ACT Science: Radioactive Decay and Half-Life—Carbon Dating and Age Calculation

Half-life (t_half) is the time for half of a radioactive sample to decay. After one half-life, 50% remains. After two half-lives, 25% remains. After three, 12.5%, and so on. Formula: N=N₀(0.5)^(t/t_half), where N=amount remaining, N₀=initial amount, t=elapsed time. Half-life is a constant for each isotope; it does not change based on initial amount or temperature. Carbon-14 half-life is 5,730 years (used for dating organic materials). Uranium-238 half-life is 4.5 billion years (used for dating rocks).

Example: Carbon-14 sample has 12.5% of original ¹⁴C. How many half-lives have passed? 100% → 50% → 25% → 12.5% = 3 half-lives. Age = 3 × 5,730 = 17,190 years.

Scenario 1 (Find remaining amount): A sample of ¹⁴C has mass 100 g. After 11,460 years (2 half-lives), how much remains? N=100(0.5)²=100(0.25)=25 g. Scenario 2 (Find age): A fossil has 6.25% of original ¹⁴C. How old? 6.25% = 1/16 = (0.5)⁴. So 4 half-lives have passed. Age = 4 × 5,730 = 22,920 years. Scenario 3 (Find half-life): A sample decays to 12.5% in 1,200 years. What is the half-life? (0.5)^(1200/t_half) = 0.125 = (0.5)³. So 1200/t_half = 3 → t_half = 400 years. Always count the number of half-lives (how many times the sample halves) and calculate from there.

Verify: After each half-life, amount should be multiplied by 0.5.

Problem 1: A ¹⁴C sample of 50 g decays for 17,190 years (3 half-lives). Amount remaining? N=50(0.5)³=50(0.125)=6.25 g. Problem 2: A sample shows 25% of original ¹⁴C. How many half-lives? 25%=0.25=(0.5)². So 2 half-lives. Age=2×5,730=11,460 years. Problem 3: An isotope with half-life 30 years decays from 1000 g to 31.25 g. How many years? 31.25%=1/32=(0.5)⁵. So 5 half-lives. Time=5×30=150 years. Complete all three daily until half-life calculations are automatic.

Remember: Half-life means cutting in half. After n half-lives, remaining = initial × (0.5)^n.

Half-life and radioactive decay questions appear in 1-2 ACT Science passages. They are computational once you know the formula and understand exponential decay. Investing 20 minutes in half-life yields 1-2 guaranteed points because the concept is straightforward and the formula is provided.

Master half-life one day before the test. By test day, dating and decay problems become routine.