SELF-DIFFUSION AND INTERDIFFUSION IN BINARY SYSTEMS
2565
Self-Diffusion and Interdiffusion in Complex-Forming Binary Systems
by P. C. Carman National Chemical Research Laboratory, South African Council for Scientific and Industrial liesearch, Pretoria, South Africa (Received January id, 1967)
Darken’s relationship between interdiffusion and self-diff usion coefficients in binary systems can be modified by allowing for formation of association polymers and compounds without departing from his basic assumption that each type of molecule has an intrinsic mobility. Agreement between observed and calculated results for urea-water, nitromethane-carbon tetrachloride, and acetone-chloroform systems is very good and it is greatly improved in the sucrose-water system, compared to use of the unmodified equation. The Darken theory is thus able to account for experimental results in a number of binary nonelectrolyte systems, both ideal and, as in this paper, very nonideal. In metallic solid solutions, where it has wide application, it is associated with a mechanism of activated jumping, and it would therefore appear that this plays a considerable role in liquid diffusion. A comparison is made with diffusion theories based on relative diffusive velocities between pairs of components.
Introduction
pi = qi*
Darken’s1 theory of diffusion for a binary mixture assumes that each component i has an intrinsic mobility pi which is independent of the diffusive flux of the other component. Thus, if Ji is the intrinsic diffusive flux of i, ci is concentration in moles per cubic centimeter, v i its diffusive velocity, and Vpi the gradient of its molar chemical potential pi vi
= J i / c i = - q . A1 p . 1
(1)
On this basis, if uncombined molecules of A and B are the only diffusing entities in the system, the interdiffusion coefficient is given (see Appendix A) by
where Ni is the mole fraction of i. Further, if some of the molecules of i in a given composition are labeled, e.g., by an isotopic tracer, and the mobility of the labeled molecules is qi*, then the self-diffusion coefficient of i a t that composition is given by Di* = qi*RT (3) Assuming identity of properties of labeled and mlabeled i, it follows from Darken’s theory that, as the mobility of i is determined solely by the nature and relative proportions of A and B molecules
(4)
Consequently
where ai is the molar activity of i. As originally derived, eq 5 assumed equal partial molar volumes, but the same result is obtained when this is not the c a ~ e ~ - ~ (see also Appendix A). There is much evidence for the validity of Darken’s equation eq 5, in metallic solid solutions, where only movement of simple, uncombined atoms is i n ~ o l v e d , ~ but results in mixtures of liquid nonelectrolytes have been conflicting, especially in very nonideal (1) L. S. Darken, Trans. A m . Inst. Mining Met. Engrs., 175, 184 (1948). (2) G. S. Hartley and J. Crank, Trans..Faraday SOC..45, 801 (1949). (3) S. Prager, J . Chem. Phys., 21, 1345 (1953). (4) P. C. Carman and L. S. Stein, Trans. Faraday SOC.,52, 619 (1956). (5) (a) W.Seith and A. Kottmann, Angew. Chem., 64, 376 (1952); (b) H. W. Mead and C. E. Birchenall, J . Metals, 9, 874 (1957); (c) J. E. Reynolds, B. L. Averbach, and M. Cohen, Acta Met., 5, 29 (1957); (d) J. R. Manning, Phys. Rev., 116, 69 (1959). (6) P. C. Carman and L. Miller, Trans. Faraday SOC.,55, 1838 (1959). (7) P. A. Johnson and A. L. Babb, J . Phys. Chem., 60, 14 (1956). (8) A. P. Hardt, D. K. Anderson, R. Rathbun, B. W. Mar, and A. L. Babb, J. Phys. Chem., 63, 2059 (1959). (9) A. P. Hardt, Dissertation Abstr., 17, 1968 (1957).
