The half-life of radium is 1622 years. How long will it take for seven-eighth of a given amount of radium to d

Question

The half-life of radium is 1622 years. How long will it take for seven-eighth of a given amount of radium to decay?

Accepted Answer

1. **Determine Fraction Remaining:** The problem states that $7/8$ of the radium decays. Therefore, the fraction of the nuclei remaining undecayed is:
   $$ \frac{N}{N_0} = 1 - \frac{7}{8} = \frac{1}{8} $$
2. **Apply Radioactive Decay Law:** Radioactive decay follows first-order kinetics. The amount of substance remaining $N$ after $n$ half-lives is given by the formula $N = N_0 (\frac{1}{2})^n$ .
3. **Calculate Number of Half-lives ($n$):**
   $$ \frac{1}{8} = \left(\frac{1}{2}ight)^n \Rightarrow \left(\frac{1}{2}ight)^3 = \left(\frac{1}{2}ight)^n $$
   Comparing the powers, we get $n = 3$ half-lives.
4. **Calculate Total Time ($t$):**
   The total time elapsed is the number of half-lives multiplied by the half-life period ($T_{1/2}$).
   $$ t = n 	imes T_{1/2} = 3 	imes 1622 	ext{ years} = 4866 	ext{ years} $$ .

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