A radioisotope P has a half-life of 3 s. At t = 0, a given sample of this isotope contains 8000 atoms. Calculate (i) its decay constant, (ii) average life, (iii) the time t1 when 1000 atoms of the isotope P remain in the sample, and (iv) number of decays per second in the sample at t = t1 s.
Please post your answers to the questions you've already asked. Do not post any more questions.
I looked through this list of questions. They are cookbook questions and the recipes are in the student's text.
for example if we know a half life in seconds and it is exponential decay
then amount = original amount *e^-kt
if amount/original amount = 1/2
then
1/2 = e^-kT where T is the half life
ln .5 = - k T
so
k = -ln .5 / T
To answer these questions, we can use the formula for radioactive decay:
N(t) = N₀ * e^(-λt)
Where:
N(t) = the number of atoms remaining at time t
N₀ = the initial number of atoms
λ = decay constant
t = time
(i) To find the decay constant (λ), we can use the half-life (T½) of the isotope:
T½ = 0.693 / λ
Given that the half-life is 3 s, we can substitute T½ = 3 s into the equation:
3 s = 0.693 / λ
Rearranging the equation, we get:
λ = 0.693 / 3 s⁻¹
λ ≈ 0.231 s⁻¹
So, the decay constant (λ) is approximately 0.231 s⁻¹.
(ii) The average life (Tav) is related to the decay constant by the equation:
Tav = 1 / λ
Substituting the value of λ we found in part (i), we get:
Tav = 1 / 0.231 s⁻¹
Tav ≈ 4.33 s
So, the average life (Tav) is approximately 4.33 s.
(iii) To find the time (t1) when 1000 atoms of the isotope P remain in the sample, we can rearrange the decay equation:
N(t) = N₀ * e^(-λt)
Substituting N(t) = 1000 and N₀ = 8000, we get:
1000 = 8000 * e^(-0.231t1)
Divide both sides by 8000:
0.125 = e^(-0.231t1)
Take the natural logarithm (ln) of both sides:
ln(0.125) = -0.231t1
Solving for t1, we get:
t1 = ln(0.125) / -0.231 s
t1 ≈ 11.46 s
So, approximately 11.46 seconds have passed when 1000 atoms of the isotope P remain in the sample.
(iv) The number of decays per second at t = t1 can be found by calculating the rate of decay using the decay constant:
Decay rate = λ * N(t)
Substituting λ ≈ 0.231 s⁻¹ and N(t) = 1000, we get:
Decay rate = 0.231 s⁻¹ * 1000
Decay rate ≈ 231 decays/s
So, there are approximately 231 decays per second in the sample at t = t1 seconds.