The relation between half life (T) and decay constant (λ) is:

CONCEPT:

• As per radioactive decay law, the total number of nuclei of radioactive compounds after radioactive decay in the sample is given by given equation

\(N=N_0 e^{-λ t}\)

where N is the number of nuclei of radioactive compounds after radioactive decay, N₀ is the number of nuclei of radioactive compounds initially, λ is the decay constant and t is the time of radioactive decay.

• Half-Life: Half-life is the time required for a radioactive substance to reduce to its half initial value.

CALCULATION:

• From the definition of half-life, radioactive decay in time T (half-life) will be half of its initial value

N = N₀ / 2

so at time t = T, no of nuclei is N = N₀ / 2

where N₀ is the initial number of nuclei.

From radioactive decay law:

\(⇒ N=N_0 e^{-λ t}\)

\(⇒ \frac{N_0}{2}=N_0 e^{-λ T}\)

\(⇒ \frac{1}{2}= e^{-λ T}\)

\(⇒ e^{λ T} = 2\)

Taking log on both the sides

⇒ λ T = log_e 2

• So the correct answer will be option 3.