Page 85 - Essentials of physical chemistry

The Kinetic Molecular Theory of Gases 47

√ 2 V
V

V
FIGURE 3.7 Assuming two particles (atoms or molecules) are traveling at an average speed of hVi and
ffiffiffi
p
approach each other at right angles, their effective speed of approach is 2hVi. Random angles of approach can
be resolved into right-angle components, so a reasonable approximation of the average rate of approach is
ffiffiffi
p
2hVi in all directions.
spheres are in motion with the average speed, to get the number of binary collisions, we should
multiply the Z 1 collisions by the concentration of targets n*, but since it takes two spheres to cause one
collision, we should divide by 2. This ignores the fact that each sphere should not count a collision
with itself but that is a small number, which can be neglected compared with the total number of
binary collisions. We also note for future discussion of gas kinetics that the possibility of a three-body
collision is so improbable that it can be neglected. Thus, we have for the binary collisions

Z 1 n*
:
2
Z 11 ffi
This is still theoretical, so we need some sort of measurement to confirm these assumptions.
Surprisingly, it is possible to construct a gas viscometer similar to the Ostwald viscometer for
liquids. If a sufficiently small diameter capillary tube is used, the time to force a given volume of
gas through that tiny cylinder will vary with the type of gas and gas viscosity can be measured
(Table 3.2). Such viscometers can be standardized using dry air with a relationship derived by the
former U.S. National Bureau of Standards [1].
7
(145:8 10 )T 3=2
poise:
h ¼
T þ 110:4
Although, the type of viscometer in Figure 3.8 [2] has been used for years in teaching laboratories,
concern regarding the accompanying vapor pressure of mercury has eliminated this apparatus from
many laboratories. The main point of interest here is the tiny capillary tube through which the gas
sample is forced to flow by the weight of the mercury. The volume between marks ‘‘a’’ and ‘‘b’’ is
designed to be exactly 100 mL.
Although, it is interesting to consider the collision properties of gases, the reason we have
emphasized the Boltzmann average speed is to get to the measurable quantity of gas viscosity.
Consider a rectangular box as in Figure 3.9 that has two slits on the side for inlets from a gas source,
which has a temperature gradient in the vertical z-direction, so that the gas entering the upper slit is
warmer (faster average speed) than the gas entering the lower slit. Let us assume that the two
entrance slits are only two mean free path lengths apart and the temperature gradient along the left
side produces a velocity gradient as you increase the z-coordinate (dv=dz).
Assume the right side of the box is open to exhaust the gas. Now, consider the plane between two
(1 cm 1 cm) sheets of gas coming through the slits. With apologies to the IUPAC Committee, we