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.