Physical Chemistry , 1st ed.

This expression can be made more accurate by recognizing that a collision

of two masses minvolves the motion of two particles with respect to each other.

As with any such motion, we should consider the reduced mass of the par-

ticles involved, not the absolute mass m. For any two equivalent masses m, the

reduced mass is m/2. (Show this.) Therefore, substituting m/2 for min the

above expression, we get a more realistic equation for the average collision fre-

quency:

z (19.41)

The totalnumber of collisions per second per unit volume, symbolized by

Z, is related to ztimes the density of the gas. However, simply multiplying z

and counts all collisions for all particles. This overcounts the total collisions

by 2, since each collision represents a collision of two particles. Including a fac-

tor of^12 to make up for this overcounting, we have for Z:

Z

1

2

z (19.42)

The 2 from the 2 ^1 term cancels with 

16 in the numerator to leave 

4 , which

equals 2. Because the density unit here is 1/m^3 , the units on Zare 1/sm^3.

Example 19.7

Xenon has a very large hard-sphere diameter of 4.00 Å. For a 1-mole sample

of Xe gas with a volume of 0.02271 m^3 , at conditions of 1.000 bar and

273.15 K, determine the following:

a.The average collision rate

b.The total collision rate per cubic meter

c.The total collision rate

Solution

This example asks for z,Z, and Ztimes the total volume (to get a collision

rate for all of the gas particles in the sample). The density is (6.02  1023

molecules)/(0.02271 m^3 ) 2.65  1025 m^3 and the mass of a xenon atom

is (0.1313 kg)/(6.02  1023 ) 2.181  10 ^25 kg.

a.We g e t

2 ^2 d^2  kT



m

d^2

16 kT



m

670 CHAPTER 19 The Kinetic Theory of Gases

z

(2.65  1025 m^3 )(4.00  10 ^10 m)^2 16(1.3 81  10 ^23 J/K)(273.15 K)



2.1 81  10 ^25 kg

Consider the units in the radicals in both the numerator and denominator:

they reduce to J/ kg, and upon decomposing the unit J into more funda-

mental units, become kgm^2 /s^2 / kg m^2 /s^2 m/s. All of the meter

units in the numerator cancel, leaving s^1 as the only remaining unit (as it

should be for a frequency).

z3.95  109 s^1

or almost 4 billion collisions per second.

b.The total collision rate per unit volume is therefore

Z^12 z 2 ^1 (3.95  109 s^1 )(2.65  1025 m^3 )

Z5.23  1034 m^3 s^1

which says that in every cubic meter of sample, 5.23  1034 collisions occur

per second.