A 6.95 gram sample of radioactive nobelium-259 has a half-life of 58 minutes. How much nobelium-259 remains after 2 hours and 54 minutes? Hint: Find the number of half-lives that have passed, then reduce the initial amount by half (divide by 2) for every half-life that passed. Or complete the table: Amount of Amount of Ratio of isotopes # of half-lives parent isotope (grams) daughter isotope (grams) (parent: daughter) Total time passed (minutes) 0 0 0 1.74 grams 3.48 grams 0.434 grams 0.869 grams
The Correct Answer and Explanation is:
Step 1: Calculate the number of half-lives
The given half-life of nobelium-259 is 58 minutes. The total time passed is 2 hours and 54 minutes, which is:
2×60+54=174 minutes2 \times 60 + 54 = 174 \text{ minutes}
The number of half-lives is:
17458=3 half-lives\frac{174}{58} = 3 \text{ half-lives}
Step 2: Apply the half-life formula
Each half-life reduces the original amount by half. The formula for remaining quantity after n half-lives is:
Final amount=Initial amount×(12)n\text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n
Substituting the given values:
6.95×(12)36.95 \times \left(\frac{1}{2}\right)^3
6.95×186.95 \times \frac{1}{8}
=0.86875 grams= 0.86875 \text{ grams}
Explanation:
In radioactive decay, the half-life is the time required for half of the radioactive material to decay. Nobelium-259 undergoes decay, transforming into daughter isotopes, while reducing in quantity every 58 minutes. The calculation above shows that after 3 half-lives (174 minutes), only 0.869 grams of nobelium-259 remain.
At the first half-life (58 minutes), half of the nobelium-259 decayed, leaving:
6.95÷2=3.48 grams6.95 \div 2 = 3.48 \text{ grams}
At the second half-life (116 minutes), half of 3.48 grams remains:
3.48÷2=1.74 grams3.48 \div 2 = 1.74 \text{ grams}
At the third half-life (174 minutes), half of 1.74 grams remains:
1.74÷2=0.869 grams1.74 \div 2 = 0.869 \text{ grams}
Thus, after 2 hours and 54 minutes, approximately 0.869 grams of nobelium-259 remains. This predictable pattern of radioactive decay helps scientists understand nuclear reactions, radiometric dating, and the behavior of unstable isotopes in nature and laboratories.
