Answer : The time passed by the sample is, [tex]1.2\times 10^2\text{ min}[/tex]
Explanation :
Half-life = 30.0 min
First we have to calculate the rate constant, we use the formula :
[tex]k=\frac{0.693}{t_{1/2}}[/tex]
[tex]k=\frac{0.693}{30.0\text{ min}}[/tex]
[tex]k=0.0231\text{ min}^{-1}[/tex]
Now we have to calculate the time passed.
Expression for rate law for first order kinetics is given by:
[tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = [tex]0.0231\text{ min}^{-1}[/tex]
t = time passed by the sample = ?
a = initial amount of the reactant = 4.5 mg
a - x = amount left after decay process =0.25 mg
Now put all the given values in above equation, we get
[tex]t=\frac{2.303}{0.0231}\log\frac{4.5}{0.25}[/tex]
[tex]t=125.15\text{ min}=1.2\times 10^2\text{ min}[/tex]
Therefore, the time passed by the sample is, [tex]1.2\times 10^2\text{ min}[/tex]
