Physical Chemistry , 1st ed.

The answers to these questions are the topic of one of Einstein’s seminal

manuscripts that were published in 1905. (Other topics include a rationaliza-

tion of the photoelectric effect in terms of Planck’s theory of light, and special

relativity). In trying to understand the phenomenon called Brownian motion,

Einstein applied kinetic theory and determined an expression for the average

displacement of a particle due to interparticle collisions and mean free paths.

In considering the one-dimensional net displacement of gas particles from

their starting point, we need to recognize that they can travel in either a posi-

tive or a negative direction, so that the average one-dimensional displacement,

xavg, is simply zero. To get around that, we will consider the average of the

squareof the displacement, (x)^2 avg, since by squaring the displacement we

make all values positive.

For the average of the square of the one-dimensional displacement, Einstein

derived the expression

(x)^2 avg 2 Dt (19.55)

where Dis the diffusion coefficient from Fick’s law of diffusion and tis the

time. This one-dimensional diffusion equation is called the Einstein-

Smoluchowski equation.(Marian Smoluchowski was a Polish physicist who also

considered the theoretical basis of Brownian motion.) The units on (x)^2 avgare

m^2 (if meters are the units used in D), so by taking the square root of (x)^2 avg

we get a root-mean-square average distance that a gas particle travels from an

initial point over some time t.Since the total distance is the sum of the dis-

placements in the x, the y, and the zdimensions, it should be easy to general-

ize equation 19.55 to all dimensions, add them, and get an average three-

dimensional displacement as

(3-D displacement)^2 avg 6 Dt (19.56)

Under controlled conditions where there is no convection, gas particle displace-

ments are not as large as one might think, as the following example shows.

Example 19.9

The diffusion coefficient D 12 of NH 3 in air is about 0.219 cm^2 /s at normal at-

mospheric pressures and room temperature. A container of ammonia is

opened at the front of a lecture hall. Assuming that the air is perfectly still

and that diffusion alone accounts for the transport of NH 3 in the air, how

long before ammonia molecules can be expected to diffuse 20.0 m away from

the source?

Solution

In this example, we are solving for the time tthat it takes for gaseous ammo-

nia molecules to travel 20.0 meters in three dimensions. Using equation 19.56:

(20.0 m)^2  6 0.219 

cm

s

2

 t 

10

1

0

m

cm



2

The last term in the expression above is needed to convert cm to m. Solving

for time:

t3.04  106 s

It would take over a month for NH 3 molecules to diffuse 20.0 meters! This

example illustrates the importance of convection, rather than diffusion, in the

transport of gas molecules under real conditions.

676 CHAPTER 19 The Kinetic Theory of Gases