A sample of radioactive element has a mass of 10 gm at an instant $t = 0$. The approximate mass of this elemen

Question

A sample of radioactive element has a mass of 10 gm at an instant $t = 0$. The approximate mass of this element in the sample after two mean lives is:

Accepted Answer

1. **Formula:** The mass $M$ of a radioactive substance remaining after time $t$ follows the exponential decay law: $M = M_0 e^{-\lambda t}$, where $M_0$ is the initial mass and $\lambda$ is the decay constant .
2. **Mean Life:** The mean life ($	au$) is the reciprocal of the decay constant: $	au = 1/\lambda$ .
3. **Substitution:** We are given $t = 2	au = 2/\lambda$.
   Substituting this into the mass equation:
   $$ M = 10 e^{-\lambda (2/\lambda)} = 10 e^{-2} $$
4. **Calculation:** Using the value of Euler's number $e \approx 2.718$:
   $$ M = \frac{10}{(2.718)^2} \approx \frac{10}{7.389} \approx 1.353 	ext{ gm} $$
5. **Conclusion:** The approximate mass remaining is 1.35 gm.

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