The half-life of thorium-227 is 18.72 days. How many days are required for three-fourths of a given amount to decay?
Answer (3)
t½=18.72days therefore t¾=18.72+½ of 18.72 we have 18.72+9.36=28.08days
"The correct answer is approximately 37.44 days are required for three-fourths of a given amount of thorium-227 to decay.
To solve this problem, we need to understand the concept of half-life. The half-life of a radioactive substance is the time required for half of the substance to decay. If we denote the half-life by t 2 1 ,[/tex] then after one half-life ,[tex] 2 1 of the substance remains, after two half-lives, 4 1 remains, and so on.
Given that the half-life of thorium-227 is 18.72 days, we want to find out how many half-lives are needed for three-fourths of the substance to decay. When three-fourths of the substance has decayed, one-fourth remains. This is equivalent to one half-life, because 4 1 is half of 2 1 , and the definition of half-life is the time it takes for half of the substance to decay.
So, if after one half-life 4 1 of the substance remains, and we know that 4 3 has decayed, we can conclude that it takes one half-life for 4 3 of the substance to decay.
Therefore, the time required for three-fourths of a given amount of thorium-227 to decay is simply the half-life of thorium-227, which is 18.72 days.
However, the question asks for the time required for three-fourths of the substance to decay, not for one-fourth to remain. Since one half-life results in one-fourth remaining, two half-lives would result in one-eighth remaining, which means that three-fourths of the substance would have decayed after two half-lives.
To find the time for two half-lives, we simply multiply the half-life by 2:
18.72 days × 2 = 37.44 days
Thus, approximately 37.44 days are required for three-fourths of a given amount of thorium-227 to decay."
Approximately 37.44 days are required for three-fourths of a given amount of thorium-227 to decay, as this occurs after two half-lives of 18.72 days each. Thus, the calculation involves multiplying the half-life by 2. This means that after 37.44 days, only one-fourth of the original amount remains.
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