Contrary to expectations, gas diffusion is rather slow. For example, in one
minute the average displacement of NH 3 molecules under the above condi-
tions is only about 9 cm. The total distancethat the ammonia gas molecule
travels in its random walk among the other gas particles, however, is over
36 km! (This can be estimated by calculating an average speed of NH 3 and
multiplying by the total time, which is 60 seconds.) That is, only 0.0002% of
the distance traveled has gone into actually moving away from the original
starting point. Although this may seem strange, it is consistent with our
understanding of gas behavior on the basis of the kinetic theory.
Finally, note that both equation 19.51, defining effusion, and equations
19.53 and 19.54, which relate to diffusion, are inversely proportional to the
square root of the mass of the gas particle (or the reduced mass of the two-
component system). This idea, expressed as
rate of gas effusion or diffusion (19.57)is called Graham’s law.Scottish scientist Thomas Graham discovered this rela-
tionship in 1831, almost 30 years before the development of the kinetic theory.
(Among other things, Graham also studied and defined colloids and proposed
the idea of “denaturing” alcohol so it would not be ingested.) Graham’s law is
a good generality but is often overused, in part because conduction and con-
vection in fluids can so easily and effectively overwhelm pure effusion and dif-
fusion—as Example 19.9 demonstrates. Also, most “examples” of Graham’s
law (like the classic HCl/NH 3 vapors-in-a-tube demonstration) use only the
masses of the individual gases themselves. This would be accurate if the ex-
periment were demonstrating effusion. However, as equation 19.54 shows, the
reduced mass must be used in cases where diffusion through another gas is
considered.
19.6 Summary
One of the goals of physical chemistry is to develop models that explain the
behavior of chemical phenomena. Since quantum mechanics is a crucial model
in chemistry, it is ironic that the physical behavior of gases can be understood
by using only classical mechanics. By assuming that gas particles are constantly
moving, and by treating them as hard spheres, we can apply classical concepts
and calculate how fast they are moving (on average) through space and how
fast they move through other gases. As part of that understanding, we are able
to determine how often gas particles collide with each other, roughly how far
they travel before they collide, and how quickly they propagate from a system
(effusion) or within a system (diffusion). Although this chapter has focused on
the behavior of gases, we can recognize that some of these ideas are also ap-
plicable to condensed phases. Indeed, the behavior of liquids and solids can be
partially understood by applying classical mechanics, also. Such applications
can be found in more advanced texts. The point of this chapter is that the
physical behavior of gases is one of the better-understood phenomena in
chemistry.
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