6.3. MOLECULAR DIFFUSION IN LIQUIDS

6.3A. Introduction

Diffusion of solutes in liquids is very important in many industrial processes, especially in such separation operations as liquid–liquid extraction or solvent extraction, gas absorption, and distillation. Diffusion in liquids also occurs in many situations in nature, such as oxygenation of rivers and lakes by the air and diffusion of salts in blood.

It should be apparent that the rate of molecular diffusion in liquids is considerably slower than in gases. The molecules in a liquid are very close together compared to a gas. Hence, the molecules of the diffusing solute A will collide with molecules of liquid B more often and diffuse more slowly than in gases. In general, the diffusion coefficient in a gas will be on the order of magnitude of about 105 times greater than in a liquid. However, the flux in a gas is not that much greater, being only about 100 times faster, since the concentrations in liquids are considerably higher than in gases.

6.3B. Equations for Diffusion in Liquids

Since the molecules in a liquid are packed together much more closely than in gases, the density and the resistance to diffusion in a liquid are much greater. Also, because of this closer spacing of the molecules, the attractive forces between molecules play an important role in diffusion. Since the kinetic theory of liquids is only partially developed, we write the equations for diffusion in liquids similar to those for gases.

In diffusion in liquids an important difference from diffusion in gases is that the diffusivities are often quite dependent on the concentration of the diffusing components.

1. Equimolar counterdiffusion

Starting with the general equation (6.2-14), we can obtain for equimolal counterdiffusion, where NA = −NB, an equation similar to Eq. (6.1-11) for gases at steady state:

Equation 6.3-1


where NA is the flux of A in kg mol A/s · m2, DAB the diffusivity of A in B in m2/s, cA1 the concentration of A in kg mol A/m3 at point 1, xA1 the mole fraction of A at point 1, and cav defined by

Equation 6.3-2


where cav is the average total concentration of A + B in kg mol/m3, M1 the average molecular weight of the solution at point 1 in kg mass/kg mol, and ρ1 the average density of the solution in kg/m3 at point 1.

Equation (6.3-1) uses the average value of DAB, which may vary some with concentration, and the average value of c, which may also vary with concentration. Usually, the linear average of c is used, as in Eq. (6.3-2). The case of equimolar counterdiffusion in Eq. (6.3-1) only occurs very infrequently in liquids.

2. Diffusion of A through nondiffusing B

The most important case of diffusion in liquids is that where solute A is diffusing and solvent B is stagnant or nondiffusing. An example is a dilute solution of propionic acid (A) in a water (B) solution being contacted with toluene. Only the propionic acid (A) diffuses through the water phase, to the boundary, and then into the toluene phase. The toluene–water interface is a barrier to diffusion of B and NB = 0. Such cases often occur in industry (T2). If Eq. (6.2-22) is rewritten in terms of concentrations by substituting cav = P/RT, cA1 = pA1/RT, and xBM = pBM/P, we obtain the equation for liquids at steady state:

Equation 6.3-3


where

Equation 6.3-4


Note that xA1 + xB1 = xA2 + xB2 = 1.0. For dilute solutions xBM is close to 1.0 and c is essentially constant. Then Eq. (6.3-3) simplifies to

Equation 6.3-5


EXAMPLE 6.3-1. Diffusion of Ethanol (A) Through Water (B)

An ethanol (A)–water (B) solution in the form of a stagnant film 2.0 mm thick at 293 K is in contact at one surface with an organic solvent in which ethanol is soluble and water is insoluble. Hence, NB = 0. At point 1 the concentration of ethanol is 16.8 wt % and the solution density is ρ1 = 972.8 kg/m3. At point 2 the concentration of ethanol is 6.8 wt % and ρ2 = 988.1 kg/m3 (P1). The diffusivity of ethanol is 0.740 × 109 m2/s (T2). Calculate the steady-state flux NA.

