Question

Recrystallization is a thermally activated process. As such we can take a look at a series of temperatures to see the fraction of recrystallized material. Please determine the activation energy for the recrystallization process using values of the 50% recrystallized portion at each temperature.

Answer 1

Complete Question

Re crystallization is a thermally activated process. As such, can be characterized by the Arrhenius expression (Equation 1). As a first approximation we can treat
`t^(-1)_R` where
`t_R` is the time necessary to fully recrystallize the micro structure. For a 50% cold-worked aluminum alloy, t_R is 100 hours at 250° C and 10 hours at 280°C .Calculate the activation energy for the Re crystallization process

Equation 1

`rate = Ce^(-Q/RT)`

Answer:

The Activation energy is
`Q = 1.85 *10^(5) J/mol`

Step-by-step explanation:

From the question we are given the Arrhenius expression for the growth rate(Z) as

`Z = Ce^(-Q/RT)`

Where Q is the activation energy

C is known as the pre-exponential constant

R is the universal gas constant

T is the absolute temperature

From the question according to the first approximation the rate is inverse of time to fully recrystallize the micro structure(t_R)

We are told that at 250°C that
`t_R` = 100 hours

Now substituting into the Arrhenius expression

`Z = Ce^(-Q/RT)`

`(1)/(100h)   = Ce^(-Q/R(250+273)K)`

`0.010h^(-1) = Ce^(-Q/R(523K))` -----(2)

We are told that at 280°C that
`t_R` = 10 hours

Now substituting into the Arrhenius expression

`(1)/(10h)   = Ce^(-Q/R(280+273)K)`

`0.10h^(-1) = Ce^(-Q/R(553K))` ----(3)

Dividing the second equation by the first one

`(0.010)/(0.10h) = (Ce^(-Q/R(523K)))/( Ce^(-Q/R(553K)))`

`ln (0.01)   = (-Q (1/523K)- (1/553K))/(R)`

`Q = (Rln(0.10))/((1/523K) - (1/553K))`

Substituting
`8.314 J/mol K` for R

`Q = ((8.314 )ln(0.10))/((1/523K) - (1/553K))`

`Q = 1.85 *10^(5) J/mol`