Find the half-life of $U^{238}$, if one gram of it emits $1.24 * 10^4$ $\alpha - particles$ per second. (Avogadro's Number = $6.023 * 10^{23}$).

Given,
$\frac{dN}{dt}$ $ = 1.24 * 10^{4}$
Half Life $(T_{1/2}) = ?$

But,
238 gm of U contains $6.023 * 10^{23}$ atoms

1 gm of U contains $\frac{6.023 \; * \; 10^{23}}{238}$ $ = 2.51 * 10^{21}$

⇒ $N = 2.51 * 10^{21}$ atoms

We have,

$\frac{dN}{dt} = \lambda \; N = \frac{ln(2)}{T_{1/2}} * N$

$T_{1/2} = \frac{0.693 \; * \; N}{\frac{dN}{dt}} = \frac{0.693 \; * \; 2.51 \; * \; 10^{21}}{1.24 \; * \; 10^4}$  

        $ = 1.41 * 10^{17}\; Sec = 4.5 * 10^9\; years$

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