The half-life of a radioactive substance is $T$ . The time taken for disintegrating $\frac{7}{8}$ part of its original mass will be:
A
$T$
B
$2T$
C
$3T$
D
$8T$

Question

The half-life of a radioactive substance is T. The time taken for disintegrating 78 part of its original mass will be:

Accepted Answer

Given: The half-life of the radioactive substance is $T$ .

Find: The time required for $\frac{7}{8}$ of the original mass to disintegrate.

Radioactive decay after $n$ half-lives is given by

N = N_0 \left(\frac{1}{2}\right)^n

If $\frac{7}{8}$ of the substance has disintegrated, then the remaining fraction is

1 - \frac{7}{8} = \frac{1}{8}

So,

N = \frac{N_0}{8}

Using the decay relation,

\frac{N_0}{2^n} = \frac{N_0}{8}

Therefore,

2^n = 8 = 2^3

Hence,

n = 3

Since one half-life is $T$ , the total time is

\text{Time} = 3T

Therefore, the correct option is C.

Answer