A sample of radioactive isotope I is to be used for medical diagnosis of the kidneys. The isotope has a half-life of 8.0 days, and the sample is required to have an activity of 8 x 10^5 per second at the time it is given to the patient. Calculate the mass of the I-131 present in the sample.
a) at the time it is given (which I correctly found)
b) when it is prepared 24 hours earlier
c) 24 hours after being given
I have found the decay constant to be approx 1.00 x 10^-6
Half life is 691200
Initial no. of atoms: 7.98 x 10^11
Mass at the time given : 1.74 x 10^-10
To calculate the mass of the I-131 present in the sample, we need to use the decay constant and the initial number of atoms.
a) At the time it is given:
Since the decay constant (λ) of I-131 is approximately 1.00 x 10^-6, we can use the formula:
N(t) = N₀ * e^(-λt)
where:
N(t) = number of atoms at time t
N₀ = initial number of atoms
e = mathematical constant approximately equal to 2.71828
λ = decay constant
t = time in seconds
Given:
N(t) = 8 x 10^5 atoms
N₀ = 7.98 x 10^11 atoms
Substituting the values into the formula:
8 x 10^5 = 7.98 x 10^11 * e^(-1.00 x 10^-6 * t)
To solve for t, we need to take the natural logarithm (ln) of both sides:
ln(8 x 10^5) = ln(7.98 x 10^11) + (-1.00 x 10^-6 * t)
Using ln(8 x 10^5) ≈ 13.5924 and ln(7.98 x 10^11) ≈ 27.8420:
13.5924 = 27.8420 - 1.00 x 10^-6 * t
Rearranging the equation:
1.00 x 10^-6 * t = 27.8420 - 13.5924
1.00 x 10^-6 * t ≈ 14.2496
t ≈ 14.2496 / (1.00 x 10^-6)
t ≈ 1.42496 x 10^7 seconds
Now, using this value of t, we can find the mass of I-131:
The atomic mass of I-131 is approximately 130.91 grams/mole.
The number of moles (n) of I-131 is given by:
n = N₀ / Avogadro's number
where Avogadro's number is approximately 6.022 x 10^23.
n = 7.98 x 10^11 / (6.022 x 10^23)
n ≈ 1.3254 x 10^-12 moles
The mass (m) of I-131 is given by:
m = n * molar mass
m = 1.3254 x 10^-12 moles * 130.91 grams/mole
m ≈ 1.7354 x 10^-10 grams
Therefore, the mass of I-131 present in the sample at the time it is given to the patient is approximately 1.7354 x 10^-10 grams.
For parts b) and c), we need to calculate the time difference from the given time of 8.0 days (691200 seconds).
b) When it is prepared 24 hours earlier:
We need to subtract 24 hours (86400 seconds) from 691200 seconds to calculate the new time.
New time (t_b) = 691200 - 86400
t_b = 604800 seconds
Using the formula N(t) = N₀ * e^(-λt), similar to part a), we can substitute the new time (t_b) to find N(t_b).
Then, we can use the same method as in part a) to calculate the mass of I-131 present at this time.
c) 24 hours after being given:
We need to add 24 hours (86400 seconds) to the time given in part a) to calculate the new time.
New time (t_c) = 1.42496 x 10^7 + 86400
t_c ≈ 1.4336 x 10^7 seconds
Using the formula N(t) = N₀ * e^(-λt), similar to part a), we can substitute the new time (t_c) to find N(t_c).
Then, we can use the same method as in part a) to calculate the mass of I-131 present at this time.
To calculate the mass of the I-131 present in the sample, we need to use the equation:
N = N0 * e^(-λ*t)
Where:
- N is the number of atoms at a given time
- N0 is the initial number of atoms
- λ is the decay constant
- t is the time in seconds
a) At the time it is given:
Given:
Activity (A) = 8 x 10^5 per second
Since activity is defined as the rate of decay, we can write:
A = λ * N
Rearranging, we get:
N = A / λ
Substituting the values, we have:
N = (8 x 10^5) / (1.00 x 10^-6) = 8 x 10^11 atoms
Now, to find the mass, we need to know the molar mass of I-131. Let's assume it is M grams per mole.
Using Avogadro's number (6.02 x 10^23 atoms per mole), we can calculate the mass (m) as follows:
m = (N / Avogadro's number) * M
Substituting the values, we get:
m = (8 x 10^11 / 6.02 x 10^23) * M = 1.33 x 10^-12 * M grams
Therefore, the mass of I-131 present in the sample at the time it is given is approximately 1.33 x 10^-12 * M grams.
b) When it is prepared 24 hours earlier:
To calculate the mass 24 hours earlier, we need to calculate the number of atoms at that time. Since the half-life of I-131 is 8.0 days, which is equivalent to 691200 seconds, we can divide 24 hours (86400 seconds) by the half-life to calculate the number of half-lives.
Number of half-lives = (t / half-life) = (86400 / 691200) = 0.125
Now, we can calculate the number of atoms at that time:
N = N0 * e^(-λ*t)
N = (7.98 x 10^11) * e^(-1.00 x 10^-6 * 0.125)
N ≈ 7.82 x 10^11 atoms
Using the same procedure as before, substituting N in the mass equation, we can find the mass of I-131 present in the sample when it is prepared 24 hours earlier.
c) 24 hours after being given:
To calculate the number of atoms 24 hours after being given, we need to use the equation:
N = N0 * e^(-λ*t)
N = (8 x 10^11) * e^(-1.00 x 10^-6 * 86400)
N ≈ 7.65 x 10^11 atoms
Again, using the mass equation mentioned earlier, we can calculate the mass of I-131 present in the sample 24 hours after being given by substituting N.