Beryllium-7 has a half-life of 53 D. A sample was observed for one minute and there were 26,880 decays.

a.) what is the activity level of the sample?
b.) what will the activity level of the sample be after 265 d?
c.) after how many days will be activity level of the sample be 112 Bq?
d.) what was activity level 106 d before the sample was observed?
e.) how many days earlier was activity level 8 times greater than the observed level?

a. The rate of decay = 26,880 dpm (decays/min)

b.
k = 0.693/t1/2
Solve for k and substitute into the below equation.
ln(No/N) = kt
No = 26880
N = ?
k from above
t = 265 days

c.
1 Bq = 1 dps
Your sample is 26,880 dpm so
26,880 decays/min x (1 min/60 sec) = about 448 dps.
ln(No/N) = kt
No = 448
N = 112
k from above
Solve for t in days.

d.
ln(No/N) = kt
No = ?
N = 26,880
k from above
t = 106

e. see d.

To answer these questions, we can use the formula for radioactive decay:

N(t) = N(0) * (1/2)^(t/T)

Where:
N(t) is the current number of radioactive atoms
N(0) is the initial number of radioactive atoms
t is the time that has passed
T is the half-life of the radioactive substance

a.) To find the activity level of the sample, we need to calculate the decay constant (λ) first. The decay constant is defined as λ = ln(2) / T.

T = 53 D = 53 days
λ = ln(2) / 53
activity level = λ * N(0) = λ * (26,880 decays / 60 seconds)
Note: We've converted the observation time from minutes to seconds.

b.) To find the activity level after 265 days, we can use the same formula for radioactive decay:

activity level = λ * N(0) * (1/2)^(t/T)

Where:
t = 265 D = 265 days

c.) To find the number of days needed for the activity level to reach 112 Bq, we can again use the formula for radioactive decay:

112 = (λ * N(0)) * (1/2)^(t/T)

Solve for t.

d.) To find the activity level 106 days before the sample was observed, we need to calculate the new initial number of radioactive atoms, N(0). We can do this by using the formula:

N(0) = N(t) * (2)^(t/T)

Where:
t = 106 D = 106 days
N(t) = 26,880 decays / 60 seconds

Then we can calculate the activity level using the formula:
activity level = λ * N(0)

e.) To find how many days earlier the activity level was 8 times greater than the observed level, we can solve the equation:

8 = (λ * N(0)) / ((1/2)^(t/T))

Solve for t.