Question

Cobalt-60 is a strong gamma emitter that has a half- life of 5.26 yr. The co balt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75% of the original sample. If an original sample was purchased in June 2016, when will it be necessary to replace the cobalt-60?

Answer 1

Answer: The sample of Cobalt-60 isotope must be replaced in January 2027

Explanation:

The equation used to calculate rate constant from given half life for first order kinetics:

$t_{1/2}=\frac{0.693}{k}$

where,

$t_{1/2}$ = half life of the reaction = 5.26 years

Putting values in above equation, we get:

$k=\frac{0.693}{5.26yrs}=0.132yrs^{-1}$

Rate law expression for first order kinetics is given by the equation:

$k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}$

where,

k = rate constant = $0.132yr^{-1}$

t = time taken for decay process = ? yr

$[A_o]$ = initial amount of the sample = 100 grams

[A] = amount left after decay process =  (100 - 75) = 25 grams

Putting values in above equation, we get:

$0.132=\frac{2.303}{t}\log\frac{100}{25}\\\\t=10.5yrs$

The original sample was purchased in June 2016

As, June is the 6th month of the year, which means the time period will be $2016+\frac{6}{12}=2016.5$

Adding the time in the original time period, we get:

$2016.5+10.5=2027$

Hence, the sample of Cobalt-60 isotope must be replaced in January 2027