Concentration and Temperature Dependence of Liquid Diffusion Coefficients John 1. Gainer Department of Chemical Engineering, University of Virginia, Charlottesville, Vu. 22901
Prediction of the concentration dependence of the liquid diffusion coefficient has required the use of activity data in most previous equations and correlations. A method is presented here, based on a modified absolute rate theory approach, to estimate the variation of the diffusivity with concentration changes using only the physical properties of the liquids involved, such as viscosity and molar volume. This method is tested on three types of systems, containing both associated and nonassociated compounds, good predictions resulting in all cases. The variation of the diffusivity with temperature is also discussed, using the modified rate theory equations.
D I F F c S I O S COEFFICIEXTS for a pair of liquids are usually meawred for a case in which one liquid is present in a very small amount. Most correlations and equations for the prediction of liquid diffusivities are also valid only a t this infinite dilution state (Gainer and Metzner, 1965; Wilke and Chang, 1955). However, it is frequently of more interest to know the diffuqivity a t other concentration levels. The most prevalent method for prediction of diffusivities a t concentrations other than the infinite dilution case is to use a thermodynamic correction factor such as (Johnson and Babb, 1956)
(
DAB= DAB" 1 idlnYA) This equation is obviously restrictive. It requires that DAB"= D B A " , which is seldom the case. It has also been proposed (Cullinan, 1966; Vignes, 1966) that the concentration dependence of the diffusivity be given by
Dullieii (1969) has recently shown that Equation 2 is not estremely accurate for a number of systems, and a new equation has been proposed by Cullinan and Leffler (1969) :
It has been proposed that this equation is more accurate than Equation 2. The diffusivities at infinite dilution may be estimated using the Wilke-Chang equation (1955) for systems containing solvents with lower viscosities (up to 3 centipoises), or using the modified absolute rate theory equations proposed by Gainer and lletzner (1965). However, activity coefficients must be known in order to estimate the diffusivity a t higher concentrations. Clearly, then, a method for estimating these diffusivitieu using only the known physical properties of the pure liquids such as molar volume, viscosity, molecular weight, and heat of vaporization would be helpful. A modification of the available diffusivity equations would appear to be a hopeful way of doing this. Any such modification can be accomplished only if the equations implicitly contain a factor which accounts for the iritermolecular forces
present in a liquid-liquid system. The Wilke-Chang equation contains an associat'ion factor which appears to account, for intermolecular forces of the solvent in some cases. The equations proposed by Gainer and Xetaner contain a method of estimating activation energies for liquid-liquid systems. Therefore, bot'h equations might be modified to take into account changes in intermolecular forces. This paper presents a modification of the absolute rate theory approach, in order to predict' the concentration dependence of the diffusivity. The modified rate theory equat,ions are (Gainer and Xetzner, 1965)
where
E p , - ED,,
=
2
-
(k)X
T h e viscosity activation energy, from absolute rate theory, is determined using Equation 4:
Using Equations 4, 5 , and 6, the ratio of the two infinite dilution diffusivities for the case of 5~ = { E becomes
Using Equation 7 , Equation 2 may be rewritten as
and Equation 3 may be rewritten as
DAB^
=
+
d In Y A D A B " ~ B ~ B ~(AI ~ A K ~ A d In X A
where
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
381
/-
5.0
I .2
0
1
1
I
I
1
0.2
0.4
0.6
0.8
1.0
0
CONCENTRATION, Mole Froction Ethonol Figure 1. Diffusivity of ethanol-water at 10°C A. Solute
c 0
0.2
0A
CO N CE NTRAT IO N
6. Solvent
Figure 3.
0 Experimental
Diffusivity
,
0.6
0.8
1.0
Mole Fract ion Benzene
of ethanol-benzene at 25°C
0 Experimental
It is apparent that Equations 8 and 1 are equivalent if K = 1, and Equations 9 and 1 are equivalent if K = 1 and
=f i = ~ I.(. Since these physical conditions will be true for many ordinary binary liquid systems, attempts to prove the superiority of one of the equations (Equation 1, 8, or 9) may prove difficult. Returning to the original idea of providing an equation to estimate diff usivities over a range of concentrations without using thermodynamic data, it is now proposed to use Equations 4 , 5 , and 6. I n order to use these equations over the entire concentration range, it will be necessary to make some assumptions. Such arbitrary assumptions might be t h a t the system is to be considered a pseudo-binary where the solute is designated as the component present in the lesser amount (on a mole per cent basis), the solvent is a single component having properties which are those of the solution at any concentration in question, and the value of 5~ for the pseudosolvent is assumed to be t h a t for the component present in the larger amount (on a mole per cent basis), while ( A is taken to be the larger value (6 or 8) for the over-all system. Assumptions similar to these have been made (Kamal and Canjar, 1962). Thus i t is possible to calculate diffusivities over the whole concentration range. These are shown in Figures 1, 2, and 3 for three systems (Hammond and Stokes, 1953; Irani and Adamson, 1960; Rodwin et al., 1965). Since the viscosity activation energy is the basis for the accounting of intermolecular forces in the modified rate theory approach, it is interesting to see if the variation in viscosity over the concentration range is in any way reflected in the variation of diffusivity with concentration. Figure 2 shows the diffusivity increasing linearly with concentration while the viscosity is going through a minimum, yet the calculated PA
0.95C
0.900 u)
0.
