Why things do not decay evenly
Imagine 100 grams of a radioactive sample with a half-life of 5 years. After 5 years you have 50 grams left. A common intuition says the next 5 years removes another 50 grams, but that is wrong: you have 25 grams, not zero. Each half-life removes half of what remains, not half of the original. This pattern, exponential decay, governs radioactive isotopes, drug clearance from the bloodstream, The half-life calculator computes how much remains after any elapsed time, or how long it takes to reach a target amount.
The decay formula
Formula: N(t) = N0 x (1/2)^(t / half-life)
N0 is the starting amount, t is the elapsed time, and the half-life is the time for half the material to decay. The exponent t / half-life counts how many half-lives have passed, and raising 1/2 to that power gives the surviving fraction. Both t and the half-life must be in the same time unit.
How to use the calculator
- Enter the initial amount N0 in grams, becquerels, or any unit. The unit carries through to the answer.
- Enter the half-life and the elapsed time, both in the same unit (years, days, hours).
- Read the remaining amount and the decayed fraction.
- For the reverse question, how long until only X remains, enter the target fraction and read the required time.
Worked example, step by step
A sample starts at 100 g with a 5-year half-life. How much remains after 10 years?
Step 1: Count the half-lives. 10 / 5 = 2 half-lives.
Step 2: Apply the fraction. (1/2)^2 = 1/4 = 0.25.
Step 3: Multiply by the start. 100 x 0.25 = 25 g.
Step 4: Check the intuition path. After 5 years: 50 g. After 10 years: half of 50 = 25 g. Consistent.
Step 5: Try a non-integer case. After 7.5 years: exponent = 7.5 / 5 = 1.5, so (1/2)^1.5 = 0.3536, and 100 x 0.3536 = 35.36 g. The calculator handles these fractional exponents where mental math gets tedious.
Decay problems often sit next to gas and solution chemistry, so the STP calculator is kept as the related chemistry page, and amount-of-substance conversions go through the grams to moles calculator and the mole fraction calculator.
Practical uses
Archaeologists date organic finds with carbon-14 (half-life 5,730 years) by measuring the surviving fraction. Doctors use biological half-lives to schedule drug doses so the concentration stays in the effective window. Nuclear waste planning uses half-lives of thousands of years to model long-term hazard. Even coffee cooling follows a half-life-like law (Newton's law of cooling), which is why a hot drink loses most of its excess heat in the first few minutes.
Common mistakes to avoid
- Subtracting half each time from the original. The decay acts on the current amount, not the starting amount. Linear thinking gives zero after two half-lives; the correct answer is a quarter.
- Mixing time units. A half-life in years with an elapsed time in days must be converted first, or the exponent is meaningless.
- Confusing half-life with mean lifetime. The mean lifetime is half-life / 0.693, about 44 percent longer. Both appear in textbooks, and swapping them skews every downstream number.
A note on limits
The formula describes the average behavior of a large population of atoms or molecules. With a handful of atoms, decay is statistical and the smooth curve breaks down into discrete random events. It also assumes a single decay process with one half-life; real samples with multiple isotopes are a sum of exponentials, which the single-formula calculator does not model.