Final answer:
Calculating the time to decay from 9.0 grams of beryllium-11 to 0.28 grams involves determining the number of half-lives by using logarithms and then multiplying by the half-life duration, which results in approximately 71.00 seconds.
Step-by-step explanation:
The time needed for an initial amount of 9.0 grams of beryllium-11 to decay to 0.28 grams can be calculated using the concept of half-lives. Given the half-life of beryllium is 13.81 seconds, we can determine the number of half-lives needed for the initial amount of beryllium to reduce to 0.28 grams. This requires setting up and solving a mathematical equation involving exponentiation.
After each half-life, the amount of beryllium remaining is halved. To find the number of half-lives (n) that have passed, one can use the equation:
final amount = initial amount × (1/2)^n
Substituting the given values into the equation, we get:
0.28 g = 9.0 g × (1/2)^n
Dividing both sides by 9.0 g and then taking the logarithm to solve for n:
n = log(0.28/9)/log(0.5) = log(0.0311111)/log(0.5) ≈ 5.14
Now, multiply the number of half-lives by the half-life duration:
5.14 × 13.81 seconds = 70.99434 seconds
Thus, it takes approximately 71.00 seconds (rounded to the nearest hundredth of a second) for 9.0 grams of beryllium-11 to decay to 0.28 grams.