Half-Life Calculator
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Understanding Half-Life Calculations
Half-life is the time required for a quantity to reduce to half its initial value. It's commonly used in nuclear physics, chemistry, medicine, and other fields to describe exponential decay processes.
1. The Half-Life Formula
The fundamental formula for half-life calculations is:
Where:
- N = remaining quantity
- N₀ = initial quantity
- t = elapsed time
- t½ = half-life
Practical Example: Radioactive Decay
If a radioactive isotope has a half-life of 5 years, and you start with 100g:
After 5 years: 100 × (1/2)^(5/5) = 50g
After 10 years: 100 × (1/2)^(10/5) = 25g
After 15 years: 100 × (1/2)^(15/5) = 12.5g
2. Calculating Half-Life (t½)
When you know the initial and remaining quantities and the elapsed time, you can calculate the half-life:
Where ln is the natural logarithm.
Example: Determining Half-Life
If you start with 200g of a substance, and after 3 years you have 50g remaining:
t½ = 3 × ln(2) / ln(200/50) ≈ 1.5 years
The half-life is approximately 1.5 years
3. Calculating Remaining Quantity (N)
When you know the initial quantity, half-life, and elapsed time:
Example: Medicine Elimination
A drug with a half-life of 8 hours in the body:
Dose: 400mg
After 24 hours: 400 × (1/2)^(24/8) = 50mg
After 24 hours (3 half-lives), 50mg remains in the body
4. Calculating Elapsed Time (t)
When you know the initial and remaining quantities and the half-life:
Example: Archaeological Dating
Carbon-14 has a half-life of 5730 years. If a sample has 25% of its original C-14:
t = 5730 × ln(1/0.25) / ln(2) ≈ 11460 years
The sample is approximately 11,460 years old
Common Half-Life Applications
1. Radioactive Dating
Different isotopes are used to date materials of different ages:
- Carbon-14: 5,730 years (archaeology, geology)
- Potassium-40: 1.25 billion years (geology)
- Uranium-238: 4.5 billion years (geology)
2. Medical Applications
Radioactive isotopes are used in diagnosis and treatment:
- Technetium-99m: 6 hours (diagnostic imaging)
- Iodine-131: 8 days (thyroid treatment)
- Cobalt-60: 5.27 years (cancer treatment)
3. Pharmacology
Drug half-life determines dosing schedules:
- Aspirin: 3-4 hours
- Ibuprofen: 2-4 hours
- Diazepam: 20-50 hours
Frequently Asked Questions
Q: What does half-life mean?
A: Half-life is the time it takes for half of the radioactive atoms present to decay. After one half-life, half of the original atoms remain. After two half-lives, one quarter remain, and so on.
Q: How is half-life related to decay rate?
A: Half-life (t½) and decay constant (λ) are related by: t½ = ln(2)/λ. The shorter the half-life, the faster the decay rate.
Q: Can half-life be changed?
A: Normally, half-life is a constant for a given isotope and can't be changed by physical or chemical means. However, some nuclear processes can affect decay rates slightly.
Q: What's the difference between biological half-life and radioactive half-life?
A: Radioactive half-life is the time for half the atoms to decay. Biological half-life is the time for half the substance to be eliminated from the body by biological processes.
Q: How accurate is carbon dating?
A: Carbon-14 dating is typically accurate to about ±30-50 years for samples up to 10,000 years old, with decreasing accuracy for older samples due to uncertainties in historical atmospheric C-14 levels.