A sample of
131
I was purified and then analyzed for iodine content. It was found that the
sample was 99.8% pure. 25 hours later the sample was tested again and it was found that the
sample was now just 78.2% pure. What is the half
-
life of
131
I?
ln(No/N) = kt
No = 99.8
N = 78.2
k = solve for this
t = 25 hours
Then k = 0.693/t1/2
Solve for t1/2
You probably do better to write I-131.
Half-life of I-131 is 71.03hours
To determine the half-life of iodine-131 (131I), we can use the information provided about the purity of the sample at two different time points. The concept of half-life can be explained as the time it takes for half of a radioactive substance to decay.
In this case, the initial purity of the sample was 99.8%, and 25 hours later, the purity dropped to 78.2%. The difference between the initial purity (99.8%) and the final purity (78.2%) represents the portion of 131I that has decayed in the given time period.
Step 1: Calculate the Fraction Remaining
To determine the fraction of iodine-131 remaining after 25 hours, we need to find the difference in purity and express it as a decimal:
Fraction remaining = final purity / initial purity
Fraction remaining = 78.2% / 99.8%
Fraction remaining = 0.782 / 0.998
Fraction remaining ≈ 0.7828
Step 2: Calculate the Decay Constant
The decay constant (λ) represents the probability of decay per unit of time. It can be found using the following formula:
Fraction remaining = e^(-λ * time), where e is the base of the natural logarithm.
Since we know the fraction remaining (0.7828) and the time (25 hours), we can rearrange the formula to solve for the decay constant:
e^(-λ * 25) = 0.7828
Take the natural logarithm (ln) of both sides:
(-λ * 25) = ln(0.7828)
Solve for the decay constant (λ):
λ = -(ln(0.7828)) / 25
λ ≈ 0.0208 hours^(-1)
Step 3: Calculate the Half-Life
The half-life (t1/2) can be obtained using the formula:
t1/2 = (ln(2)) / λ
Substitute the value of the decay constant (λ) into the formula to calculate the half-life:
t1/2 = ln(2) / (0.0208)
t1/2 ≈ 33.3 hours
Therefore, the half-life of 131I is approximately 33.3 hours.