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

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

Step-by-step explanation:

For Avrami equation:

`y = 1-e ^((-kt^n)) \\ \\ e^((-kt^n)) = 1-y\\ \\ -kt^n = In(1-y) \\ \\ k = (-In(1-y))/(t^n)`

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

`k = (-In(1-0.40))/(200^(2.5))`

`k = 9.030 * 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 =[ (-In(1-y))/(k)]^{^(1/n)}`

`t_(0.5) =[ (-In(1-0.4))/(9.030 * 10^(-7))]^{^(1/2.5)}`

= 200.00183 min

The rate of reaction for Avrami equation is:

`rate = (1)/(t_(0.5))`

`rate = (1)/(200.00183)`

rate = 0.00499 / min

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