Volume 7 1 , hTumber 8 July 1967
P. C. CARMAN
2566
ie., where thermodynamic factor b In a A / b In N A eq 4 is not valid. Thus, self-diffusion coefficients can departs widely from unity. This can be well underonly be introduced into eq 2 to give a modified form of stood when it is realized that, in such systems, diffusion eq 5 by taking account of the ratios q A / q A * and q B / q B * . may not only involve free molecules of A and B, but Association of Component A also (i) dimers or higher association polymers of one Suppose that component A , with a total monomeric or even both components and (ii) compounds between of c A , forms a series of association polyconcentration A and B. This would mean that three or more inmers in which the fraction of A as n-mer is an,Le. trinsic mobilities are involved instead of only two. Since the various species are in equilibrium, it is in Can= 1 (7) n=l principle possible to modify the derivation of eq 5 to take this into account. Thus, it was shown recently‘O If the molar concentration is cn, then that the relationship between D A B and self-diff usion ffn coefficients in aqueous solutions of urea could be acCn = -CA n counted for very satisfactorily on the assumption that urea is partly dimerized. The total flux of A is given by I n the following, some cases are selected where there JA = C n J n are particularly strong deviations and where it is pos(9) n i l sible to make at least a rough estimate of the proporwhere tions of association polymers or addition compounds present. A very significant point is that, in studies of eq 5 , “the thermodynamic factor appears always to overcorrect.” I n other words, positive deviations from Raoult’s law corresponding to b In U A / ~In N A < 1 in accord with Darken’s basic assumption in eq 1. Now, lead to calculated values of D A B which are too small as all polymers are in equilibrium and negative deviations corresponding to b In a A / Ani1 A n AI (11) b In N A > 1 lead to values which are too large. Xow, positive deviations can be correlated with a tendency so that to form association polymers” and, as will be shown, Vpnil = V p n VPI (12) these result in a modified Darken equation which gives larger calculated values of DABclose to observed values. from which it can be easily deduced that Conversely, negative deviations suggest compound for1 mation and the resulting modified Darken equation V p A = vpl = ’/2Vc(2 = - V p n (13) n gives smaller calculated values of D A B . The principles employed to modify the Darken equaSubstituting, one finds tion have been previously applied by A n d e r ~ o n . ~ ~ ~ ’ ~ J A -C A V ~C An f f n q n (14) If, for example, transport of component A takes place n-1 partly by free molecules and partly by complexes so that the “effective mobility” q A in eq 6 is given by which are in equilibrium with the free molecules, it is still possible to write the intrinsic flux of A as
+
+
n=1
J A = -cAqAVpA (6) But q A is now an effective mobility which contains the mobilities of both free and combined A and their relative proportions. Nevertheless, with such effective mobilities for the two primary components, eq 2 is still valid and. A n d e r s ~ n ~has ~ J extensively ~ explored the resulting variations of DABwith composition. His analysis, however, was not extended to selfdiffusion. It is also possible to write DA* = q A * R T (34 where q A * is another effective mobility, but the two effective mobilities q A and q A * are no longer equal, i e . , The JOUTW~ of Physical Chemiatry
and eq 2 becomes
If a self-diff usion experiment is carried out, labeled (10) P. C. Carman, J . Phys. Chem., 7 0 , 3356 (1966). (11) (a) F. Dolezalek, Z . Physik. Chem. (Frankfurt), 64, 727 (1908); (b) J. H. Hildebrand and R. L. Scott, “Solubility of Nonelectrolytes,” 3rd ed, Reinhold Publishing Corp., New York, N. Y., 1950, Chapter 11. (12) D . K.Anderson, Dissertation AbstT., 21, 1388 (1961). (13) D. K. Anderson and A. L. Babb, J. Phys. Chem., 65, 1281 (1961); 66, 1363 (1962).
SELF-DIFFUSION AND INTERDIFFUSION IN BINARY SYSTEMS
molecules of A in a given composition are evenly distributed over all forms of A,14 so that the effectiye mobility q A * can at once be written as
2567
DAB
(from eq 5 ) = 0.22 X
When, however, eq 16 with f(a) = 2 is used, we obtain
DAB(calcd) Thus, when self-diff usion coefficients are introduced into eq 16, the resulting modified Darken equation becomes
in which (19)
Cnanqn/Canqn
f(a>
n=l
n-1
This function must necessarily be greater than unity and so tends to give larger calculated values of DAB than eq 5 , Le., it acts in the opposite direction to the thermodynamic factor.
Formation of Dimers Here, f(a)reduces to f ( a >=
{alp1
+ 2(1 -
rul)qz)/{alql
+ (1 -
a1)q*)
(194 I n the case of urea-water solutions, up to a molality of 4, the highest concentration studied, thermodynamic properties could be interpreted in terms of an equilibrium between monomer and dimer, both in ideal s01ution.l~ An estimate of a1 could thus be made. By using a ratio of qz/ql s 0.7 for mobilities of dimer and monomer, calculated values of D A B were found to agree very well with observed values.1° The choice of q2/q1 was not critical, as f(a)was almost unchanged over the range 0.6-0.8. Another system for consideration is that of nitromethane-carbon tetrachloride a t 20".'j From spectroscopic evidence at 18", de et al., have shown that nitromethane is completely dimerized at concentrations as low as 0.15 M. In this case, it may be noted that thermodynamic properties are not in accord with an ideal solution of dimer. At the equimolar concentration studied, eq 19 simply becomes
f(4= 2
(19b)
= 0.36 X
The agreement is perhaps to some extent fortuitous, since the only value obtainable for b In U A / ~ In N A is from the data of Brown and Srnithl'j at 45", so that it should have a somewhat smaller value a t 20". Nevertheless, within the accuracy of the self-diffusion measurements and the very large variations in DAB,due to nonideality of the system, it provides strong evidence for the validity of eq 18.
