Respuesta :
Answer:
Therefore 25 grams will be present in 135.62 years.
Step-by-step explanation:
The model for radioactive decay is
[tex]y=y_0e^{-kt}[/tex]
y=remaining amount of radioactive element.
[tex]y_0[/tex] = initial amount of radioactive element
Given that, the half life of the radioactive element is 200 years.
For half life, [tex]y=\frac{1}{2}y_0[/tex], t=200 years
[tex]y=y_0e^{-kt}[/tex]
[tex]\Rightarrow \frac12 y_0=y_0e^{-k.200}[/tex]
[tex]\Rightarrow \frac12 =e^{-k.200}[/tex]
Taking ln both sides
[tex]\Rightarrow ln|\frac12 |=ln|e^{-k.200}|[/tex]
[tex]\Rightarrow ln|\frac12 |={-k.200}[/tex]
[tex]\Rightarrow -200k=ln|\frac12|[/tex]
[tex]\Rightarrow k=\frac{ln|\frac12|}{-200}[/tex]
⇒ k= 0.0034657
Given that,
y= 25 gram, y₀= 45 grams
[tex]25=40e^{-0.0034657t}[/tex]
[tex]\Rightarrow \frac{25}{40}=e^{-0.0034657t}[/tex]
Taking ln both sides
[tex]\Rightarrow ln|\frac{25}{40}|=ln|e^{-0.0034657t}|[/tex]
[tex]\Rightarrow ln|\frac{25}{40}|={-0.0034657t}[/tex]
[tex]\Rightarrow t=\frac{ln|\frac{25}{40}|}{-0.0034657}[/tex]
⇒ t= 135.62 years.
Therefore 25 grams will be present in 135.62 years.
