- Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus.
- How is the mean life of a given radioactive nucleus related to the decay constant?
CBSE OUTSIDE DELHI - SET 1 NORTH 2016
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- Let there be $\text{N}_{0}$ radioactive nuclei at t =0.
$\frac{-\text{dN}}{\text{dt}}\propto\text{N}$
Or $\frac{-\text{dN}}{\text{dt}} = \lambda\text{N}(\lambda =\text{decay constant })$
$\therefore\frac{\text{dN}}{\text{N}} = -\lambda\text{dt}$
Or $l \text{nN} = - \lambda\text{t} + constant $
$\therefore \text{ At } \text{t} = 0 , $ we have
$l\text{nN}_{0} = constant $
$l \text{nN} = - \lambda\text{t} + l\text{nN}_{0}$
Or $l\text{n}(\frac{\text{N}}{\text{N}_{0}}) = -\lambda\text{t}$
$\therefore \text{N} = \text{N}_{0}\text{e}^{-\lambda\text{t}}$
- Mean life $ = \frac{1}{\text{decay constant }}$
$\tau = \frac{1}{\lambda}$.
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