Final answer:
To calculate the time needed for the decay of 15/16 of radium-226, with a half-life of 1599 years, we determine that it takes 4 half-lives, resulting in 6396 years for this amount of decay.
Step-by-step explanation:
To calculate the time required for the decay of 15/16 of a given amount of radium-226, we utilize its half-life, which is 1599 years. The process of radioactive decay is exponential, meaning after each half-life, half of the remaining radium-226 would have decayed. To decay 15/16 of the original amount, we need to calculate the number of half-lives required to leave 1/16, since 15/16 would have decayed at that point.
To find the number of half-lives needed, we can use the formula:
1/2n = Remaining fraction of the substance
n = Number of half-lives
In this case, we have:
1/2n = 1/16
n = 4 (because 1/24 = 1/16)
Thus, it will take 4 half-lives for 15/16 of the radium-226 to decay.
Since one half-life is 1599 years:
Time = number of half-lives × half-life = 4 × 1599 years = 6396 years.
So, it will take 6396 years for 15/16 of the given amount of radium-226 to decay.