Respuesta :
Given Information:
constant = n = 1.7
transformation time 50% completion = tâ â = 100 s
Required Information:
transformation time 99% completion = tââ = ?
Answer:
transformation time 99% completion = [tex]t_{90} = 202.75[/tex] [tex]seconds[/tex]
Step-by-step explanation:
The Avrami equation is used to model the transformation of solids that is from one phase to another provided that temperature is constant.
The equation is given by
[tex]y=1 - e^{{-kt}^{n}}[/tex]
Where t is the transformation time in seconds and n, k are constants.
Let us first find the constant k, since after 100 s transformation is 50% complete,
[tex]0.50=1 - e^{{-k*100}^{1.7}}[/tex]
[tex]0.50 - 1= - e^{{-k*100}^{1.7}}[/tex]
[tex]-0.50= - e^{{-k*100}^{1.7}}[/tex]
[tex]0.50 = e^{{-k*100}^{1.7}}[/tex]
Take ln on both sides,
[tex]ln(0.50) = ln(e^{{-k*100}^{1.7}})[/tex]
[tex]-0.693 = -k*100}^{1.7}[/tex]
[tex]0.693 = k*100}^{1.7}[/tex]
[tex]k = 0.693/100}^{1.7}[/tex]
[tex]k = 2.759*10^{-4}[/tex]
Now we can find out the time when the transformation is 99% complete.
[tex]0.90=1 - e^{{-kt}^{n}}[/tex]
[tex]0.90 - 1= - e^{{-k*t}^{n}}[/tex]
[tex]-0.10= - e^{{-kt}^{n}}[/tex]
[tex]0.10 = e^{{-kt}^{n}}[/tex]
Take ln on both sides,
[tex]ln(0.10) = ln(e^{{-k*t}^{n}})[/tex]
[tex]-2.303 = -kt}^{n}[/tex]
[tex]\frac{2.303}{k} = t^{n}[/tex]
[tex]\frac{2.303}{2.759*10^{-4} } = t^{n}[/tex]
Again take ln on both sides
[tex]ln(\frac{2.303}{2.759*10^{-4} }) =ln( t^{n})[/tex]
[tex]9.03 = nln(t)[/tex]
[tex]\frac{9.03}{n} = ln(t)[/tex]
[tex]\frac{9.03}{1.7} = ln(t)[/tex]
[tex]5.312 = ln(t)[/tex]
Take exponential on both sides
[tex]e^{5.312} = e^{ln(t)}[/tex]
[tex]202.75 = t[/tex]
[tex]t = 202.75[/tex] [tex]seconds[/tex]
