Page 397 - Materials Science and Engineering An Introduction

10.4 Metastable Versus Equilibrium States • 369

The problem statement provides us with the value of y (0.30) at some time t (20 min), and also
the value of n (3.1) from which data it is possible to compute the value of the constant k. In
order to perform this calculation, some algebraic manipulation of Equation 10.17 is necessary.
First, we rearrange this expression as follows:

n
exp(-kt ) = 1 - y
Taking natural logarithms of both sides leads to
n
-kt = ln(1 - y) (10.17a)
Now, solving for k,

ln(1 - y)
k = -
t n
Incorporating values cited above for y, n, and t yields the following value for k:

ln(1 - 0.30)
k = - 3.1 = 3.30 * 10 -5
(20 min)
At this point, we want to compute t 0.5 —the value of t for y 0.5—which means that it is
necessary to establish a form of Equation 10.17 in which t is the dependent variable. This is
accomplished using a rearranged form of Equation 10.17a as
ln(1 - y)
n
t = -
k
From which we solve for t
ln(1 - y) 1>n
t = c - d
k
And for t t 0.5 , this equation becomes

ln(1 - 0.5) 1>n
t 0.5 = c - d
k
Now, substituting into this expression the value of k determined above, as well as the value of
n cited in the problem statement (viz., 3.1), we calculate t 0.5 as follows:

ln(1 - 0.5) 1>3.1
t 0.5 = c - -5 d = 24.8 min
3.30 * 10
And, finally, from Equation 10.18, the rate is equal to
1 1 -2 -1
rate = = = 4.0 * 10 (min)
t 0.5 24.8 min

10.4 METASTABLE VERSUS EQUILIBRIUM STATES
Phase transformations may be wrought in metal alloy systems by varying temperature,
composition, and the external pressure; however, temperature changes by means of heat
treatments are most conveniently utilized to induce phase transformations. This corre-
sponds to crossing a phase boundary on the composition–temperature phase diagram as
an alloy of given composition is heated or cooled.