Question

The kinetics of some phase transformations obey the Avrami relationship. Using the fraction transformed-time data given below, determine the total time required for the transformation to go to 92% completion. Fraction transformed Time (s) 0.2 12.6 0.7 25.7 Enter your answer in accordance to the question statement s

Answer 1

Solution :

Avrami relationship

$1-y=exp(-kt^n)$

$\ln(1-y)=-kt^n$

$-\ln(1-y)=kt^n$

$\ln\left[\ln\left(\frac{1}{1-y}\right)\right]=\ln k + n \ln t$

The fraction transformed is 0.2 at 12.6 s,

$\ln\left[\ln\left(\frac{1}{1-0.2}\right)\right]=\ln k + n \ln (12.6)$

$-1.5 = \ln k + 2.533 n$  .........(i)

The fraction transformed is 0.7 at 25.7 s,

$\ln\left[\ln\left(\frac{1}{1-0.7}\right)\right]=\ln k + n \ln (25.7)$

$0.5 = \ln k + 3.24 n$   ..............(ii)

Subtract (ii) from (i),

-2 = -0.71 n

n = 2.81

Therefore, from (i),

$-1.5 = \ln k + 2.533 (2.81)$

k = 0.000181

Now if the fraction transformed is 0.92, then

$kt^n=-\ln (1-y)$

$t=\left[\frac{-\ln (1-y)}{k}\right]^{\frac{1}{n}}$

$t=\left[\frac{-\ln (1-0.92)}{0.000181}\right]^{\frac{1}{2.81}}$

t = 29.8528 s