Half-Life & Radioactive Decay: Complete Guide

What Is Half-Life?

Half-life is the time required for a quantity to reduce to half of its initial value. It's a fundamental concept in nuclear physics, chemistry, medicine, and many other fields. The beauty of half-life is its consistency—no matter how much material you start with, it always takes the same amount of time to lose half.

Key Insight

If you start with 100 grams of a substance with a half-life of 10 years, after 10 years you'll have 50 grams. After another 10 years (20 total), you'll have 25 grams. After 30 years, 12.5 grams, and so on.

Why Is Half-Life Constant?

In radioactive decay, each atom has the same probability of decaying in any given time period, regardless of how long it has existed. This statistical property leads to exponential decay, where a fixed percentage of atoms decay per unit time rather than a fixed number.

Think of it like a casino where every atom is rolling dice each second. If any atom rolls a specific number, it decays. With trillions of atoms, the statistical behavior becomes very predictable—even though we can't predict when any individual atom will decay.

Understanding Exponential Decay

Half-life decay is an example of exponential decay, where the rate of decrease is proportional to the current amount. This creates a distinctive curved graph that approaches but never reaches zero.

Half-Lives Elapsed	Remaining (%)	Fraction	Example (100g start)
0	100%	1	100 g
1	50%	1/2	50 g
2	25%	1/4	25 g
3	12.5%	1/8	12.5 g
4	6.25%	1/16	6.25 g
5	3.125%	1/32	3.125 g
10	0.098%	1/1024	0.098 g

After 10 half-lives, less than 0.1% of the original substance remains. This is why radioactive materials eventually become "safe"—though technically, they never reach zero, the amount becomes negligibly small.

The Half-Life Formula

Several equivalent formulas describe exponential decay. Choose the one that fits your problem:

Primary Half-Life Formula

N(t) = N₀ × (1/2)^t/t½

Where N(t) = amount at time t, N₀ = initial amount, t½ = half-life

Exponential Form

N(t) = N₀ × e^-λt

Where λ = ln(2)/t½ ≈ 0.693/t½ is the decay constant

Relationship Between λ and t½

t½ = ln(2)/λ ≈ 0.693/λ

Convert between decay constant and half-life

Understanding the Decay Constant (λ)

The decay constant λ represents the probability per unit time that a single atom will decay. A larger λ means faster decay and shorter half-life. The relationship λ = ln(2)/t½ comes from the definition of half-life.

Calculating Remaining Amount

The most common half-life calculation: given initial amount, half-life, and elapsed time, find how much remains.

Example 1: Radioactive Iodine-131

A hospital has 400 mg of Iodine-131 (half-life = 8 days). How much remains after 24 days?

N(t) = N₀ × (1/2)^t/t½

N(24) = 400 × (1/2)^24/8

N(24) = 400 × (1/2)³

N(24) = 400 × 1/8

N(24) = 50 mg

Answer: 50 mg remains after 24 days (3 half-lives)

Example 2: Non-Integer Half-Lives

You have 200 grams of a substance with a 5-year half-life. How much remains after 12 years?

N(12) = 200 × (1/2)^12/5

N(12) = 200 × (1/2)^2.4

N(12) = 200 × 0.189

N(12) = 37.9 grams

Answer: About 37.9 grams remain after 12 years

Finding Time from Remaining Amount

Sometimes you know how much remains and need to find how long has passed. This requires using logarithms to solve for time.

Formula for Finding Time

t = t½ × log₂(N₀/N)

Or equivalently: t = -t½ × log₂(N/N₀)

Or: t = t½ × ln(N₀/N) / ln(2)

Example: How Long Has Passed?

A sample originally had 1000 atoms of a radioactive isotope (half-life = 20 minutes). Now it has 62.5 atoms. How much time has passed?

62.5 = 1000 × (1/2)^t/20

62.5/1000 = (1/2)^t/20

0.0625 = (1/2)^t/20

log₂(0.0625) = t/20

-4 = t/20

t = 80 minutes

Answer: 80 minutes (4 half-lives: 1000 → 500 → 250 → 125 → 62.5)

Radioactive Decay in Physics

Radioactive decay occurs when unstable atomic nuclei release energy by emitting radiation. There are several types of radioactive decay:

Alpha Decay (α)

The nucleus emits an alpha particle (2 protons + 2 neutrons). Common in heavy elements like uranium and radium. Alpha particles are stopped by paper but dangerous if inhaled or ingested.

