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The half-life of a radioactive element is defined as the amount of time required for the amount of the element to decrease by half. If the half-life of an element is 3 years, how long will it take for the element to be reduced to one-eighth of its original amount?
- a)3 years
- b)9 years
- c)21 years
- d)24 years
- e)27 years
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The half-life of a radioactive element is defined as the amount of tim...
Let us first understand the given information.
The question states the following about half-life:
- Half-life is defined as the time taken for the mass of a radioactive substance to reduce to half.
The interesting thing to notice here is that the Half-life apparently is not dependent on the initial mass of the radioactive substance.
In other words, if a particular radioactive substance’s half-life is 2 years, then the time taken by it
- To reduce from 1000 kg to 500 kg is 2 years
- To reduce from 500 kg to 250 kg is 2 years
- To reduce from 250 kg to 125 kg is 2 years and so on.
This makes it easier for us to calculate the time taken by the radioactive substance to decay because we will be simply counting the number of half-lives taken.
We need to find the approximate time taken for a X grams block of the radioactive element to reduce to about X/8 grams.
As we noted earlier,
In 1 half-life, the amount of element left = ½ * X
In 2 half-lives, the amount of element left = ½ * ½ * X = (½)2 * X
Let’s say the number of half-lives taken = n.
Then in n half-lives, the amount of element left = (½)n * X
Given that the mass left is 1/8 th of the original amount.
Therefore amount of element left = X/8
This gives us (½)n * X = X/8
- (½)n = 1/8
- (½)n = (½)3
- n = 3
Therefore the time taken = 3 half-lives = 3*3 = 9 years.
Correct Answer: B
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