The Foundations of Chemistry

We arrange the Arrhenius equation for Ea.

ln  


T

1

1



T

1

2

 so Ea

Substituting,

Rln 

k

k

2

1





T

1

1



T

1

2



Ea



R

k 2



k 1

Compare this approach to that

described in the earlier Enrichment

section for determining k.

16-8 Temperature: The Arrhenius Equation 687

Ea2.09 105 J/mol

8.314 mo

J

lK

(5.98)



2.38 10 ^4 K^1

8.314 mo

J

lK

ln


6

1

.

.

3

6

6

0





1

1

0

0





3

5

s

s





1

 1 




 60

1

0.K



7

1

00

.K

You should now work Exercise 54.

The determination of Eain the manner illustrated in Example 16-12 may be subject to
considerable error because it depends on the measurement of kat only two temperatures.
Any error in either of these kvalues would greatly affect the resulting value of Ea. A more
reliable method that uses many measured values for the same reaction is based on a graph-
ical approach. Let us rearrange the single-temperature logarithmic form of the Arrhenius
equation and compare it with the equation for a straight line.

ln k


ln A

AA

gggg

y  mx
 b

The value of the collision frequency factor, A,is very nearly constant over moderate
temperature changes. Thus, ln Acan be interpreted as the constant term in the equation
(the intercept). The slope of the straight line obtained by plotting ln kversus 1/Tequals
Ea/R.This allows us to determine the value of the activation energy from the slope
(Figure 16-14). Exercises 57 and 58 use this method.

1



T

Ea



R

Figure 16-14 A graphical method for determining activation energy, Ea. At each of several
different temperatures, the rate constant, k,is determined by methods such as those in
Sections 16-3 and 16-4. A plot of ln kversus 1/Tgives a straight line with negative slope.
The slope of this straight line is Ea/R.Use of this graphical method is often desirable,
because it partially compensates for experimental errors in individual kand Tvalues.

1n

k

1/T

Ea

slope  (^) R
 
or 209 kJ/mol