<u>Answer:</u> The sample of Cobalt-60 isotope must be replaced in January 2027
<u>Explanation:</u>
The equation used to calculate rate constant from given half life for first order kinetics:
![t_{1/2}=\frac{0.693}{k}](https://tex.z-dn.net/?f=t_%7B1%2F2%7D%3D%5Cfrac%7B0.693%7D%7Bk%7D)
where,
= half life of the reaction = 5.26 years
Putting values in above equation, we get:
![k=\frac{0.693}{5.26yrs}=0.132yrs^{-1}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B0.693%7D%7B5.26yrs%7D%3D0.132yrs%5E%7B-1%7D)
Rate law expression for first order kinetics is given by the equation:
![k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B2.303%7D%7Bt%7D%5Clog%5Cfrac%7B%5BA_o%5D%7D%7B%5BA%5D%7D)
where,
k = rate constant = ![0.132yr^{-1}](https://tex.z-dn.net/?f=0.132yr%5E%7B-1%7D)
t = time taken for decay process = ? yr
= 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](https://tex.z-dn.net/?f=0.132%3D%5Cfrac%7B2.303%7D%7Bt%7D%5Clog%5Cfrac%7B100%7D%7B25%7D%5C%5C%5C%5Ct%3D10.5yrs)
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](https://tex.z-dn.net/?f=2016%2B%5Cfrac%7B6%7D%7B12%7D%3D2016.5)
Adding the time in the original time period, we get:
![2016.5+10.5=2027](https://tex.z-dn.net/?f=2016.5%2B10.5%3D2027)
Hence, the sample of Cobalt-60 isotope must be replaced in January 2027