Which of the following is the correct equation for the law of radioac…

The correct answer is Option (2) → \( N(t) = N_{0}e^{-\lambda t} \)

The law of radioactive decay says that the rate of decay is proportional to the number of undecayed nuclei present, with the decay constant \( \lambda \) as the constant of proportionality.

    \( \displaystyle \frac{dN}{dt} = -\lambda N \)

Integrating from \( N_0 \) at \( t = 0 \) to \( N \) at time \( t \):

    \( \displaystyle \ln\frac{N}{N_0} = -\lambda t \;\Rightarrow\; N(t) = N_{0}e^{-\lambda t} \)

Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.

NCERT (Pre-rationalised edition, 2022-23):
Law of Radioactive Decay – “… Here N₀ is the number of radioactive nuclei in the sample at some arbitrary time t₀ and N is the number of radioactive nuclei at any subsequent time t. Setting t₀ = 0 and rearranging Eq. (13.12) gives us

    \( \displaystyle \ln\frac{N}{N_0} = -\lambda t \qquad (13.13) \)

which gives

    \( \displaystyle N(t) = N_0\, e^{-\lambda t} \qquad (13.14) \)

… The decay of radionuclides follows quite a different law, the law of radioactive decay represented by Eq. (13.14).”