Page 69 - Modern physical chemistry

58 Gases and Collective Properties

at the interparticle distance r from the origin. Parameter J1 is called the reduced mass.
The relative speed of the particles
dr
V=- [3.96}
dt
is represented by the radial speed of the model particle. The argument follows the treat-
ment in examples 11.1 and 12.1.
But for the mean speed of a molecule, we have

[3.97}

Introducing Maxwell distribution law (3.94) and integrating, following example 3.11, yields

m
- _ 4 [ -- ) 3/2 roo 3 -mv 2 /2kT d _ 4 [ m )3/2 1 2kT )2 [ 8kT )112 [3.98}
_
[
JI v e
v -
v -
--
7r
2rr:kT 0 7r -- - 2 m ---
2rr:kT
mn
We consider the model particle to behave as a molecule of an ideal gas. So for the
mean relative speed of molecules at equilibrium, we have
112
[
__ 8kT )
Vrel- -- with [3.99}
7rJ1.
With respect to molecule B, molecule A is moving on the average at speed Vrel' But
for A to strike B, its center must hit within the area
a=7r(rA +rBt [3.100}
centered on molecule B. Here r A is the effective radius of molecule A, r B the effective
radius of molecule B. Recall figure 2.6.
In time dt, molecule A will sweep out volume (J'Vre1dt. The average number of B's per
unit volume is N.jV. So the average number of collisions per unit time effected by one A
molecule is (J'vre/V.jV. But the number of A molecules per unit volume on average is N/V.
So the mean collision rate density between 1\s and B's is

ZAB = a( 8kTJ1I2 Nl N 2 . [3.1Ol}
7rJ1. V V

If every A reacted on colliding with B we would have

[3.102}

Here NA is Avogadro's number while [AJ and [BJ are the molar concentrations of A and
Band kAB is the specific reaction rate.
When molecules A and B are identical, the above expressions count each collision
twice. Furthermore, the reduced mass is

mm m
J1.=--=-. [3.103}
m+m 2