Continuous Association Strongly polar molecules in a relatively nonpolar solvent show strong association to a continuous series of polymers, a typical and well-studied instance being the methanol-benzene s y ~ t e m . ~ , Characteristic 9 features in these systems are a very rapid decrease in both the interdiffusion coefficient and of the selfdiffusion coefficient of the polar constituent as the concentration of the latter is increased, followed by a leveling off, while the self-diffusion coefficient of the nonpolar component does not exhibit wide variations. The unmodified Darken equation, eq 5 , yields calculated values of DABwhich can be only a small fraction of the observed value in the middle ranges of composition. The thermodynamic and spectroscopic behavior of such systems is consistent with the view that polymers of unlimited size can be formed" and the diffusive behavior is qualitatively consistent with this. As concentration increases, the self-diffusion coefficient of the polar component corresponds to the high mobility of the monomer at first, but rapidly decreases due to formation of relatively slow-moving polymers. On the other hand, the effect of the high values of n in the numerator of f(a) = xncrnqn/xanqn is to give n=l
n=l
large values to !(a), with the result that calculated values of DABby eq 18 are several times larger than by eq 5 . Quantitative discussion will not be undertaken here, since it cannot be done briefly. Furthermore, a number of assumptions must be made about the mode of polymerization, the ideality of the solutions, and the
The data obtained were
x ~= * 1.36 x
DA*
D DAB
=
2.33
10-5 10-5
(obsd) = 0.35 X b-In a A - 0.12 b In N A
(14) R.H.Stokes, J . Phys. Chem., 69. 4012 (1965). (15) P.A. D.de Maine, M. M. de Maine, and A . G. Goble, Trans. Faraday Soc., 53, 409 (1957). (16) I. Brown and F. Smith, Australian J . Chem., 8, 501 (1955). (17) (a) H. Kempter and R. Mecke, 2. Physik. C h m . (Leipsig), B46, 229 (1940); (b) E. C . Hoffmann, ibid., B53, 179 (1943); (c) 1. Prigogine, V. Mathot, and A . Desmyter, Bull. SOC.Chim. Belges, 58, 547 (1949); (d) L. SarolBa-Mathot, Trans. Faraday SOC.,49, 8 (1953).
Volume 71, Number 8 July 1967
P. C. CARMAN
2568
mobility of the polymers, all of which permit introduction of arbitrary parameters. It is not therefore difficult to make a plausible selection of parameters to fit the experimental data.
+
CAVPA CBVPB = 0
(28)
Then substituting from eq 20, 21, 26, 27, and 28 in eq 24 and 25 and thence in eq 22 and 6, the expression for qA is obtained
Compound Formation For a system which shows a negative deviation from Raoult’s law over the whole range of composition, the simplest interpretation of the thermodynamic data is that a 1 : l compound is formed and that the two free components and the compound form an ideal ternary mixture.” This is applicable to the acetone-chloroform system, but, in the sucrose-water system, we must consider hydrated sugar molecules containing h molecules of water. This is a much more complex case, since one must consider equilibria between a series of molecules AB, where n can vary from 1 to h. Furthermore, when h > 1, even assumption of ideal solutions does not give a negative deviation from Raoult’s law over the whole range of composition. The following derivation deals with the most general case. Suppose that concentrations of free components and of the compounds AB, are cl, CZ, c3,. Then
The true mole fraction of A, ignoring compound formation, is N A = CA/(CA CB) and apparent mole fractions Ni are defined by
+
N i = ci/(cA
(21)
CB)
The total fluxes of A and B are given by
By similar reasoning
These are the quantities which should be employed in eq 2. The self-diffusion mobilities qA* and qB* can be written down directly as proportionated between the free and combined forms of components A and B. (31)
From this
Thus, substituting qA and qB into eq 2, with Di* in place of qi*, we obtain
DAB =
b In aA [NBDA* b In N A ~
h
+ CJ3n JZ+ CnJ3, h
JA = JI
(22)
n-1 h
JB =
(23)
n-1
where each type of molecule has its own intrinsic mobility in accord with eq 1. Thus, in eq 22
Ji = -ciqiVki
(24)
J3n = -~3nq3nV~an and we wish to obtain an apparent mobility whole of component A as defined by eq 6. Owing to equilibria vP1 =
