Answer: The time required will be 15 seconds.
Explanation:
All the radioactive reactions follows first order kinetics.
The equation used to calculate half life for first order kinetics:
[tex]k=\frac{0.693}{t_{1/2}}[/tex] .....(1)
Rate law expression for first order kinetics is given by the equation:
[tex]k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}[/tex] ......(2)
where,
k = rate constant
t = time taken for decay process = 5 sec
[tex][A_o][/tex] = initial amount of the reactant = 112 mg
[A] = amount left after decay process = 56 mg
Putting values in above equation, we get:
[tex]\frac{0.693}{t_{1/2}}=\frac{2.303}{5}\log\frac{112}{56}\\\\t_{1/2}=5s[/tex]
Now, calculating the rate constant from equation 1, we get:
[tex]k=\frac{0.693}{5}=0.1386s^{-1}[/tex]
To calculate the time taken when 14 mg of amount remains, we use equation 2, we get:
[tex]t=\frac{2.303}{5}\log \frac{112}{14}\\\\t=15s[/tex]
Hence, the time required will be 15 seconds.
