Half-life Relation

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Half-life T1/2 in terms of λ equals?

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1. **Definition**: Half-life ($T_{1/2}$) is the time taken for half of the initial radioactive nuclei to decay.
2. **Equation**: Set $N = N_0 / 2$ in the decay law $N = N_0 e^{-\lambda t}$.
   - $\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$
   - $1/2 = e^{-\lambda T_{1/2}}$
3. **Taking Natural Log**: $\ln(1/2) = -\lambda T_{1/2} \implies -0.693 = -\lambda T_{1/2}$.
4. **Result**: $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$.

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Half-life Relation

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