VPA, v 112 = VPB
Vr3, = V P I
+~ V P Z
and, from the Duhem-Margules equation The Journal of Physical Chemiatry
+ NADB* -
(25) PA
for the
(26) (27)
n=l
2 - ( n - 1)“-1>1 (35) NB
In 1:1 compounds, the third term in the square brackets reduces to -2q3N3RT, which is always negative, so that lower calculated values of DAB are obtained than with eq 5 , thereby acting in the opposite direction to the thermodynamic factor, which is greater than unity. To apply eq 35 quantitatively, of course, estimates must be made of q3 and N3, but discussion of this will be confined to the specific case of acetone-chloroform. The case of 1:h compounds is more complex, since the third term can become positive as the ratio N A / N B increases, but so also does the deviation from Raoult’s law if compound formation is the sole source of nonideality. The practical application of this case requires determination of h values of q3, and of Nan, so that it is only possible to deal with simplified cases as will be discussed for sucrose-water.
SELF-DIFFUSION AND INTERDIFFUSION IN BINARY SYSTEMS
2569
Acetone-Chloroform System at 25" This is the best known example of a negative deviation from Raoult's law and its thermodynamic data can be interpreted in terms of a 1 : l c ~ m p o u n d . ~ ~ J The ' ~ algebra required for the operation is omitted here. The uncertainties in N 3 and 43 have not as great an Thus, if C represents chloroform, eq 35 becomes effect as might be expected. Calculations were reb In a A peated using values of K between 1.2 and 1.6 and the DAG = ___ [NcDA* NAD,* - 2q3N&T] (35a) b In N A factor for q3 between 0.6 and 0.8, without seriously affecting the general agreement between observed and Accurate measurements of activities for 25" are availcalculated values in Table I. able18p19and, apart from giving the thermodynamic factor, these can be used to calculate the equilibrium constant K cv 1.5, where
+
Table I : Comparison of Calculated and Observed Values of DACin the Acetone-Chloroform System
and hence N 3 from
Equation 36 assumes activities equal to true mole fractions, e.g., for uncombined A, the true mole fraction is ( N A - N B ) / ( N A Nc - N3) = (NA- N a ) / ( l - N d , and similarly for the others. That this is a very approximate assumption is shown by the fact that the plot of b In aA/b In N A vs. NA should be symmetrical, whereas it is decidedly unsymmetric. Clearly, even if most of the deviation from Raoult's law can be accounted for by compound formation, it is not the sole factor. It follows that the calculation of N3 is attended with some degree of uncertainty. To estimate 43, we can assume that it is 0.7 of the average of the mobilities of the two free components, i.e.
+
93
0.35(qi
+ qz)
(38)
This is in line with the previous assumption used for the urea-water system that qdimer
s 0.7qmonomer
The use of an arithmetic mean mobility can be justified by the fact that acetone and chloroform differ little in molar volume and have similar mobilities. If the differences were large, it would be better to average resistances, i e . , to use the harmonic mean of q1 and q2 instead of the arithmetic mean, though this would greatly increase the labor of calculation. To get 43 in terms of DA* and Dc*, it is necessary to eliminate q1 and q 2 from eq 38 and from
DA*/RT
=
(1 -
= q1
z) -
+ q3-N3 NA
(39)
NA
Na
DA* X 1Oa
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.057 0.107 0.149 0.174 0.183 0.174 0.149 0.107 0.057
2.41 2.54 2.72 2.86 3.02 3.23 3.49 3.80 4.20
DEI* X 1Oa
2.67 2.83 2.89 2.99 3.05 3.21 3.42 3.60 3.62
X 10-7
-DAC Darken In N A eq 5,
Obsd
eq 35a,
1.20 1.40 1.54 1.59 1.53 1.43 1.33 1.21 1.10
2.67 2.97 3.18 3.37 3.44 3.47 3.52 3.56 3.60
2.66 3.06 3.26 3.36 3.30 3.36 3.52 3.63 3.66
Calcd
'lnaA
2.94 3.70 4.26 4.60 4.65 4.63 4.60 4.40 3.98
Measurements of DAChave been given by Anderson, Hall, and Babb,20while Hardt, et a1.,8 have published values of DA* and Dc* using the capillary method and HardtQhas also given values for DA* which were decidedly larger in the acetone-rich region. Nore recently, measurements of DA* and Dc* have been made by McCall and DouglassZ1using the nmr spin-echo method and it is these values which have been used in Table I.