Solution: The diffusivity is DAB = 0.740 × 109 m2/s. The molecular weights of A and B are MA = 46.05 and MB = 18.02. For a wt % of 6.8, the mole fraction of ethanol (A) is as follows when using 100 kg of solution:


Then xB2 = 1 − 0.0277 = 0.9723. Calculating xA1 in a similar manner, xA1 = 0.0732 and xB1 = 1 − 0.0732 = 0.9268. To calculate the molecular weight M2 at point 2,


Similarly, M1 = 20.07. From Eq. (6.3-2),


To calculate xBM from Eq. (6.3-4), we can use the linear mean since xB1 and xB2 are close to each other:


Substituting into Eq. (6.3-3) and solving,



6.3C. Diffusion Coefficients for Liquids

1. Experimental determination of diffusivities

Several different methods are used to determine diffusion coefficients experimentally in liquids. In one method, unsteady-state diffusion in a long capillary tube is carried out and the diffusivity determined from the concentration profile. If the solute A is diffusing in B, the diffusion coefficient determined is DAB. Also, the value of diffusivity is often very dependent upon the concentration of the diffusing solute A. For liquids, unlike gases, the diffusivity DAB does not equal DBA.

In a fairly common method a relatively dilute solution and a slightly more concentrated solution are placed in chambers on opposite sides of a porous membrane of sintered glass as shown in Fig. 6.3-1. Molecular diffusion takes place through the narrow passageways of the pores in the sintered glass while the two compartments are stirred. The effective diffusion length is τδ, where the tortuosity τ > 1 is a constant and corrects for the fact that the path is actually greater than δ cm. In this method, discussed by Bidstrup and Geankoplis (B4), the effective diffusion length is obtained by calibrating with a solute such as KCl having a known diffusivity.

Figure 6.3-1. Diffusion cell for determination of diffusivity in a liquid.


To derive the equation, quasi-steady-state diffusion in the membrane is assumed:

Equation 6.3-6


where c is the concentration in the lower chamber at a time t, c' is the concentration in the upper chamber, and ε is the fraction of area of the glass open to diffusion. Making a balance on solute A in the upper chamber, where the rate in = rate out + rate of accumulation, making a similar balance on the lower chamber, using volume V = V', and combining and integrating, the final equation is

Equation 6.3-7


where 2εA/τδV is a cell constant that can be determined using a solute of known diffusivity, such as KCl. The values c0 and are initial concentrations and c and final concentrations.

2. Experimental liquid diffusivity data

Experimental diffusivity data for binary mixtures in the liquid phase are given in Table 6.3-1. All the data are for dilute solutions of the diffusing solute in the solvent. In liquids the diffusivities often vary quite markedly with concentration. Hence, the values in Table 6.3-1 should be used with some caution when outside the dilute range. Additional data are given in (P1). Values for biological solutes in solution are given in the next section. As noted in the table, the diffusivity values are quite small and in the range of about 0.5 × 109 to 5 × 109 m2/s for relatively nonviscous liquids. Diffusivities in gases are larger by a factor of 104–105.

Table 6.3-1. Diffusion Coefficients for Dilute Liquid Solutions
SoluteSolventTemperatureDiffusivity [(m2/s)109 or (cm2/s)105]Ref.
°CK
NH3Water122851.64(N2)
  152881.77 
O2Water182911.98(N2)
  252982.41(V1)
CO2Water252982.00(V1)
H2Water252984.8(V1)
Methyl alcoholWater152881.26(J1)
Ethyl alcoholWater102830.84(J1)
  252981.24(J1)
n-Propyl alcoholWater152880.87(J1)
Formic acidWater252981.52(B4)
Acetic acidWater9.7282.70.769(B4)
  252981.26(B4)
Propionic acidWater252981.01(B4)
HCl (9 g mol/liter)Water102833.3(N2)

(2.5 g mol/liter)

 102832.5(N2)
Benzoic acidWater252981.21(C4)
AcetoneWater252981.28(A2)
Acetic acidBenzene252982.09(C5)
UreaEthanol122850.54(N2)
WaterEthanol252981.13(H4)
KClWater252981.870(P2)
KClEthylene glycol252980.119(P2)

