Page 70 - Modern physical chemistry

3.15 The Collision Rate Density 59

So the mean collision rate density between molecules of the same kind is

[3.104]

If every A reacted on colliding with another A we would have

[3.105]

Considering r A and rB to be definite implies that molecules A and B act as hard spheres.
Nevertheless, equations (3.102) and (3.105) yield fair upper limits to actual specific reac-
tion rates. Only some monatomic molecules are spherical. Even these are relatively soft.
Furthermore, an energy barrier must be surmounted with the reactants properly oriented
with respect to each other. These effects can greatly reduce the reaction rates, as we will
see later.

Example 3.13
Calculate the cross section, the mean relative speed, and the collision rate density
for molecular oxygen at 25° C and 1.000 bar pressure, where the effective radius of
O 2 is 1.S05 A.
Substitute the effective radius into formula (3.100):

2
C1 = 1r( 3.61 x 10- 10 m t = 4.094 x 10- 19 m .
Insert the temperature and the reduced mass into (3.99):

Vrel = [ SkT )112 = [16RT)1I2 = [16(S.31451 J K- mor X29S.15 K)J1I2 = 62S.13 m S-l.
1
1
1M /2 7rM 1r( 31.999 x 10-3 kg mOl-i)
For the number density, we have

5
N =!!....- = 1.000 x 10 N m- 2 = 2.429 x 1025 m-3.
V kT (1.3S07x10- 23 JK-1X29S.15 K)

Multiplying these as in line (3.104) leads to

2
1 -
ZAA = 2"C1Vrel V = 7.588 x 10 m s .
-3-1
34
[
N )
Example 3.14
Calculate the hypothetical maximum specific reaction rate kAA for the process in
example 3.13.