Physical Chemistry Third Edition

10.2 Transport Processes 451

2 1.5 2 1.0 2 0.5 0 0.5 1.0 1.5

Concentration

z/cm

t 5 4 hours

t 5 2 hours

t 5 1 hour

Figure 10.4 Concentration as a Function of Position in a Diffusing System with an

Initial Thin Layer of Solute, as in Eq. (10.2-16).

Since all of the molecules of substance 2 started out atz0, we can use Eq. (10.2-16)

to study their average displacement in thezdirection. Consider a thin slab of the system

lying betweenz′andz′+dz, wherez′is some value ofz. The fraction of the molecules

of substance 2 in the slab at timetis

(fraction in slab)

c 2

(

z′,t

)

dz

∫∞

−∞

c 2 (z,t)dz



c 2

(

z′,t

)

dz

n 0

(10.2-17)

The mean value of the coordinatezat timetis given by

〈z(t)〉

1

n 0

∫∞

−∞

zc 2 (z,t)dz 0 

1

n 0

n 0

2

√

πD 2 t

∫∞

−∞

ze−z

(^2) / 4 D 2 t
dz 0 (10.2-18)
The mean value〈z(t)〉vanishes because the integrand is anodd function.Iff(x)isan
odd function, thenf(−x)−f(x). The fact that〈z(t)〉0 corresponds to the fact
that for every molecule that has moved in the positivezdirection, another has moved
the same distance in the negativezdirection.
Theroot-mean-square valueof thezcoordinate is a measure of the magnitude of
the distance traveled in thezdirection by an average molecule. The root-mean-square
value is the square root of themean-square value,

〈

z^2

〉

:

zrms

〈

z^2

〉 1 / 2



[

1

n 0

∫∞

−∞

z^2 c 2 (z,t)dz

] 1 / 2



[

1

n 0

n 0

2

√

πD 2 t

∫∞

−∞

z^2 e−z

(^2) / 4 D 2 t
dz

] 1 / 2

zrms[ 2 D 2 t]^1 /^2 (10.2-19)

where we have looked up the integral in Appendix C. The root-mean-square displace-

ment is proportional to the square root of the elapsed time and to the square root of

the diffusion coefficient. This behavior is similar to that of arandom walk,^2 which is

(^2) L. E. Reichl,A Modern Course in Statistical Physics, University of Texas Press, Austin, 1980, p. 151ff.