Answer : The amount of the sample left is, 50.25 grams
Solution : Given,
As we know that the radioactive decays follow first order kinetics.
First we have to calculate the rate constant of a radioisotope.
Formula used : [tex]t_{1/2}=\frac{0.693}{k}[/tex]
[tex]8.07days=\frac{0.693}{k}=3.820[/tex]
[tex]k=0.0858days^{-1}[/tex]
The expression for rate law for first order kinetics is given by :
[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = [tex]0.0858days^{-1}[/tex]
t = time taken for decay process = 16.14 days
a = initial amount of the reactant = 200 g
a - x = amount left after decay process = ?
Putting values in above equation, we get the value of amount left.
[tex]0.0858days^{-1}=\frac{2.303}{16.14days}\log\frac{200g}{a-x}[/tex]
[tex]a-x=50.25g[/tex]
Therefore, the amount of the sample left is, 50.25 grams
