Physical Chemistry , 1st ed.

and concentrating on the flow of P 1 into the right side of the system. (The

equation we will derive will apply to either gas, ultimately.)

An understanding of diffusion therefore centers on determining the diffu-

sion coefficient D. We can actually define two types of diffusion coefficients.

The first one describes the diffusion of a gas particle through itself (the case in

which the gases represented by P 1 and P 2 are actually the same chemical

species). This is called self-diffusion.There is also the case in which two gases

have different identities, and they diffuse into each other. This is called mutual

diffusion.

In either case, we expect that the diffusion coefficient is related to the aver-

age velocity of the gas particles as well as its mean free path. This is indeed the

case. Without going through the detailed derivation, it can be shown that the

self-diffusion coefficient Dis

D

3

1

6



v

8 d

3

 (^2) 

R

M

T

 (19.53)

where all variables have been previously defined. Experimental measurements

of diffusion coefficients can be used with the equation to estimate the hard-

sphere diameter,d, for polyatomic gas molecules. As for accuracy, equation

19.53 results compare fairly well with experimental diffusion coefficients, when

using hard-sphere diameters determined from other methods.

For mutual diffusion coefficients, the derivation is even more complicated

because there are three mean free paths to consider, between like gas particles

(there are two like-particle mean free paths, one for each gas) and between dif-

ferent gas particles. The final answer is

D 12 

3

8



2

R

T





(r 1 

1

r 2 )^2 tot

 (19.54)

where is the reduced (molar) masses of the two gases,r 1 and r 2 are the hard-

sphere radii of P 1 and P 2 , and totis the total particle density of the gases.

Equation 19.54 shows the curious fact, observed experimentally, that the dif-

fusion coefficient does not depend on the mole fractions of each gas in the sys-

tem, as one might expect.

Many diffusion coefficients for gases are on the order of 10^1 cm^2 /s.

Diffusion coefficients can also be defined for liquid and solid phases. Although

the kinetic theory of gases does not apply directly to these phases, there are

some conceptual similarities. However, diffusion coefficients for condensed

phases are much lower than for gases, especially for solids at normal tempera-

tures. Diffusion coefficients for solids are typically in the range of 10^19 to

10 ^25 cm^2 /s.

From the concept of a mean free path, it should be understood that as gas

particles diffuse, they do not travel in a straight line into a new region. Rather,

as they collide with other gas particles, their direction changes continuously

but—as the concentration gradient requires—they ultimately end up moving

in the direction of lower concentration. Such a path is called a random walk

and is illustrated (in two dimensions) in Figure 19.11. In reality, random walks

of individual gas particles are three-dimensional, but the overall result is the

same: a net displacement from one part of the system (of high concentration)

to another part of the system (of low concentration). How long does it take

any one particle to move a certain distance, given this random-walk descrip-

tion of its motion? That is, can we determine the net displacement of a gas

particle?

19.5 Effusion and Diffusion 675

Initial
position

Final

position

Net (1-D) distance traveled

Figure 19.11 Over time, a gas particle travels
some net distance. However, in doing so the
particle doesn’t take a direct path. Rather, its
actual travel is a complicated “dance” in three-
dimensional space. The true path of any one gas
particle is called a random walk.