Answer
In first approximation, Eq. (5.17) would be directly applicable. From Table 5.3 we
derive that for sucroseD¼ 4 : 7? 10 ^10 m^2 ?s^1 .Awould equal 0: 25 ?pr^2 —where the
factor 0.25 derives from the limited void fraction—which makes 0:0007 m^2. The
concentration gradient can be taken as the concentration differenceð100 kg?m^3 Þ
over the thickness of the diskð0.003 mÞ. This yields for dm/dtabout 11? 10 ^9 kg?s^1.
To transport 0.001 kg then takes about 90,000 s, or about one day.
Notes Inspection of Figure 2.3 shows that the sucrose activity coefficient
equals unity in a 10%solution. Nevertheless, correction is needed. In the
first place, the diffusion coefficient will be smaller than assumed, because of
the increased viscosity of the liquid. From Table 5.2 it can be estimated that
this would amount to a factor of about 1.4 for 10%sucrose. In the second
place, the concentration gradient will be smaller than assumed, because the
channels in the porous glass will be tortuous. Altogether the transport may
take almost two days rather than one. See also Figure 5.17.Question 2
The same setup as in Question 1 is used to study the diffusion rate of sodium dodecyl
sulfate. In one experiment, the first compartment contains an 8 millimolar solution,
the other one water. A diffusion coefficient of about 5? 10 ^10 m^2 ?s^1 is observed. In
a control experiment, one compartment contains 16 millimolar and the other one 8
millimolar SDS. Now the diffusion coefficient turns out to be smaller by a factor of
about 5, despite the concentration gradient being the same. How is this to be
explained?
Answer
Inspection of Figure 2.8 shows that 8 millimolar SDS would equal the critical
micellization concentration. In the first experiment, the concentration gradient of the
SDS molecules (or ions, rather) would thus have been as assumed. In the second
experiment, both compartments would have contained the same concentration of
free SDS, the concentration gradient being merely be due to SDS in micellar form.
The micelles clearly have a larger hydrodynamic radius than free SDS species, hence
the slower diffusion.
Note There are some other complications, such as ion activity coefficients
being< 1 :