Physical Chemistry Third Edition

448 10 Transport Processes

The bottom of the slab is located atzz′. The area of the large face of the slab

isAand the thickness of the slab is∆z. Assume that substance numberiis diffusing

in the positivezdirection. The net amount of substanceientering the slab per second

from below is

influxAJiz(z′) (10.2-5)

whereJiz(z′) is the diffusion flux of substanceievaluated atz′. The net amount leaving

the slab per second through its top surface is proportional to the diffusion flux evaluated

atz′+∆z(the top of the slab):

effluxAJiz(z′+∆z) (10.2-6)

The rate of change ofni, the amount of substanceiin the slab, is equal to the influx

minus the efflux:

dni

dt

A

[

Jiz(z′)−Jiz(z′+∆z)

]

(10.2-7)

The molar concentration of substancei, denoted byci, is the amount of substanceiper

unit volume. The volume of the slab isA∆z, so that the rate of change ofciis

∂ci

∂t



(dni/dt)

A∆z



Jiz(z′)−Jiz(z′+∆z)

∆z

(10.2-8)

We write∂ci/∂tas a partial derivative because it is taken at a fixed value ofz. Now

we take the limit that∆zapproaches zero so that the right-hand side of Eq. (10.2-8)

becomes the negative of a derivative. The result is the one-dimensional version of the

equation of continuity:

∂ci

∂t

−

∂Jiz

∂z

(equation of continuity

in one dimension)

(10.2-9)

The physical content of this equation is the conservation of matter. It is equivalent

to saying that rate of change in the concentration is just the difference between what

arrives (the influx) and what leaves (the efflux). The three-dimensional version of the

equation of continuity is

∂ci

∂t

−

(

∂Jix

∂x

+

∂Jiy

∂y′

+

∂Jiz

∂z

)

−∇·Ji (equation of continuity) (10.2-10)

where∇·Jiis thedivergenceofJi, defined in Eq. (B-44) of Appendix B. The diver-

gence is a measure of the rate at which “stream lines” of a vector quantity diverge from

each other. If it is positive, the stream lines move away from each other and the concen-

tration of the substance decreases as one follows the flow. This physical interpretation

of the divergence explains why the name was chosen.

Exercise 10.2

Derive Eq. (10.2-10) by considering a small cube in a fluid system instead of a thin slab.