u
y
0.85C
0.8N
1.2
P
3
2 102
0.8
0
Figure 2. at 25°C
382
0.2
0.4
0.6
0.0
I.o
CONCENTRATION, Mole Fraction Cyclohexane Diffusivity of cyclohexane-carbon tetrachloride
Ind. Eng. Cham. Fundam., Vol. 9, No. 3, 1970
diffusivities are close in value t o the experimental ones. Figure 3 shows t h a t viscosity decreases over the concentration range, but diffusivity goes through a minimum over the same range. These systems were chosen because one contains two hydrogen-bonding compounds (Figure 1), one contains two hydrocarbons (Figure 2), and one contains a hydrogenbonding compound and a pure hydrocarbon (Figure 3). This latter case has been dealt with in special ways (Rathbun and Babb, 1966; Vignes, 1966), and the data presented here might be estimated more accurately if the ethanol were considered to be present in complexes of several molecules, as has been done before (Vignes, 1966). However, the general trend of the data is evident and, without considering this system to be a special case, the accuracy is good over the concentration range for which the mole fraction of benzene is less than or equal to 0.8. I n general, although only a limited amount of data has been investigated, it would appear t h a t the variation of diffusivity in binary liquid systems with concentration may be estimated using a modified absolute rate theory approach. Temperature Dependence
Most earlier equations and correlations such as the wellknown Stokes-Einstein or Wilke-Chang equations result in the following dependence of the diffusivity on temperature:
Use of Equation 4 would predict t h a t
(
DABPB ED,, exp RT
)
EfiB
= constant
(11)
The activation energies calculated according to Equation 6 change with temperature variations, probably because one of the assumptions used in developing Equation 4, that the partition functions for the normal and activated states cancel each other (Gainer and Metzner, 1965; Olander, 1963), is not true over a range of temperatures. Some recently published data (Unver and Himmelblau, 1964) offer a striking comparison of Equations 10 and 11 for a highly nonideal system, t h a t of the diffusion of 1-butene into water. These are shown in Table I, where Equation 10 refers to calculations made using the Wilke-Chang equation, and 11 to those made using the theory of Gainer and RIetzner. S o t only does Equation 4 better account for the nonideality of the system, but Equation 11 predicts the diffusivity to double over the temperature range shown in Table I, which is shown to be the actual case. Prior-art equations would have predicted a fourfold increase over the same temperature range. Only for the special case of self-diffusion will Equation 11 be equivalent to Equation 10. Again, these calculations are
Table 1. T,
O C
Exptl.
7
5.16 6.61 9.49 12.5
25 40 60
Calculations DAB X Eq. 1 1
6.50 9.00 10.8 13.5
lo5 Cm2/Sec Eq. 10
0.67 1.15 1.64 2.44
limited, but the indication of the applicability of absolute rate theory is evident. Nomenclature
diffusivity of solute A through solvent 13 at infinite dilution of X in B diffusivity of solute X thiough iolveiit U a t concentration C A viscosity activation energy due to hydrogen bonding viscosity activation energy due to dispersion forces energy of vaporization = A H \ a D - RT Boltzmann constant molecular weight distance between molecules X and Y absolute temperature molar volume mole fraction of A activity coefficient of A in solution geometrical parameter obtained from self-diffusion data (usually equal to 6, but sometimes has a value of 8) viscosity Literature Cited
Cullinan, H. T., Jr., IKD. ENG.C H m . F E S D ~5X, 251 . (1066). Cullinan, H. T., Jr., Leffler, J . E., 6jth National Meeting, Akmerican Institute of Chemical Engineers, Cleveland, Ohio, May 4-7, 1969.
Dullien, F. A . L., 65th Sational Meeting, American Inatitute of Chemical Engineers, Cleveland, Ohio, May 4-7, 1969. Gainer, J. L., RIetzner, A. B., Proceedings of A.I.Ch E.-I. Chem. E. Meeting, London, 1965, T'ol. 6, p. 74, 1965. Hammond, B. R., Stokes, R . H., Trans. Faraday Soc 49, 890 (1953). Irani, R. R., Adamson, A . W., J . Phys. Chern. 64, 199 (1960). Johnson, P. A,, Babb, A . L., Chem. Revs. 5 6 , 357 (19*56). Kamal, M .R., Canjar, L. S . ,-4.I.Ch.E. J . 8, 329 (1962). Olander. D. R.. A.I.Ch.E. J . 9. 207 (1963'1. Rathbun, R. E., Babb, A. L., Ind. Eng. Chem. Process Des. Develop. 5 , 273 (1966). Rodwin, - - - L., Harpst, J. A,, Lyons, P. A., J . Phys. Chem. 69, 2783 (1Yti5 1. Unver, A. A., Himmelblau, D. AI., J . Chenz. Eng. Data 9, 428 (1964). ENG.CHEM.FUNDIM. 5 , 189 (1966). Vignes, A,, IND. Wilke, C. R., Chang, P., d.Z.Ch.E. J . 1, 264 (10%). RECEIVED for review J U ~22, V 1969 ACCEPTEDMarch 15, 1970
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