First slide

Kinetics of nuclear reactions or nuclear chemistry

Question

The nuclei of a certain radioactive element are produced at a constant rate of $α$ per second. This element has a decay constant $λ$ . Initially $N_{0}$ nuclei of the element are present.

Moderate

Question

What is the number N of nuclei of this element at a time t?

Solution

$\frac{d N}{d t} = (α - λ N)$
$\int_{N_{0}}^{N} \frac{d N}{α - λ N} = \int_{0}^{t} d t$
$\frac{l n (\frac{α - λ N}{α - λ N_{0}})}{- λ} = t$
$λ t = l n (\frac{α - λ N_{0}}{α - λ N})$
$e^{λ t} (α - λ N) = α - λ N_{0}$
$α e^{λ t} - λ N e^{λ t} = α - λ N_{0}$
$N = \frac{α e^{λ t} + λ N_{0} - α}{(λ e^{λ t})} = \frac{1}{λ} [α + λ N_{0} e^{- λ t} - α e^{- λ t}]$
$N = \frac{1}{λ} [α (1 - e^{- λ t}) + λ N_{0} e^{- λ t}]$

Question

If in the above question $α = 2 λ N_{0}$ calculate the number of nuclei left in the element after one half life period of the element.

Solution

$λ t = l n (\frac{2 λ N_{0} - λ N_{0}}{2 λ N_{0} - λ N})$
$λ \frac{l n 2}{λ} = l n (\frac{λ N_{0}}{2 λ N_{0} - λ N})$
$2 = \frac{λ N_{0}}{2 λ N_{0} - λ N}$
$4 λ N_{0} - 2 λ N = λ N_{0}$
$N = \frac{3}{2} N_{0}$

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