MP Board · Class 12 · Physics · NucleiThe half-life of a radioactive substance is 30 days. Calculate the time taken for $7/8^{\text{th}}$ of the original mass of the substance to disintegrate.

Given data:

• Half-life ($T_{1/2}$) = 30 days
• Fraction disintegrated = $\frac{7}{8}$ \nStep 1: Find the fraction of the substance remaining undecayed ($N$):\nRemaining fraction $\frac{N}{N_0} = 1 - \frac{7}{8} = \frac{1}{8}$ \nStep 2: Relate the remaining fraction to the number of half-lives ($n$):\nWe know the radioactive decay law formula: $\frac{N}{N_0} = \left(\frac{1}{2}\right)^n$ \nSubstitute the remaining fraction: $\frac{1}{8} = \left(\frac{1}{2}\right)^n$\nSince $\frac{1}{8} = \left(\frac{1}{2}\right)^3$, we get: $n = 3$ \nStep 3: Calculate the total time ($t$):\nThe total time is the product of the number of half-lives and the half-life period: $t = n \times T_{1/2}$ $t = 3 \times 30 \text{ days} = 90 \text{ days}$ \nThus, the time taken for $7/8^{\text{th}}$ of the substance to disintegrate is 90 days.