The mechanism of diffusion in liquids was briefly discussed in Chapter 9 in terms of the mean free path and velocities of atoms or molecules in liquids. What follows is the more traditional approach to diffusion in liquids. For atoms, molecules, or particles moving in liquids, the mobility of each of these depends on the viscous drag force derived by Stokes for particles moving in a fluid, Equation 13.10.
* For example, V NiO( )=11 10. ,V MgO( )=11 26. ;V Ag( )=10 28. ,V Au( )=10 21. , and so on where V are molar volumes in cm3/mol.
13.7 Diffusion in Liquids 455 Einstein, in explaining Brownian motion,* took Equation 13.19 with B = 1/γ where, again, γ=6 r vπ µ
and derived the following equation for diffusion in liquids (Einstein 1956):
D k T r
= B
6π µ. (13.21)
This is the Stokes–Einstein equation for diffusion in liquids where r is the hydrodynamic radius of the diffusing species—atoms, ions, molecules, or actual particles. It was noted earlier in Chapter 8 that diffusion in liquids frequently follows an exponential relationship, D D e= 0 −(Q RT), and this is consistent with Equation 13.21 because the viscosity is typically temperature dependent and of the form µ µ= 0e(Q RT), as was seen in Chapter 7; that is, the viscosity decreases as the temperature increases, opposite behavior from that of the diffusion coefficient. As a result, the activation energy for viscosity and the diffusion coefficient for liquids should be the same, and for liquids that do not incorporate large molecules and intermolecular bonding, the value for Q is on the order of 10 kJ/mol.
Polymer diffusion in liquids deserves special attention because of the difficulty in defining the hydrodynamic radius and the variations that can occur depending on the type of polymer and its interaction with the solvent. This is discussed later in a separate section. What follows in the remainder of this section are examples of diffusion in liquids.
13.7.2 s
elf-D
iffusioninw
AteRIn the literature, data exist for the viscosity and diffusion coefficients for pure water over a limited temperature range (Robinson and Stokes 2002) and these data are plotted in Figure 13.6.
* Brownian motion or movement was first described by the botanist Robert Brown, who observed the random movement of pollen and other small particles in liquids. The phenomenon was studied in the late nineteenth century and is now known to be caused by the random motion of the liquid molecules creating random nonuniform forces on the small particles producing their motion. Between about 1906 and 1908, Einstein explained the phenomenon and generalized it to diffusion (Einstein 1956).
3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
4 6 8 10 12 14 16 18 20 22
Viscosity−1 (10−4 Pas)−1
D/T (10−8cm2/s-k)
1000/T (K−1)
60 50 40 30 20 10 0
Temperature (°C)
Q(viscosity) = 17.12 kJ/mole
Q(D(D2O)) = 16.90 kJ/mole Q(D(H218O)) = 16.45 kJ/mole
FIGURE 13.6 Water self-diffusion coefficients as a function of temperature measured by both 12
H O and 2
H O218 (Robinson and Stokes 2002) and the viscosity of water versus temperature (Haynes 2013) showing that they all have essentially the same activation energy as predicted by the Stokes–Einstein equation for diffusion in liquids.
The inverses of the viscosity and D/T values are plotted in units so that all three plots have simi- lar values. Note that the activation energies are very close for all three plots, about 17 kJ/mol. Also note that the diffusion coefficients for water with deuterium, D2O, as the tracer are different from those in which water with oxygen-18 was used, H218O. As a result, for the former, the radius of the water molecule calculated from Equation 13.21 is about 0.113 nm while that for the H218O diffusion is about 0.086 nm. Both of these values are smaller than those obtained from x-ray measurements, about 0.138 nm (Robinson and Stokes 2002) and 0.132 nm from gaseous diffusion (Geankoplis 1972).
So, these calculated values are not too different. An estimate for the size of a molecule or atom is to assume simple cubic packing of the atom as if they were spheres so that
d r V
N
V
d V nm
b A
b
b
= =
= ×
=
2 6 022 10
0 118
1 3
23 1 3
1 3
/ /
.
. /
(13.22)
where:
d = molecular diameter
Vb = molar volume at the boiling point (Geankoplis 1972; Bird et al. 2002).
The density of water at its boiling point is 0.95863 (Haynes 2013), so
r= M nm
=
= 0 118
2
0 118 2
18 01
0 95856 0 157
1 3 1 3
. . .
. .
/ /
ρ
which is larger than the radius of the water molecule calculated from the diffusion coefficients.
Nevertheless, these results demonstrate that the Stokes–Einstein equation gives pretty good values for the diffusion coefficient in liquids and explains the weak temperature dependence due to the temperature dependence of the liquid viscosity. It is clear that the value chosen for the radius of the diffusing molecule, atom, or particle is critical in determining the value of the calculated liquid diffusion coefficient.
