Physical Chemistry Third Edition

462 10 Transport Processes

where〈v〉is the mean speed of the molecules and where the number densityN 1 (z′−λ)

is evaluated atz′−λ, the location of the lower plane. Since the temperature of the

system is uniform, the mean speed is independent of position. The rate of molecules

of substance 1 passing downward through unit area of the center plane is given by

Eq. (9.6-6) withN 1 evaluated atz′+λ:

ν 1 (down)

1

4

N 1 (z′+λ)〈v〉 (10.3-2)

The mean speed of the molecules has the same value in both of these equations since

the temperature is uniform. The diffusion flux equals the net amount (in moles) of the

substance passing unit area per second:

J 1 z

1

NAv

[ν 1 (up)−ν 1 (down)] (10.3-3)

where the divisor of Avogadro’s constant is needed to express the flux in moles instead

of molecules. The expressions for the upward and downward rates are substituted

into Eq. (10.3-3):

J 1 z

〈v〉

4 NAv

[

N 1 (z′−λ)−N 1 (z′+λ)

]



〈v〉

4

[

c 1 (z′−λ)−c 1 (z′+λ)

]

−

〈v〉

4

[

c 1

(

z′+λ

)

−c 1

(

z′−λ

)

2 λ

]

2 λ

where the molar concentrationc 1 is equal toN 1 /NAvand where we have in the last

equation multiplied and divided by 2λ.

Ifc 1 is approximately a linear function ofz, the quotient of finite differences is

nearly equal to a derivative, and we write

J 1 z−

〈v〉λ

2

∂c 1

∂z

(10.3-4)

Comparison of this equation with Fick’s law as given in Eq. (10.2-4) gives

D 1 

〈v〉λ

2

(10.3-5)

Notice how reasonable this equation is. The diffusion coefficient is proportional to the

mean speed of the molecule and to the mean free path.

The mean free path between collisions with any kind of molecule is given by

Eq. (9.8-18):

λ

1

√

2 πd^2 Ntot

(10.3-6)

wheredis the effective hard-sphere diameter of the molecules and whereNtotis the

total number density

NtotN 1 +N 2 (10.3-7)

The total number density occurs in the formula because we include collisions with both

types of molecules. From the formulas for the mean speed and the mean free path in