Half Life & Decay Constant — N = N₀e^(-λt)
Nuclei
7
JEE Qs
8%
Hard
75
min
Master the exponential decay law and its logarithmic relations for half-life and mean life to efficiently solve problems involving time-dependent radioactive processes.
🧮 Key Formulas
✅ Key Points for JEE
- 1Radioactive decay is a first-order process, meaning the decay rate is proportional to the number of undecayed nuclei present.
- 2The decay constant (λ) represents the probability of a nucleus decaying per unit time; it's an intrinsic property of the isotope.
- 3Half-life (T_½) is the time required for half of the initial radioactive nuclei in a sample to decay, not necessarily half of the mass of the sample.
- 4Mean life (τ) is the average lifetime of an unstable nucleus; it is always greater than the half-life (τ ≈ 1.44 T_½).
- 5The exponential decay law N = N₀e^(-λt) applies to the number of undecayed nuclei, their mass, or the activity of the sample, but not the number of *decayed* nuclei directly.
⚠️ Common Mistakes
- ✕Confusing half-life with mean life or applying them interchangeably.
- ✕Assuming linear decay over time instead of exponential decay, especially when calculating remaining nuclei after non-integer multiples of half-lives.
- ✕Incorrectly interpreting 'N' as the number of *decayed* nuclei or total mass, rather than the *remaining undecayed* radioactive nuclei.
- ✕Errors in logarithmic calculations or unit conversions (e.g., using different time units for λ and t).
📝 Practice Questions
See allQ32.Choose the correct nuclear process from the below options [p: proton, n : neutron, e− : electron, e+ : positron, v : neutrino, ¯v : antineutrino] (1) n →p + e+ + ¯v (2) n →p + e+ + v (3) n →p + e−+ v (4) n →p + e−+ ¯v
Q40.A radioactive nucleus n2 has 3 times the decay constant as compared to the decay constant of another radioactive nucleus n1 . If initial number of both nuclei are the same, what is the ratio of number of nuclei of n2 to the number of nuclei of n1 , after one half-life of n1 ? (1) 1/8 (2) 8 (3) 4 (4) 1/4
Q18.Binding energy of a certain nucleus is 18 × 108 J. How much is the difference between total mass of all the nucleons and nuclear mass of the given nucleus: (1) 10μg (2) 20μg (3) 0.2μg (4) 2μg
Q19.Which of the following nuclear fragments corresponding to nuclear fission between neutron (10n) and uranium isotope (23592 U) is correct : (1) 144 56 Ba + 8936Kr + 410n (2) 14456 Ba + 8936Kr + 310n (3) 140 56 Xe + 9438Sr + 310n (4) 15351 Sb + 9941Nb + 310n
Q30.If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is _____ ×10−2MeV . (Given 1u = 931MeV/c2 , atomic mass of helium = 4.002603u )
Q30.The radius of a nucleus of mass number 64 is 4 . 8 fermi. Then the mass number of another nucleus having radius of 4 fermi is 1000 where 𝑥 is _________. 𝑥,
NCERT Chapters
- Class 12 Physics Ch 13: Nuclei