Answer :
It takes approximately 116.02 seconds for a sample of 137I to decay to 3.5% of the original atoms.
The time it takes for a sample of 137I to decay to 3.5% of the original atoms, we can use the concept of half-life.
The half-life of 137I is given as 24.13 seconds. This means that in 24.13 seconds, half of the original amount of 137I will decay.
The time required for the sample to decay to 3.5% of the original atoms, we can use the following formula:
Final amount = Initial amount × (1/2)^(number of half-lives)
The time required as "t". The final amount remaining after decay will be 3.5% of the initial amount, which can be written as 0.035 times the initial amount.
0.035 = 1 × (1/2)^(t/24.13)
Now, to solve for "t". Taking the logarithm (base 2) on both sides to isolate "t":
log2(0.035) = (t/24.13) × log2(1/2)
Simplifying further:
t/24.13 = log2(0.035) / log2(1/2)
t/24.13 ≈ -4.807
t ≈ -4.807 × 24.13
t ≈ -116.02 seconds
Since time cannot be negative, we can conclude that it takes approximately 116.02 seconds for the sample of 137I to decay such that only 3.5% of the original atoms are left.
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