Answer: The fraction that is left is [tex]\frac{1}{8}[/tex]
Explanation:
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
t = age of sample
a = let initial amount of the reactant
a - x = amount left after decay process
a) for completion of half life:
Half life is the amount of time taken by a radioactive material to decay to half of its original value.
[tex]t_{\frac{1}{2}}=\frac{0.693}{k}[/tex]
[tex]k=\frac{0.693}{5700years}=0.00012years^{-1}[/tex]
b) for 17100 years
[tex]t=\frac{2.303}{k}\log\frac{1000}{a-x}[/tex]
[tex]17100=\frac{2.303}{0.00012}\log\frac{1000}{a-x}[/tex]
[tex]\log\frac{1000}{a-x}=0.89[/tex]
[tex]\frac{1000}{a-x}=7.8[/tex]
[tex](a-x)=125g[/tex]
Fraction of the sample remained = [tex]\frac{125}{1000}=\frac{1}{8}[/tex]
The fraction that is left is [tex]\frac{1}{8}[/tex]
