Question 16
(a)
(i) Define radioactive decay constant.
[2]
(ii)
Show that the decay constant λ is
related to the half-life t1/2 of
a radioactive isotope by the expression
λ t1/2
= ln2
[2]
(b)
A small volume of solution containing the radioactive isotope
sodium-24 (2411Na) has an initial activity of 3.8 × 104 Bq. Sodium-24, of half-life 15 hours, decays to
form a stable daughter isotope.
All of the solution is
poured into a container of water. After 36 hours, a sample of water of volume
5.0 cm3, taken from the container, is found to have an activity of
1.2 Bq.
Assuming that the
solution of the radioactive isotope is distributed uniformly throughout the container
of water, calculate the volume of water in the container. [4]
[Total: 8]
Reference: Past Exam Paper – March 2018 Paper 42 Q13
Solution:
(a)
(i) Radioactive
decay constant is defined as the probability of decay (of a nucleus) per unit
time.
(ii)
A = A0 exp(-λt)
{After one half-life, the initial activity
would be halved.}
after one half-life, ½A0 = A0 exp(-λt)
½ = exp(–λt½)
and hence taking logs, ln2 = λt1/2
(b)
{Activity A at time t = A0 exp (-λt)
Note: ln 2 = λt1/2
Decay constant
λ = ln 2 /
t1/2
Activity A at time t = A0 exp (-λt)
Activity A at time t = A0 exp (- ln 2 × t / t1/2)}
Activity = 3.8 × 104 exp(–ln2
× 36 / 15)
Activity = 7200 Bq
{An activity of 1.2 Bq corresponds to a volume
of 5.0 cm3.
1.2 Bq - - > 5.0 cm3
7200 Bq - - > (7200 / 1.2) × 5.0}
volume = (7200 / 1.2) × 5.0
volume = 3.0 × 104 cm3
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