Beta Decay (β)

A neutron converts to a proton (or vice versa), emitting an electron or positron. More penetrating than alpha—stopped by aluminum foil or thick plastic.

Gamma Decay (γ)

High-energy photons emitted as the nucleus releases excess energy. Very penetrating—requires lead or concrete shielding. Often accompanies alpha or beta decay.

Common Radioactive Isotopes

Isotope	Half-Life	Primary Use
Carbon-14	5,730 years	Archaeological dating
Iodine-131	8 days	Thyroid treatment
Technetium-99m	6 hours	Medical imaging
Uranium-238	4.5 billion years	Geological dating
Radon-222	3.8 days	Indoor air quality concern
Cobalt-60	5.27 years	Cancer radiotherapy

Carbon-14 Dating

Carbon-14 dating is one of the most famous applications of half-life. It's used to determine the age of organic materials up to about 50,000 years old.

How Carbon Dating Works

Carbon-14 is created in the atmosphere when cosmic rays collide with nitrogen atoms
Living organisms absorb C-14 through food and respiration, maintaining equilibrium with atmospheric levels
When an organism dies, it stops absorbing carbon, and its C-14 begins to decay
By measuring remaining C-14, scientists calculate how long ago the organism died

Example: Dating a Wooden Artifact

A piece of ancient wood has 25% of the C-14 found in living wood. How old is it? (C-14 half-life = 5,730 years)

25% remaining = (1/2)ⁿ where n = number of half-lives

0.25 = (1/2)ⁿ

n = 2 half-lives

Age = 2 × 5,730 = 11,460 years

Answer: The wood is approximately 11,460 years old

Limitations of Carbon Dating

Maximum age: ~50,000 years (after ~10 half-lives, too little C-14 remains to measure accurately)
Only works on organic materials: Cannot date rocks, minerals, or synthetic materials
Calibration needed: Atmospheric C-14 levels have varied over time
Contamination: Samples can be contaminated by newer or older carbon

Half-Life in Pharmacology

In medicine, half-life refers to how long it takes for the concentration of a drug in the body to decrease by half. This determines dosing schedules and how long drugs remain effective.

Short Half-Life Drugs

Ibuprofen: 2 hours
Morphine: 2-3 hours
Acetaminophen: 2-4 hours

Need frequent dosing (every 4-6 hours)

Long Half-Life Drugs

Diazepam: 20-100 hours
Fluoxetine (Prozac): 1-6 days
Amiodarone: 40-55 days

Less frequent dosing, longer to clear system

Clinical Implications

Steady state: After about 5 half-lives, drug levels stabilize with regular dosing
Drug clearance: After 5-7 half-lives, the drug is essentially eliminated
Loading doses: Sometimes used for long half-life drugs to reach therapeutic levels quickly
Drug interactions: Can affect half-life by changing metabolism

Other Applications

💡 Electronics: Capacitor Discharge

When a capacitor discharges through a resistor, the voltage follows exponential decay. The time constant τ = RC determines how quickly it discharges. After 5τ, the capacitor is considered fully discharged.

☕ Physics: Cooling (Newton's Law)

Hot objects cool exponentially toward room temperature. A cup of coffee might have a "half-life" for cooling—though the formula is slightly different since it approaches room temperature, not zero.

📱 Technology: Moore's Law (Inverse)

While not decay, Moore's Law (transistor density doubling every ~2 years) is the "half-life" concept in reverse—exponential growth instead of decay.

📊 Finance: Depreciation

Some assets depreciate exponentially. A car might lose half its value every 3-4 years, following a pattern similar to radioactive decay.

Summary: Key Takeaways

Half-life is constant regardless of how much substance remains
After n half-lives, (1/2)ⁿ of the original amount remains
The decay constant λ = ln(2)/t½ ≈ 0.693/t½
Carbon-14 dating works up to ~50,000 years on organic materials
Drug half-lives determine dosing schedules—5 half-lives to steady state or clearance
Exponential decay appears in physics, chemistry, biology, medicine, and many other fields