Respuesta :
The half life of the substance is 20.1 years.
What is exponential decay of a substance?
Exponential decay is the decrease in a quantity according to the law for a parameter and constant , where is the exponential function and is the initial value.
According to the given problem,
We know,
Equation of decay: [tex]y = ae^{-kt}[/tex] where,
a = original mass of the substance
t = time in years
Mass of the substance remaining = ( 100 - 99.9 )%
= 0.1%
⇒ 0.1 % of a = [tex]ae^{-200k}[/tex]
⇒ 0.001a = [tex]ae^{-200k}[/tex]
⇒ 0.001 = [tex]e^{-200k}[/tex]
Taking log on both sides,
⇒ ln (0.001) = ln [tex]e^{-200k}[/tex]
⇒ ln ( [tex]10^{-3}[/tex] ) = ln [tex]e^{-200k}[/tex]
⇒ -3 ln 10 = -200k [ ln(e) = 1 ]
⇒ k = [tex]\frac{3 ln(10)}{200}[/tex] [equation 1 ]
Now,
Now substituting 50% of the remaining mass to find time (t),
⇒ 50% of a = 0.5a
⇒ 0.5a = [tex]ae^{\frac{3ln(10)}{200} }[/tex]
⇒ 0.5 = [tex]e^{\frac{3ln(10)}{200} }[/tex]
Taking log on both sides,
⇒ ln (0.5) = ln [tex]e^{\frac{3ln(10)t}{200} }[/tex]
⇒ ln (0.5) = [tex]\frac{3 ln(10)}{200}[/tex]t
⇒ t = ln (0.5) × 200 / 3ln(10)
⇒ t ≈ 20.1 years.
Hence, we can conclude, the half-life of the substance is 20.1 years.
Learn more about exponential decay of a substance here: https://brainly.com/question/27492127
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