Iodine-131 is used to treat hypo-thyroidism since it is preferentially absorbed by the thyroid and typically involves a total radiation dosage of 10,000,000 millirem. Iodine-131 has a half-life of eight days.
1. Set up the appropriate integral to represent the radiation dosage delivered by the absorbed iodine in eight days.
2. Use the fact that eight days is the half-life of the isotope to find the initial radiation intensity in millrems/hour.
3. To the nearest 10 millirems, how much of the total radiation is delivered in six weeks?
well, we know the intensity is exponential in time.
I = Io e^-kt
the half life is 8 days
I/Io = .5 = e^-8k
ln .5 = -8k
-.693147=-8k
k = .0866
so
I = Io e^-.0866 t
now total over 8 days, call it D for dose
D = int I dt
D = int dt Io e^-.0866 t from t=0 to t = 8
D = (Io/-.0866) [ e^-.0866(8)-e^0]
e^0 = 1 of course
e^-.6931 = .5 of course
so
D = .5 Io/.0866
or
Io = 5.77 D
D is given of course so you are there in millirems /24 hours
I will leave the doing it in hours and the six weeks thing for you
To answer these questions, we need to understand the basics of radioactive decay and exponential decay models.
1. Integration of radioactive decay:
The integral of a radioactive decay process represents the total radiation dosage delivered over a specific time period. In this case, we want to find the radiation dosage delivered by iodine-131 in eight days.
Since the half-life of iodine-131 is eight days, we can use the exponential decay model to represent the radiation dosage as a function of time:
D(t) = D₀ * e^(-kt)
Where:
- D(t) is the radiation dosage at time t
- D₀ is the initial radiation dosage
- k is the decay constant
To find the integral of D(t) over eight days, we integrate from 0 to 8:
∫[0,8] D(t) dt = ∫[0,8] D₀ * e^(-kt) dt
2. Finding initial radiation intensity:
Given that eight days is the half-life of iodine-131, we can find the initial radiation intensity in millrems/hour. The half-life of a radioactive substance is the time it takes for the radiation intensity to decrease by half.
We can use the following equation to relate the half-life (t₁/₂) with the decay constant (k):
k = ln(2) / t₁/₂
In this case, t₁/₂ = 8 days. To convert it to hours, multiply by 24 (since there are 24 hours in a day):
t₁/₂ = 8 * 24 = 192 hours
Plugging this value into the equation for k, we get:
k = ln(2) / 192
3. Radiation dosage delivered in six weeks:
To find the radiation dosage delivered in six weeks, we need to integrate the exponential decay function from 0 to six weeks (42 days):
∫[0,42] D(t) dt = ∫[0,42] D₀ * e^(-kt) dt
Now, let's use these values to calculate the answers to the questions.