Physics · nuclei · NEET
After n half-lives the fraction of nuclei still remaining is (1/2)^n. So after 1 half-life 1/2 is left, after 2 half-lives 1/4 is left, after 3 half-lives 1/8 is left, and so on. Just multiply N0 by (1/2) once for each half-life. Example: after 4 half-lives, fraction remaining = (1/2)^4 = 1/16.
Use n = t / T, where t is the total time elapsed and T is the half-life. For example, if the half-life is 5 years and the total time is 20 years, then n = 20/5 = 4 half-lives. Then plug n into N = N0 x (1/2)^n. Here N = N0 x (1/2)^4 = N0/16, so 1/16 of the sample remains.
Fraction remaining is the part still undecayed = (1/2)^n. Fraction decayed is the part that has broken down = 1 - (1/2)^n. Many students give the wrong one. Example: after 3 half-lives, fraction remaining = 1/8, so fraction decayed = 1 - 1/8 = 7/8. Always read whether the question asks for what is LEFT or what has DECAYED.
The same formula works for number of nuclei, mass, or activity, because all three are proportional. If you start with mass m0, then mass left = m0 x (1/2)^n. Example: 80 g sample, half-life 2 hours, after 6 hours: n = 6/2 = 3, so mass left = 80 x (1/2)^3 = 80/8 = 10 g.
When n is not a whole number, you cannot just halve step by step. Use the full formula N = N0 e^(-lambda t), or equivalently N = N0 x (1/2)^(t/T). For example, half-life 4 days and t = 6 days gives n = 6/4 = 1.5, so N = N0 x (1/2)^1.5 = N0 x 0.354, about 35.4% remaining.
Fraction remaining = (1/2)^n, so N = N0 x (1/2)^n. Here N0 is the starting number of nuclei and n is the number of half-lives that have passed.
n = t / T, where t is the total time and T is the half-life. This n can be a whole number or a decimal.
Yes. Number of nuclei, mass, and activity all fall by the same factor (1/2)^n, so the same formula applies to all three.
(1/2)^10 = 1/1024, which is about 0.098%. So after 10 half-lives, less than 0.1% of the original sample remains.
87.5% decayed means 12.5% = 1/8 remains. 1/8 = (1/2)^3, so n = 3 half-lives. Total time = 3 x T.
