Question

The time in which activity of an element reduces to $90\%$ of its original value is:
The half-life period of element is $1.4\times {10}^{10}$ years.

• A. $2.128\times {10}^{9}years$
• B. $1.578\times {10}^{7}years$
• C. $6.954\times {10}^{7}years$
• D. none of these

Answer 1

The correct option is A $2.128\times {10}^{9}years$

The half-life period of element is $1.4\times {10}^{10}$ years.
The decay constant is $\lambda =\frac{0.693}{t_{1/2}}=\frac{0.693}{1.4\times {10}^{10}}$
The activity of an element reduces to 90% of its original value.
$r_1=\frac{90}{100}{r}_0$
$\frac{r_0}{r_t}=\frac{100}{90}$
Rate of decay at any time is directly proortional to the number of atoms.
$\frac{r_0}{r_1}=\frac{N_0}{N_t}=\frac{100}{90}$
Now, $t=\frac{2.303}{\lambda }log\frac{N_0}{N_1}$
$=\frac{2.303\times 1.4\times {10}^{10}}{0.693}log\frac{100}{90}$ $=2.128\times {10}^{9}years$
Thus, the time in which activity of an element reduces to $90\%$ of its original value is $=2.128\times {10}^{9}years$.