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 [tex] t^{-1}_R [/tex] where [tex] t_R [/tex] 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
[tex] rate = Ce^{-Q/RT} [/tex]
Answer:
The Activation energy is [tex] Q = 1.85 *10^{5} J/mol[/tex]
Explanation:
From the question we are given the Arrhenius expression for the growth rate(Z) as
[tex] Z = Ce^{-Q/RT} [/tex]
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 [tex] t_R[/tex] = 100 hours
Now substituting into the Arrhenius expression
[tex] Z = Ce^{-Q/RT} [/tex]
[tex]\frac{1}{100h} = Ce^{-Q/R(250+273)K} [/tex]
[tex] 0.010h^{-1} = Ce^{-Q/R(523K)} [/tex] -----(2)
We are told that at 280°C that [tex] t_R[/tex] = 10 hours
Now substituting into the Arrhenius expression
[tex]\frac{1}{10h} = Ce^{-Q/R(280+273)K} [/tex]
[tex] 0.10h^{-1} = Ce^{-Q/R(553K)} [/tex] ----(3)
Dividing the second equation by the first one
[tex]\frac{0.010}{0.10h} = \frac{Ce^{-Q/R(523K)}}{ Ce^{-Q/R(553K)}} [/tex]
[tex] ln (0.01) = \frac{-Q (1/523K)- (1/553K)}{R} [/tex]
[tex] Q = \frac{Rln(0.10)}{(1/523K) - (1/553K)}[/tex]
Substituting [tex] 8.314 J/mol K [/tex] for R
[tex] Q = \frac{(8.314 )ln(0.10)}{(1/523K) - (1/553K)} [/tex]
[tex] Q = 1.85 *10^{5} J/mol[/tex]
