Question

For some transformation having kinetics that obey the Avrami equation (Equation 10.17), the parameter n is known to have a value of 1.7. If, after 100 s, the reaction is 50% complete, how long (total time) will it take the transformation to go to 99% completion

Answer 1

Given Information:

constant = n = 1.7

transformation time 50% completion = t₅₀ = 100 s

Required Information:

transformation time 99% completion = t₉₀ = ?

Answer:

transformation time 99% completion = $t_{90} = 202.75$ $seconds$

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

$y=1 - e^{{-kt}^{n}}$

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,

$0.50=1 - e^{{-k*100}^{1.7}}$

$0.50 - 1= - e^{{-k*100}^{1.7}}$

$-0.50= - e^{{-k*100}^{1.7}}$

$0.50 = e^{{-k*100}^{1.7}}$

Take ln on both sides,

$ln(0.50) = ln(e^{{-k*100}^{1.7}})$

$-0.693 = -k*100}^{1.7}$

$0.693 = k*100}^{1.7}$

$k = 0.693/100}^{1.7}$

$k = 2.759*10^{-4}$

Now we can find out the time when the transformation is 99% complete.

$0.90=1 - e^{{-kt}^{n}}$

$0.90 - 1= - e^{{-k*t}^{n}}$

$-0.10= - e^{{-kt}^{n}}$

$0.10 = e^{{-kt}^{n}}$

Take ln on both sides,

$ln(0.10) = ln(e^{{-k*t}^{n}})$

$-2.303 = -kt}^{n}$

$\frac{2.303}{k} = t^{n}$

$\frac{2.303}{2.759*10^{-4} } = t^{n}$

Again take ln on both sides

$ln(\frac{2.303}{2.759*10^{-4} }) =ln( t^{n})$

$9.03 = nln(t)$

$\frac{9.03}{n} = ln(t)$

$\frac{9.03}{1.7} = ln(t)$

$5.312 = ln(t)$

Take exponential on both sides

$e^{5.312} = e^{ln(t)}$

$202.75 = t$

$t = 202.75$ $seconds$