Answer: 229 years
Explanation:
Half life is the amount of time taken by a radioactive material to decay to half of its original value.
Half life for first order kinetics is given by:
[tex]t_{\frac{1}{2}=\frac{0.693}{k}[/tex]
[tex]t_{\frac{1}{2}[/tex] = half life = 30 years
k = rate constant =?
[tex]30years=\frac{0.693}{k}[/tex]
[tex]k=0.0231years^{-1}[/tex]
Integrated rate law for first order kinetics is given as:
[tex]A=A_0e^{-kt}[/tex]
k = rate constant of the reaction
[tex][A_o][/tex] = Concentration of reactant initially = 200 mg
[tex][A][/tex] = Concentration of reactant at time t = 1 mg
[tex]1=200e^{-0.0231\times t}[/tex]
[tex]t=229years[/tex]
Thus time take for the Cesium to decay to the point where there is only 1 milligram remaining is 229 years.
