Diffusivities of liquids are very small, usually being in the range of 0.5 # 10-9 to 2.5 # 10-9 m2/s at 25°C. On the other hand, diffusivities of gases at 25°C and 1 atm pressure lie between 0.1 # 10-4 and
1.0 # 10-4 m2/s. In contrast to gaseous diffusivities, diffusivities of liquids often change appreciably with concentration. The molecular theory of liquids has not yet reached a stage to permit precise prediction of liquid diffusivities. A list of experimental values of liquid phase diffusivities of some common substances is given in Table 2.5.
Table 2.5 Liquid phase diffusivities of some common compounds at infinite dilution at 25°C Solute Solvent D°AB # 105 cm2/s
Acetic acid Water 1.26 liquid phase diffusivity. But the accuracy of prediction is not as good as in case of gases. The reason for this is that the molecular theory of liquids has not yet reached a stage so as to permit precise prediction of liquid diffusivities.
The prediction of liquid phase diffusivity has become much more complex because of the effect of concentration on liquid diffusivity. Diffusivity of a solute in a concentrated solution may sometimes become quite different from diffusion in a dilute solution since concentrated solutions may differ widely from ideality and solute-solvent interactions may be more complex. Change in viscosity with change in concentration may also affect liquid diffusivity.
Poling et al. (2001) have given detailed account of several available equations for prediction of liquid phase diffusivity along with their applicability.
While developing expressions for prediction of liquid diffusivity, nonelectrolytes and electrolytes have been treated separately.
Nonelectrolytes
By considering Brownian movement of colloids where the particles of radius rp are very large in comparison with that of solvent B and assuming Stokes’ law of drag, Einstein developed the so called Stokes-Einstein equation (Brodkey and Hershey 1988) as
DAB = (2.57)
where,
DAB = diffusivity of A in dilute B k = Boltzmann constant
T = absolute temperature mB = viscosity of solvent rp = radius of particles of A.
Equation (2.57) fails to predict diffusivities of solutes of relatively smaller size. Sutherland (1905) found that if the particles are of like size and there is no tendency for the fluid to stick at the surface of the diffusing particle, i.e. coefficient of sliding friction is zero, then Eq. (2.57) becomes
DAB = (2.58)
For pure liquids, where the molecules are of the same size and are arranged in a closely packed cubic lattice with all molecules just touching (Bird et al. 1960)
rp = (2.59)
where, n = Avogadro’s number so that for self diffusion,
DAA = (2.60)
Wilke and Chang (1955) had modified the above equation and proposed the following semiempirical equation for prediction of liquid diffusivity, but the values are not always very reliable
DAB = 1.173 # 10-16 (zMB)0.5 (2.61) where,
DAB = diffusivity of solute A in solvent B in infinitely dilute solution, m2/s
z = association parameter for solvent (some values of z are: water = 2.26, methanol = 1.9, nonassociated solvent = 1.0)
vA = molar volume of solute at normal boiling point, m3/kmol T = absolute temperature, K
n = viscosity of solution, kg/m$s
VA = molal volume of solute A at its normal boiling point, cm3/gmol
Equation (2.61) is not dimensionally consistent and is usually accurate to within 10 to 15% in the temperature range of 10 to 30°C. Although the correlation was proposed for both aqueous and nonaqueous system, later studies have shown that it is satisfactory for most cases of organic solute diffusing in water and fails for many systems when water is diffusing through organic solvent.
However, it is the most general and requires minimum supplementary data.
Electrolytes
In dilute solutions, liquid diffusivity is given by
DAB (dil) = 2.62)
where,
DAB (dil) = diffusivity in dilute solution, cm2/s
u+ and u- = absolute velocities of cations and anions respectively, cm/s z+ and z- = valencies of cations and anions respectively
T = absolute temperature, K
R = gas constant = 8.314 # 107 ergs/(gmol) (K)
In concentrated solutions, diffusivity is obtained from the relation
DAB (conc) = DAB(dil) + (2.63)
where,
DAB (conc) = diffusivity in concentrated solution, cm2/s m = molality of solution
f = mean ionic activity coefficient referred to molality
f(m) = a correction factor for effect of concentration on ionic mobility, given by
f(m) = (2.64)
V′B = partial molal volume of water in solution n = viscosity of water
nB = viscosity of solvent
EXAMPLE 2.15 (Estimation of liquid diffusivity by Wilke-Chang equation): Estimate the diffusivity of ethyl alcohol in dilute aqueous solution at 20°C.
For water as solvent, z = 2.26
Atomic volumes of C, H and O are 14.8, 3.7 and 7.4 cm3/gmol, respectively.
Solution: Molecular weight of water, MB = 18.02, Temperature, T = 20°C = 293 K.
Molar volume of ethyl alcohol = 2 (14.8) + 6 (3.7) + 7.4 = 59.2 cm3/gmol.
= 0.0592 m3/kmol
For dilute solution, viscosity may be taken as that of water.
At 20 °C, viscosity of water, n = 0.001005 kg/m$s.
Substituting the values in Eq. (2.61),
DAB = m2/s
= 1.19 # 10-9 m2/s
Diffusivity of ethyl alcohol = 1.19 # 10-9 m2/s
2.8.2 Experimental Determination of Liquid Phase Diffusivity
Reports available on experimentation by investigators using various methods for prediction of diffusivity of different liquids are as follows:
(i) Taylor dispersion technique
1990, 1991, Ramakanth et al. 1991, Ghosh et al. 1991, Lee and Li 1991, Chin, et al. 1991; Riede and Schlüder, 1991, Ghorai, 1994). Wise and Houghton (1966) measured diffusion coefficient for dissolved gases like hydrogen, oxygen, helium, argon, methane, ethane, etc. in water by the following rate of collapse of small bubbles in gas-free water.
Amongst these, the diaphragm cell method is widely used as it is the only successful method based on Fick’s first law and combines experimental simplicity with accuracy. The apparatus as shown in Figure 2.8 used in laboratory for measurement of diffusivity in binary hydrocarbon systems, consisted of two compartments separated by a G-4 diaphragm (sintered glass or porous membrane).
The concentrated solution and the comparatively dilute solution were kept in two compartments. The liquid layers adjacent to the surfaces of the diaphragm were stirred by magnetic stirrers which was operated by the permanent bar magnets fitted outside the shell. Potassium chloride solution was used as a standard for calibration of the diaphragm cell. The same was analysed by conductivity measurement whereas the concentrations of unknown components were measured by Abbe refractometer. The diffusivity was then determined using Fick’s first law in convenient form described by Geankoplis (2005).
Figure 2.8 The diaphragm cell for measuring diffusivity of liquids.