Half-Life Calculator – Radioactive Decay
Calculate remaining amount after radioactive decay using the half-life formula.
About This Calculator
Calculate remaining amount after radioactive decay using the half-life formula. Use the calculator above for instant results.
Understanding Exponential Half-Life Decay
Half-life is the time required for an exponentially decaying quantity to fall to one-half of its current amount. The key phrase is its current amount. After one half-life, 50% remains; after two, 25% remains; after three, 12.5% remains. Equal time intervals therefore remove equal fractions, not equal absolute amounts.
Here N0 is the initial amount, N(t) is the expected remaining amount, t is elapsed time, and T1/2 is the half-life. The ratio t/T1/2 counts how many half-lives have elapsed. It does not have to be an integer. After 2.5 half-lives, the remaining fraction is (1/2)2.5 ≈ 0.1768, or 17.68%.
The same decay law can be written N(t)=N0e−λt, where the decay constant is λ = ln(2)/T1/2. This form is useful when exponential functions are written with base e. For radioactive atoms, the formula describes an expected population: individual nuclei decay randomly, but a large collection follows the exponential law very closely.
Keep time units consistent. If the half-life is in hours, elapsed time must also be expressed in hours before forming t/T1/2. The output retains the same amount unit as N0, whether that is grams, milligrams, activity units, or a count.
Half-life is independent of the starting amount in the exponential model. A 10 g sample and a 100 g sample of the same isotope lose the same fraction during one half-life, although the larger sample loses more grams. This is another way to recognize exponential rather than linear change.
Worked Examples
- 3 half-lives -> 100 x 0.5^3 = 12.5g
Answer: 12.5g
- 4 half-lives -> 1000 x 0.5^4 = 62.5 atoms
Answer: 62.5
- (0.5)^(10000/5730) = 29.8% remaining
Answer: 29.8%
An 80 mg sample has a half-life of 6 hours and decays for 18 hours.
- 18/6 = 3 half-lives.
- 80(1/2)3 = 10 mg.
Answer: 10 mg remains
For t = 2.5T1/2, the remaining fraction is (1/2)2.5.
- (1/2)2.5 ≈ 0.1768.
Answer: about 17.68% remains
Who Uses This Calculator?
Radioactive decay calculations.
Nuclear chemistry.
Medical isotope dosing.
Radiometric dating.
Common Mistakes to Avoid
❌ Units must match
Half-life and elapsed time must use the same units.
❌ Never fully reaches zero
After 2 half-lives: 25% remains. Approaches zero asymptotically.
❌ Subtracting one-half of the original amount each period
Half-life removes half of what remains. Repeated subtraction would become negative, while exponential decay approaches zero without crossing it.
❌ Mixing time units
A half-life of 30 minutes and an elapsed time of 2 hours must be converted to common units before using t/T1/2.
Frequently Asked Questions
Formula?
N(t) = N0 x (0.5)^(t/t½).
Carbon dating?
C-14 half-life 5,730 years. Used to date organic material up to ~50,000 years.
Which elements radioactive?
All elements with atomic number > 83. Common: Uranium-238, Carbon-14, Iodine-131.
Can a half-life calculation use a non-integer number of half-lives?
Yes. The exponent t/T1/2 can be any nonnegative real number. Exponential decay is continuous, so 1.4 or 2.75 half-lives is mathematically meaningful even though classroom examples often use whole numbers.
How is half-life related to the decay constant?
For N=N0e−λt, the relationship is λ = ln(2)/T1/2. A shorter half-life therefore corresponds to a larger decay constant and faster exponential decrease.
Does the amount ever become exactly zero in the formula?
No. The continuous exponential model approaches zero as time increases but never reaches zero at a finite time. For a small discrete number of radioactive nuclei, however, the actual count can eventually reach zero.
Why can a calculated number of atoms be fractional?
The exponential equation gives an expected amount for a population. An actual atom count must be an integer, but an expectation such as 62.5 means that repeated comparable samples would average to that value.
Formula Explorer connections
Interpretation: This relationship converts, summarizes or checks numerical quantities using standard arithmetic and measurement rules. Assumption: Use consistent units, preserve enough significant digits, and round only the final result unless the method states otherwise.