Q. Find the fraction of thorium atoms decayed per year if the half-life of thorium $232$ is $1.4\times 10^{10}years$ :
NTA AbhyasNTA Abhyas 2020
Solution:
As we know,
given $t_{1/2}=1.4\times 10^{10}years$
$N=N_{0}e^{- \lambda t}$
$\frac{d N}{d t}=-\lambda N\Rightarrow \frac{d N}{N}=-\lambda .dt\ldots \ldots \ldots \left(1\right)$
$\lambda =\frac{ln 2}{t_{1/2}}\ldots \ldots \ldots \left(2\right)$
thus, $\frac{d N}{N}=\frac{-ln2.dt}{t_{1/2}}=\frac{-0.693\times 1}{1 . 4 \times 10^{10}}=-4.95\times 10^{- 11}$
given $t_{1/2}=1.4\times 10^{10}years$
$N=N_{0}e^{- \lambda t}$
$\frac{d N}{d t}=-\lambda N\Rightarrow \frac{d N}{N}=-\lambda .dt\ldots \ldots \ldots \left(1\right)$
$\lambda =\frac{ln 2}{t_{1/2}}\ldots \ldots \ldots \left(2\right)$
thus, $\frac{d N}{N}=\frac{-ln2.dt}{t_{1/2}}=\frac{-0.693\times 1}{1 . 4 \times 10^{10}}=-4.95\times 10^{- 11}$