Respuesta :
Answer:
[tex]0.001199 year^{-1}[/tex] is the rate constant for this reaction.
It will take [tex]1.73\times 10^3 years[/tex] to concentration to reach 12.5% of its original value.
Explanation:
A decomposition reaction follows first order kinetics:
Half life of the reaction = [tex]t_{1/2}=578 years[/tex]
Rate constant of the reaction = k
For first order reaction, half life and rate constant are linked with an expression :
[tex]k=\frac{0.693}{t_{1/2}} [/tex]
[tex]k=\frac{0.693}{578 years}=0.001199 year^{-1}[/tex]
[tex]0.001199 year^{-1}[/tex] is the rate constant for this reaction.
Initial concentration of reactant =[tex][A_o][/tex] = x
Final concentration of reactant after time t =[tex][A][/tex] = 12.5% of x = 0.125x
The integrated law of first order reaction :
[tex][A]=[A_o]\times e^{-kt}[/tex]
[tex]0.125x=x\times e^{-0.001199 year^{-1}\times t}[/tex]
t = 1,734.31 years =[tex]1.73\times 10^3 years[/tex]
It will take [tex]1.73\times 10^3 years[/tex] to concentration to reach 12.5% of its original value.
