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`

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)`

`(2.303)/(k) = t^(n)`

`(2.303)/(2.759*10^(-4) ) = t^(n)`

Again take ln on both sides

`ln((2.303)/(2.759*10^(-4) )) =ln( t^(n))`

`9.03 = nln(t)`

`(9.03)/(n) = ln(t)`

`(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`