Question
- Write symbolically the $\beta^– $ decay process of $\frac{15}{32}P$.
- Derive an expression for the average life of a radionuclide. Give its relationship with the half $-$ life.
✓
Answer
$\beta^–$ decay process
$^{32}_{15}\text{P}\rightarrow^{32}_{16}\text{S} + \text{e}^{-} + \overline{\text{v}}$ or $^{32}_{15}\text{P}\rightarrow^{32}_{15}\text{X} + _{-1}\text{e}^{0} + \overline{\text{v}}$
Derivation of average life:
$\tau=\frac{\lambda\text{N}_{0}\int\limits_{0}^{\infty}\text{te}^{-\lambda\text{t}}\text{dt}}{\text{N}_{0}} = \lambda\int\limits_{0}^{\infty}\text{te}^{-\lambda\text{t}}\text{dt}$
$\Rightarrow\tau = 1/ \lambda$
Relation of average life with half life:
$\text{T}_{1/2} = \frac{\ell\text{n}2}{\lambda} = \tau\ell\text{n} 2$
$^{32}_{15}\text{P}\rightarrow^{32}_{16}\text{S} + \text{e}^{-} + \overline{\text{v}}$ or $^{32}_{15}\text{P}\rightarrow^{32}_{15}\text{X} + _{-1}\text{e}^{0} + \overline{\text{v}}$
Derivation of average life:
$\tau=\frac{\lambda\text{N}_{0}\int\limits_{0}^{\infty}\text{te}^{-\lambda\text{t}}\text{dt}}{\text{N}_{0}} = \lambda\int\limits_{0}^{\infty}\text{te}^{-\lambda\text{t}}\text{dt}$
$\Rightarrow\tau = 1/ \lambda$
Relation of average life with half life:
$\text{T}_{1/2} = \frac{\ell\text{n}2}{\lambda} = \tau\ell\text{n} 2$
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