Question

It is known that the kinetics of recrystallization for some alloy obey the Avrami equation andthat the value of n in the exponential is 2.5. If, at some temperature, the fraction recrystallized is0.40 after 200 min, determine the rate of recrystallization at this temperature.

Answer 1

Answer:

rate of recrystallization = 4.99 × 10⁻³ min⁻¹

Explanation:

For Avrami equation:

$y = 1-e ^{(-kt^n)} \\ \\ e^{(-kt^n)} = 1-y\\ \\ -kt^n = In(1-y) \\ \\ k = \dfrac{-In(1-y)}{t^n}$

To calculate the value of k which is a dependent variable for the above equation ; we have:

$k = \dfrac{-In(1-0.40)}{200^{2.5}}$

$k = 9.030 \times 10 ^{-7}$

The time needed for 50% transformation can be determined as follows:

$y = 1-e ^{(-kt^n)} \\ \\ e^{(-kt^n)} = 1-y\\ \\ -kt^n = In(1-y) \\ \\ t =[ \dfrac{-In(1-y)}{k}]^{^{1/n}}$

$t_{0.5} =[ \dfrac{-In(1-0.4)}{9.030 \times 10^{-7}}]^{^{1/2.5}}$

= 200.00183 min

The rate of reaction for Avrami equation is:

$rate = \dfrac{1}{t_{0.5}}$

$rate = \dfrac{1}{200.00183}$

rate = 0.00499 / min

rate of recrystallization = 4.99 × 10⁻³ min⁻¹