6.3D. Prediction of Diffusivities in Liquids

The equations for predicting diffusivities of dilute solutes in liquids are by necessity semiempirical, since the theory for diffusion in liquids is not well established as yet. The Stokes–Einstein equation, one of the first theories, was derived for a very large spherical molecule (A) diffusing in a liquid solvent (B) of small molecules. Stokes' law was used to describe the drag on the moving solute molecule. Then the equation was modified by assuming that all molecules are alike and arranged in a cubic lattice, and by expressing the molecular radius in terms of the molar volume (W5),

Equation 6.3-8


where DAB is diffusivity in m2/s, T is temperature in K, μ is viscosity of solution in Pa · s or kg/m · s, and VA is the solute molar volume at its normal boiling point in m3/kg mol. This equation applies very well to very large unhydrated and sphere-like solute molecules of about 1000 molecular weight or greater (R1), or where the VA is above about 0.500 m3/kg mol (W5) in aqueous solution.

For smaller solute molar volumes, Eq. (6.3-8) does not hold. Several other theoretical derivations have been attempted, but the equations do not predict diffusivities very accurately. Hence, a number of semitheoretical expressions have been developed (R1). The Wilke–Chang (T3, W5) correlation can be used for most general purposes where the solute (A) is dilute in the solvent (B):

Equation 6.3-9


where MB is the molecular weight of solvent B, μB is the viscosity of B in Pa · s or kg/m · s, VA is the solute molar volume at the boiling point (L2), which can be obtained from Table 6.3-2, and φ is an “association parameter” of the solvent, where φ is 2.6 for water, 1.9 for methanol, 1.5 for ethanol, 1.0 for benzene, 1.0 for ether, 1.0 for heptane, and 1.0 for other unassociated solvents. When values of VA are above 0.500 m3/kg mol (500 cm3/g mol), Eq. (6.3-8) should be used.

Table 6.3-2. Atomic and Molar Volumes at the Normal Boiling Point
MaterialAtomic Volume(m3/kg mol) 103MaterialAtomic Volume (m3/kg mol) 103
C

H

O (except as below)
14.8

3.7

7.4
Ring, 3-membered as in ethylene oxide−6
 Doubly bound as carbonyl7.44-membered 5-membered−8.5 −11.5
 Coupled to two other elements 6-membered Naphthalene ring−15 −30
  In aldehydes, ketones7.4Anthracene ring−47.5
  In methyl esters9.1  
  In methyl ethers9.9  
  In ethyl esters9.9  
  In ethyl ethers9.9 Molecular Volume (m3/kg mol) 103
  In higher esters11.0 
  In higher ethers11.0Air29.9
  In acids (−OH)12.0O225.6
 Joined to S, P, N8.3N231.2
N Br253.2
 Doubly bonded15.6Cl248.4
 In primary amines10.5CO30.7
 In secondary amines12.0CO234.0
Br  27.0 H214.3
Cl in RCHClR'24.6H2O18.8
Cl in RCl (terminal)21.6H2S32.9
F8.7NH325.8
I37.0NO23.6
S25.6N2O36.4
P27.0SO244.8
Source: G. Le Bas, The Molecular Volumes of Liquid Chemical Compounds. New York: David McKay Co., Inc., 1915.

When water is the solute, values from Eq. (6.3-9) should be multiplied by a factor of 1/2.3 (R1). Equation (6.3-9) predicts diffusivities with a mean deviation of 10–15% for aqueous solutions and about 25% in nonaqueous solutions. Outside the range 278–313 K, the equation should be used with caution. For water as the diffusing solute, an equation by Reddy and Doraiswamy is preferred (R2). Skelland (S5) summarizes the correlations available for binary systems. Geankoplis (G2) discusses and gives an equation to predict diffusion in a ternary system, where a dilute solute A is diffusing in a mixture of B and C solvents. This case is often approximated in industrial processes.

EXAMPLE 6.3-2. Prediction of Liquid Diffusivity

Predict the diffusion coefficient of acetone (CH3COCH3) in water at 25° and 50°C using the Wilke–Chang equation. The experimental value is 1.28 × 109 m2/s at 25°C (298 K).

