What do you want to calculate? Remaining Amount (N) Time Elapsed (t) Half-Life (t₁/₂) Initial Amount (N₀)

Time Unit Years Days Hours Minutes Seconds

• After 1 half-life: 50% remains (½ of original)
• After 2 half-lives: 25% remains (¼ of original)
• After 3 half-lives: 12.5% remains (⅛ of original)
• After 4 half-lives: 6.25% remains (1/16 of original)
• After 5 half-lives: 3.125% remains (1/32 of original)
• After n half-lives: (1/2)ⁿ × 100% remains

Time Elapsed Calculation

Used to find how much time has passed given initial and remaining amounts

Identify Known Values

List all the values provided in the problem: initial amount, remaining amount, half-life, or time elapsed. You need at least three to solve for the fourth.

Choose the Right Formula

Select the appropriate equation based on what you're solving for. Use N(t) = N₀ × (½)^(t/t₁/₂) for most decay problems.

Substitute Values

Plug your known values into the formula. Make sure all time units match (don't mix years with days). Double-check your substitution.

Solve & Verify

Calculate the result using logarithms if needed. Verify your answer makes sense (remaining amount should be less than initial, time should be positive, etc.).

Given Information:

• Initial amount (N₀) = 100% (we can use 100 for simplicity)
• Remaining amount (N) = 25% (or 25)
• Half-life (t₁/₂) = 5,730 years
• Time elapsed (t) = ? (what we're solving for)

Solution Steps:

Answer: The fossil is approximately 11,460 years old. This makes sense because 25% remaining means two half-lives have passed (100% → 50% → 25%), and 2 × 5,730 = 11,460 years.

Radiometric Dating

Archaeologists and geologists use half-life to determine the age of fossils, rocks, and artifacts. Carbon-14 dating (half-life 5,730 years) is used for organic materials up to 50,000 years old, while uranium-238 (half-life 4.5 billion years) dates ancient rocks.

Nuclear Medicine

Radioactive isotopes with specific half-lives are used in medical imaging and cancer treatment. Technetium-99m (half-life 6 hours) is ideal for diagnostic scans because it decays quickly, minimizing patient exposure to radiation.

Nuclear Power & Waste

Understanding half-life is crucial for nuclear power plant operations and radioactive waste management. Some nuclear waste products have half-lives of thousands of years, requiring long-term storage solutions.

Pharmacology

Drug half-life determines how long medications remain active in the body. This helps doctors determine proper dosing schedules and predict when a drug will be eliminated from a patient's system.

Environmental Science

Scientists track radioactive contamination from nuclear accidents using half-life calculations. This helps predict when areas will be safe for habitation after radioactive exposure.

Food Preservation

Food irradiation uses radioactive isotopes with known half-lives to kill bacteria and extend shelf life. Understanding decay rates ensures food receives the correct radiation dose without becoming radioactive itself.

Isotope	Half-Life	Common Use
Carbon-14 (C-14)	5,730 years	Archaeological dating
Uranium-238 (U-238)	4.5 billion years	Geological dating
Iodine-131 (I-131)	8 days	Thyroid treatment
Technetium-99m (Tc-99m)	6 hours	Medical imaging
Radon-222 (Rn-222)	3.8 days	Environmental hazard
Plutonium-239 (Pu-239)	24,100 years	Nuclear fuel
Cobalt-60 (Co-60)	5.3 years	Radiation therapy
Strontium-90 (Sr-90)	29 years	Nuclear fallout concern

What is half-life?

Half-life is the time required for a quantity to reduce to half of its initial value. In radioactive decay, it's the time taken for half of the radioactive nuclei in a sample to decay. For example, if a substance has a half-life of 10 years, after 10 years, only 50% of the original amount remains. After another 10 years (20 years total), 25% remains.

What is the formula for half-life?

The primary half-life formula is N(t) = N₀ × (½)^(t/t₁/₂), where N(t) is the remaining amount at time t, N₀ is the initial amount, t is time elapsed, and t₁/₂ is the half-life. Alternatively, you can use the exponential form: N(t) = N₀ × e^(-λt), where λ = 0.693/t₁/₂ is the decay constant.

How do you calculate remaining amount after radioactive decay?

To calculate the remaining amount, use the formula N(t) = N₀ × (1/2)^(t/t₁/₂). Divide the elapsed time by the half-life to get the number of half-lives, raise 0.5 to that power, and multiply by the initial amount. For example, if you start with 100 grams, and 2 half-lives have passed: 100 × (0.5)^2 = 100 × 0.25 = 25 grams remaining.

What happens after one half-life?

After one half-life, exactly 50% of the original radioactive material remains, and 50% has decayed into other elements or isotopes. After two half-lives, 25% remains (half of the remaining 50%). After three half-lives, 12.5% remains. The pattern continues exponentially, with the amount halving after each half-life period.

Can you calculate how old something is using half-life?

Yes, half-life is the foundation of radiometric dating. By measuring the ratio of remaining radioactive isotopes to decay products, scientists can calculate how many half-lives have passed and determine the age. Carbon-14 dating is used for organic materials up to 50,000 years old, while uranium-lead dating can date rocks billions of years old.

Does half-life mean the substance is completely gone?

No, half-life only refers to the time for half to decay. Theoretically, the substance never completely disappears; it approaches zero asymptotically. However, after about 10 half-lives, less than 0.1% remains, which is often considered negligible for practical purposes. The substance continues to decay following the exponential pattern indefinitely.

What factors affect half-life?

Radioactive half-life is a nuclear property that remains constant regardless of external conditions. Temperature, pressure, chemical state, or physical form do not affect the half-life of radioactive isotopes. This constancy makes radioactive decay perfect for dating and timing applications. Each isotope has its own characteristic half-life determined by its nuclear structure.

How is half-life used in medicine?

In medicine, half-life determines how long drugs remain active in the body and helps set dosing schedules. Radioactive isotopes with specific half-lives are used for medical imaging (like Technetium-99m with a 6-hour half-life) and cancer treatment. Short half-lives minimize patient radiation exposure, while longer half-lives provide sustained therapeutic effects.

Match Your Time Units

Always ensure the time elapsed and half-life use the same units. If the half-life is in years, express elapsed time in years. Mixing units (like years and days) will give incorrect results.

Use Logarithms for Complex Problems

When solving for time or half-life, you'll need logarithms. Remember: log(N/N₀) / log(0.5) gives the number of half-lives. Most calculators have a log button for base-10 logarithms.

Check Reasonableness

After calculating, verify your answer makes sense. The remaining amount should always be less than the initial amount. Time should be positive. If you get strange results, recheck your calculations.

Memorize Key Percentages

Remember common decay patterns: 50% after 1 half-life, 25% after 2, 12.5% after 3, 6.25% after 4. These benchmarks help you quickly estimate answers and verify calculations.