exponential decay process. The mathematics are identical.
❓ What's the longest half-life known?
Tellurium-128 has an extremely long half-life of about 2.2 × 10^24 years—far longer than the age of the universe (13.8 billion years). Bismuth-209 was long thought stable but has a half-life of about 1.9 × 10^19 years.
❓ When would I use this calculator in real life?
Use for: carbon dating artifacts, calculating medication dosing schedules, nuclear safety planning, environmental contamination assessment, medical imaging planning, waste storage calculations, archaeology research, geology studies, chemistry homework, physics problems, or any exponential decay analysis.

Start Calculating Half-Life

Whether you're dating archaeological artifacts, calculating medication dosing, planning nuclear safety, studying physics, or analyzing environmental contamination, this comprehensive half-life calculator handles all your decay analysis needs. Fast, accurate, and completely free.

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