Nuclear medicine technologists use the Iodine isotope I-131, with a half-life of 8 days,to check thyroid function of patients. After ingesting a tablet containing the iodine, the isotopes collect in a patient's thyroid, and a special camera is used to view its function. Suppose a person ingests a tablet contaiing 9 microcuries of I-131. To the nearest hour, how long will it be until there are only 2.8 microcuries in the person's thyroid?
To find out how long it will take for the amount of I-131 to decay to 2.8 microcuries, we can use the following equation:
N(t) = N₀ * (1/2)^(t/h)
Where:
N(t) is the amount of I-131 at time t
N₀ is the initial amount of I-131
t is the time passed
h is the half-life of I-131
In this case:
N₀ = 9 microcuries
N(t) = 2.8 microcuries
h = 8 days
Now, let's rearrange the equation to solve for t:
(1/2)^(t/h) = N(t)/N₀
Taking the logarithm with base 1/2 on both sides gives:
log[(1/2)^(t/h)] = log(N(t)/N₀)
Using the property of logarithms, we can bring down the exponent:
(t/h) * log(1/2) = log(N(t)/N₀)
Now, we can solve for t by dividing both sides by log(1/2):
t/h = log(N(t)/N₀) / log(1/2)
Plugging in the values, we get:
t/8 = log(2.8/9) / log(1/2)
Using a calculator to evaluate the logarithm gives:
t/8 ≈ -0.8239
To find t, multiply both sides by 8:
t ≈ -0.8239 * 8
t ≈ -6.59
Since time cannot be negative, we can take the absolute value:
|t| ≈ 6.59
Rounding to the nearest hour, it will take approximately 7 hours for the amount of I-131 to decay to 2.8 microcuries in the person's thyroid.