Indium-117 has a half-life of 43 minutes. If you start with 500 grams of indium-117, how much time will it take to have 31.25 grams of indium-117 left in the sample?
A. 172 minutes
B. 86 minutes
C. 215 minutes
D. 129 minutes
The amount of indium-117 remaining in the sample can be modeled by the equation:
A(t) = A(0) * (1/2)^(t/h)
where A(t) is the amount remaining after time t, A(0) is the initial amount, t is the time elapsed, and h is the half-life.
Using this equation, we can plug in the values:
31.25 = 500 * (1/2)^(t/43)
Dividing both sides by 500:
0.0625 = (1/2)^(t/43)
Taking the logarithm of both sides:
log(0.0625) = log[(1/2)^(t/43)]
Using the logarithm property:
log(0.0625) = (t/43) * log(1/2)
Dividing both sides by log(1/2):
(t/43) = log(0.0625) / log(1/2)
Simplifying:
t = 43 * log(0.0625) / log(1/2)
Using a logarithm calculator, we find:
t ≈ 129.174
Therefore, it will take approximately 129 minutes to have 31.25 grams of indium-117 left in the sample.
The correct answer is D. 129 minutes.
To solve this question, we can use the half-life formula:
N = N0 * (1/2)^(t/t1/2)
Where:
N = the final amount of the substance
N0 = the initial amount of the substance
t = time passed
t1/2 = half-life of the substance
We have the initial amount of indium-117 (N0) as 500 grams and the final amount (N) as 31.25 grams. The half-life (t1/2) is given as 43 minutes.
Substituting these values into the formula, we get:
31.25 = 500 * (1/2)^(t/43)
To solve for t, we can take the logarithm of both sides of the equation:
log(31.25) = log(500) + (t/43) * log(1/2)
Using logarithm properties, we can simplify further:
log(31.25) - log(500) = (t/43) * log(1/2)
Now, we can isolate the t term:
t/43 = (log(31.25) - log(500)) / log(1/2)
Dividing both sides by 43, we get:
t = (log(31.25) - log(500)) / log(1/2) * 43
Using a calculator, we find:
t ≈ 128.79
Therefore, it will take approximately 128.79 minutes to have 31.25 grams of indium-117 left in the sample. Rounded to the nearest whole number, the answer is 129 minutes.
Therefore, the correct option is D. 129 minutes.