The Energy of Activation
245
PROBLEM:
The rate constants at several different temperatures are obtained for the rearrangement of cyclopropane (A) to propylene (B):
H
H
H H "
\
/
I
I
I
H-C
C-H-* H—C —C=C
/
C s x
H H H
H
H
The data are the following.
1 (°C)
470
485
500
510
519
530
Msec"
1 )
1.10 x 10~
4
2.61 -x 10-<
5.70 x I0~*
1021 x I0~<
(6.4* 10~
4
28.6 x \Q~'
(a) Determine the enthalpy of activation for this reaction, and (b) calculate the
fraction of molecules having the requisite enthalpy of activation for entering
into reaction at SOOT.
SOLUTION:
(a) Transform the data to the form needed for the application of Equation 15-17
and a straight-line plot, as follows (with T in K).
IT
1.346 x IOJ
1.319 x 10"
1
1.294 x 10~'
1.277 x 10"'
1.263 x lfl->
1 245 'x 1Q-'
log*.
-3.959
-3.583
-3.244
-2.991
-2.785
-2.544
When these data are plotted as in Figure 15-7, you get a straight line whose
slope is -1.407 x 10
4 K, and whose y intercept is 14.98, with a correlation coefficient of 0.9998. The slope is equal to -A// a /2.3fl. Therefore,
A// a = -(2.30)(fl)(slope)
= -(2.30) (l.987
C
.
a
' V-1.407 x 10
4 K) = 6.43 x W -^j\
mole K/
mole
= 64.3 kcal/mole
\H.,
(b) The fraction of molecules with any given energy is obtained from 10
2 3Rr .
Therefore, at 500°C (773 K), the fraction is
64.300 cal
(2.30)1 1.987
,' I (773 K)
10
V
'
= 6.29 x 10~
19
This is a very small fraction indeed!