radioactive decay law

Question

Number remaining after time t: N =? (Initial N0, decay constant λ)

RankGuru · Accepted Answer

1. **Differential Form**: The rate of decay is proportional to the number of radioactive nuclei present:
   - $\frac{dN}{dt} = -\lambda N$
2. **Separation of Variables**: $\frac{dN}{N} = -\lambda dt$.
3. **Integration**: $\ln(N) = -\lambda t + C$.
4. **Boundary Condition**: At $t=0$, $N=N_0$, so $C = \ln(N_0)$.
5. **Exponential Form**: $\ln(N/N_0) = -\lambda t \implies N = N_0 e^{-\lambda t}$.
6. **Conclusion**: Radioactive decay follows an **exponential law**.

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radioactive decay law

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