Solution: From Appendix A.2 the viscosity of water at 25.0°C is μB = 0.8937 × 103 Pas and at 50°C, 0.5494 × 103. From Table 6.3-2 for CH3COCH3 with 3 carbons + 6 hydrogens + 1 oxygen,


For water the association parameter φ = 2.6 and MB = 18.02 kg mass/kg mol. For 25°C, T = 298 K. Substituting into Eq. (6.3-9),


For 50°C or T = 323 K,


6.3E. Prediction of Diffusivities of Electrolytes in Liquids

Electrolytes in aqueous solution such as KCl dissociate into cations and anions. Each ion diffuses at a different rate. If the solution is to remain electrically neutral at each point (assuming the absence of any applied electric-potential field), the cations and anions diffuse effectively as one component, and the ions have the same net motion or flux. Hence, the average diffusivity of the electrolyte KCl is a combination of the diffusion coefficients of the two ions. Its value is in between the diffusivity values for the two ions.

The well-known Nernst–Haskell equation for dilute, single-salt solutions can be used at 25°C to predict the overall diffusivity DAB of the salt A in the solvent B (R1):

Equation 6.3-10


where is in cm2/s, n+ is the valence of the cation, n is the valence of the anion, and λ+ and λ are the limiting ionic conductances in very dilute solutions in (A/cm2)(V/cm)(g equiv./cm2). Values of the conductances are given in Table 6.3-3 at 25°C. The value of T = 298.2 in Eq. (6.3-10) when using values of λ+ and λ at 25°C.

Table 6.3-3. Limiting Ionic Conductances in Water at 25°C (R4)
AnionλCationλ+
OH197.6H+349.8
Cl76.3Li+38.7
Br78.3Na+50.1
71.4K+73.5
40.973.4
8059.5
68.053
  69.5
Ionic conductances in (A/cm2)(V/cm)(g equiv./cm2)

The diffusion coefficient of an individual ion i at 25°C can be calculated from

Equation 6.3-11


Then Eq. (6.3-10) becomes as follows:

Equation 6.3-12


To correct for temperature, first calculate at 25°C from Eq. (6.3-10) using the λ+ and λ values given in Table 6.3-3 at 25°C. Then, multiply this at 25°C by T/(334 μW), where μW is the viscosity of water in cp at the new T (R1).

EXAMPLE 6.3-3. Diffusivities of Electrolytes

Predict the diffusion coefficients of dilute electrolytes for the following cases:

  1. For KCl at 25°C, predict and compare with the value in Table 6.3-1.

  2. Predict the value for KCl at 18.5°C. The experimental value is 1.7 × 105 cm2/s (S9).

  3. For CaCl2 predict DAB at 25°C. Compare with the experimental value of 1.32 × 105 cm2/s (C10). Also predict Di of the ion Ca2+ and of Cl and use Eq. (6.3-12).

Solution: For part (a) from Table 6.3-3, λ+(K+) = 73.5 and λ(Cl) = 76.3. Substituting into Eq. (6.3-10),


The experimental value in Table 6.3-1 is 1.87 × 105 cm2/s, which is reasonably close considering that this value is not at infinite dilution.

For part (b) the correction factor for temperature is T/(334 μw). For T = 273.2 + 18.5 = 291.7 K from Table A.2-4, μw = 1.042 cp. Then T/(334 μw) = 291.7/(334 × 1.042) = 0.8382. Correcting the value of at 25°C to 18.5°C,


This compares with the experimental value of 1.7 × 105 cm2/s.

For part (c), for CaCl2, from Table 6.3-3, λ+(Ca2+/2) = 59.5, λ(Cl) = 76.3, n+ = 2, and n = 1. Again, using Eq. (6.3-10),


This compares well with the experimental value of 1.32 × 105 (C10).

To calculate the individual ion diffusivities at 25°C using Eq. (6.3-11),


Substituting into Eq. (6.3-12) for 25°C,


Hence, one can see that the diffusivity of the salt is in between that of the two ions.


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