13.7.3 o
theRD
iffusione
xAMPles13.7.3.1 Self-Diffusion in Water at 300 K
From Section 13.7.1, r(H2O) ≅ 0.157 ≅ 10–9 m, μ ≅ 10–3 Pa s. Therefore,
D H O k T r D H O
2 B
23
10 3
2
6
1 38 10 300 6 1 57 10 1 10
( )
= =(
×) ( )
(
×) (
×)
−
− −
π µ π . .
(( )
=1 40 10. × −9m s2/ =1 46 10. × −5cm s2/compared to tracer diffusion coefficients with deuterium and 18O-containing water molecules of around 2.5 × 10–5 cm2/s in Figure 13.6 (Robinson and Stokes 2002).
13.7.3.2 Diffusion in Liquid Copper at Its Melting Point
For copper, Tmp = 1083°C = 1356 K. The following data were obtained from the literature (Brandes and Brook 1992):
D Cul D e e cm s
Q
( )
= 0 −RT=1 46 10. × −3 −8 314 1356.40 700,× =3 95 10. × −5 2 −1µ Cul µe e Pa s
Q
( )
= 0 −RT=0 3 10. × −3 −8 314 1356.30 500,× =4 5 10. × −3 .13.7 Diffusion in Liquids 457 Again from the literature, r(Cu) = 1.28 × 10–10 m (Brandes and Brook 1992; Emsley 1998). So, at the
melting point of copper, the diffusion coefficient calculated from the Stokes–Einstein equation is
D=
(
×) ( )
m s(
×) (
×)
= ×−
− −
1 38 10 1356 −
6 1 28 10 4 5 10 1 72 10
23
10 3
9 2
.
. . . /
π ==1 72 10. × −5cm s2/
which is a little more than a factor of 2 smaller than D calculated from the experimentally determined values of D0 and Q, 3.95 × 10–5 cm2/s but, again, reasonably close given the simplicity of the model.
13.7.3.3 Diffusion in Liquid Lead at Its Melting Point
For lead, Tmp = 327°C = 600 K. Again, taking data from the literature (Brandes and Brook 1992):
D Pbl D e e cm s
Q
( )
= 0 −RT =2 37 10. × −4 −8 314 600.24 700,× =1 67 10. × −6 2/µ Pbl µe e Pa s
Q
( )
= 0 −RT =0 4636 10. × −3 −8 314 600.8610× =2 6 10. × −3 .Again from the literature, r(Pb) = 1.75 × 10–10 m (Brandes and Brook 1992; Emsley 1998). So, at the melting point of lead, the diffusion coefficient calculated from the Stokes–Einstein equation is
D=
(
×) ( )
m s(
×) (
×)
= ×−
− −
1 38 10 600 −
6 1 75 10 2 6 10 9 65 10
23
10 3
10 2
.
. . . /
π ==9 65 10. × −6cm s2/ .
In this case, the calculated value from the Stokes–Einstein relation is about a factor of 5, too high compared to the measured diffusion data.
13.7.3.4 Oxygen Diffusion in Water
The above three examples were self-diffusion in a liquid. An example of a solute diffusing in a solvent is oxygen diffusing in water at room temperature, 25°C. From the data used to plot Figure 13.6, the viscosity of water is µ =0 89 10. × −3Pa s. From tables of collision diameters for gas diffusion (Cussler 1997) 2r O
( )
2 =0 3467. nm soD k T
O H O Br
2 2
6
1 38 10 298 6 1 73 10 0 89 10
23
10 3
−
−
− −
= =
(
×) ( )
(
×) (
×)
π µ π .
. .
D
DO2−H O2 =1 42 10. × −9m s2/ =1 42 10. × −5cm s2/
which is only about 70% of the literature value of D=2 10 10. × −5cm s2/ (Cussler 1997).
13.7.3.5 Summary
From the above calculations several things are worth noting. First, in liquids including a polar molecular liquid, such as water, around room temperature, and liquid metals, at considerably higher temperatures, the diffusion coefficients are all in the 10–4–10–5 cm2/s range. Second, if one had to estimate the value of a diffusion coefficient in a liquid for a ‘back of the envelope” calculation, choos- ing one of these values (e.g., D≅10−5cm s2/ ) would be pretty close. Third, these examples were used because data were relatively available. The same approach could be applied to molten salts and other molecular liquids. However, the data are sometimes harder to locate, whereas the literature holds considerable data on aqueous (Haynes 2013) and metallic liquids (Brandes and Brook1992), respec- tively. Fourth, although the agreement between measured diffusion coefficients and those calculated is not bad, it is not perfect. As a result, there exist many additional models in the literature to gener- ate more accurate models than the simple Stokes–Einstein relation of Equation 13.21.