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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.
E X A M P L E 10.15
a. If a substance has D 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 by a:
d rms
√
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
r rms
6D 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:
r rms 0.0046 m N 1/2 (6 × 10 −10 m)
N
0.0046 m
6 × 10 −10 m
2
5.9 × 10 13 steps in 1.00 hour
time for one step
3600 s
5.9 × 10 13 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:
F f −f v
(10.4-1)
where F f is the frictional force, v is the velocity of the molecule or ion, and where f
is called the friction 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 6πηr eff
(10.4-2)
and use this relation to define r eff as an effective radius of 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
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.
E X A M P L E 10.15
a. If a substance has D 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 by a:
d rms
√
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
r rms
6D 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:
r rms 0.0046 m N 1/2 (6 × 10 −10 m)
N
0.0046 m
6 × 10 −10 m
2
5.9 × 10 13 steps in 1.00 hour
time for one step
3600 s
5.9 × 10 13 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:
F f −f v
(10.4-1)
where F f is the frictional force, v is the velocity of the molecule or ion, and where f
is called the friction 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 6πηr eff
(10.4-2)
and use this relation to define r eff as an effective radius of 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