Physical Chemistry Third Edition

468 10 Transport Processes

into an adjacent cage. Whereas a molecule in a typical gas might undergo a collision

every 10−^10 to 10−^9 s, a molecule in a typical liquid might undergo a collision with its

neighbors every 10−^13 to 10−^11 s (see Example 9.21). However, only a small fraction

of the collisions leads to motion into another cage. The molecule might move to a new

cage every 10−^11 to 10−^9 second.

EXAMPLE10.15

a.If a substance hasD 1. 0 × 10 −^9 m^2 s−^1 , find the root-mean-square distance diffused

in 1.00 hour.

b.In the random walk model an object repeatedly takes a step of fixed length in a randomly

chosen direction. The root-mean-square distance traveled in one dimension is equal to

the square root of the number of steps,N, times the length of a step, denoted bya:

drms

√

Na. If a molecule moves 6× 10 −^10 m each time it moves to a new cage and if

it requires 1 hour to diffuse the distance found in the previous example, estimate the time

required to move to a new cage.

Solution

a.From Eq. (10.2-20) for motion in three dimensions

rrms

√

6 D 2 t

√

6(1. 0 × 10 −^9 m^2 s−^1 )(3600 s) 0 .0046 m 0 .46 cm

b.We calculate the number of steps in 1.00 hour:

rrms 0 .0046 mN^1 /^2 (6× 10 −^10 m)

N

(

0 .0046 m

6 × 10 −^10 m

) 2

 5. 9 × 1013 steps in 1.00 hour

time for one step

3600 s

5. 9 × 1013

 6. 1 × 10 −^11 s

In some approximate theories of liquid transport it is found that the motion of a

molecule or ion through a fluid is on the average impeded by an effective frictional

force that is approximately proportional to the negative of the average velocity of the

molecules or ions:

Ff−fv (10.4-1)

whereFfis the frictional force,vis the velocity of the molecule or ion, and wheref

is called thefriction coefficient. This equation is similar to Stokes’ law, Eq. (10.2-27),

which applies to a macroscopic sphere moving through a continuous fluid. We now

write the analogue of Eq. (10.2-27):

f 6 πηreff (10.4-2)

and use this relation to definereff as aneffective radiusof the molecule or ion.

Reasonable values for effective radii of molecules and hydrated ions are obtained from

experimental data. Macromolecules (molecules of large molecular mass) and colloidal