The radioactivity of a sample is R_1 at a time T_1 and R_2 at a time T_2. If the half-life of the sp — Nuclear Chemistry and Radioactivity Chemistry Question

💡 Solution & Explanation

Step 1 - Radioactivity and Number of Active Atoms $$R = \lambda N \implies N = \frac{R}{\lambda}$$ At time $T_1$: $N_1 = \dfrac{R_1}{\lambda}$; at time $T_2$: $N_2 = \dfrac{R_2}{\lambda}$ Step 2 - Atoms Disintegrated in Interval $(T_2 - T_1)$ $$\Delta N = N_1 - N_2 = \frac{R_1}{\lambda} - \frac{R_2}{\lambda} = \frac{R_1 - R_2}{\lambda}$$ Step 3 - Substitute $\lambda = 0.693/T$ $$\Delta N = \frac{R_1 - R_2}{0.693/T} = \boxed{\frac{(R_1 - R_2)\,T}{0.693}}$$ Step 4 - Evaluate Options - **(A) $(R_1 T_1 - R_2 T_2)$** — Incorrect. Multiplying activity by absolute time is dimensionally wrong for first-order decay. - **(B) $(R_1 - R_2)$** — Incorrect. This is the change in activity (units: s$^{-1}$), not the number of atoms. - **(C) $(R_1 - R_2)/T$** — Incorrect. Dividing by half-life gives the wrong relationship. - **(D) $\dfrac{(R_1 - R_2) T}{0.693}$** — Correct. Directly derived from $N = R/\lambda$ and $\lambda = 0.693/T$.