Sucrose-Water System at 25" The data of Irani and Adamson22for this system show that over a wide range of concentration (NwDs* NsDw*) alone is greater than D ~ wso , that the thermodynamic factor in the unmodified Darken equation (eq 5 ) acts in the wrong direction. As first pointed out by Scatchard,2s however, the marked negative
+
(18) H. Rock and W. Schroder, Z. Phusik. Chem. (Frankfurt), 11, 41 (1957). (19) C. R. Mueller and E. R. Kearns, J . Phys. Chem., 62, 1441 (1958). (20) D.K. Anderson, J. R. Hall, and A. L. Babb, ibid., 62,404 (1958). (21) D.W.McCall and D. C. Douglass, ibid., 71, 987 (1967). (22) R. R. Irani and A. W. Adamson, ibid., 62, 1517 (1958). (23) G. Scatchard, J . Am. Chem. SOC.,43, 2406 (1921).
Volume 71 Number 8 July 1967
P. C. CARMAN
2570
Table 11: Comparison of Calculated and Observed Values of DEWin the Sucrose-Water System
x 106
x
108
h
a In U S a In Ns
4.26 3.76 3.18 2.47 1.96 1.76 1.56 1.46 1.34 1.20
21.5 15.3 8.76 3.50 1.14 0.61 (0.32)' (0.22)' (0.15)" (0.06)"
5.0 5.0 5.0 5.0 4.5 4.2 3.9 3.8 3.6 3.3
1.03 1.10 1.26 1.50 1.74 1.85 1.95 2.00 2.04 2.11
Ds*
CS
0.2 0.5 1.0 1.5 2.0 2.2 2.4 2.5 2.6 2.8
N8
0.0038
0.0100 0.0224 0.0383 0.0600 0.0704 0.0825 0.0895 0.0968 0.112
' By extrapolation.
Dw*
' Reduced values.
- 2h-N s
Nw
h(h
- 1Ix2} Nw
+ + NsDw*]
(41)
The data of Irani and Adamson, however, extend to = 0.112, i.e., to a supersaturated solution of sucrose, and application of the Robinson and Stokes calculation to these gives a steadily decreasing value of h, as would be expected when competition for water molecules increases. If we can assume that the values obtained represent average degrees of hydration and ignore differences of mobility for sucrose molecules with different degrees of hydration, eq 41 could still be regarded as a reasonable approximation to (35). Owing to the approximate nature of the calculations, only two significant figures have been used for h in Table 11. As shown in Table 11, there is a great improvement compared with the unmodified Darken equation (5)
Ns
The Journal of Phyeieal Chembtrg
Obsd,
Obsd,
eq (eq 5)
ref 22
ref 25
4.46 4.24 4.16 3.75 3.32 3.12 2.85 2.70 2.55 2.28
4.55 3.68 (2.8)* (2. 0)' 1.34 1.14 0.93 0.83 0.73 (0.52)'
4.1 3.0 2.2 1.5
Calcd, eq 41
4.31 3.85 3.30 2.44 1.68 1.42 1.14 0.99 0.88 0.67
From ref 25c.
deviation from Raoult's law in this system can be accounted for quantitatively in dilute solutions if it is assumed that every sucrose molecule is combined with about five molecules of water and that the hydrated molecules form ideal solutions. A calculation on this basis, with use of the accurate data of Robinson and Stokes,24and also their method of calculating h from water activities shows that h is almost constant a t 5.0 or 5.1 up to a molality of about 2 moles/lOOO g of water, ie., NS 'v 0.035. A reasonable starting point is thus to assume that, up to this concentration, sucrose molecules are uniformly hydrated with 5 moles of water. This assumption greatly simplifies eq 35, since there is only a single value of n = h, N3* = Ns, and N1 = 0, so that qs* = @ h , (from eq 31) or q3hRT = &*. Thus Dsw = -[NwDs*(l b In as b In N S
Dsw X lob------------. Darken
over the whole range of compositions studied. On the whole, the calculated values of DSW are still appreciably higher than observed values, but the difference is not unreasonable in the light of the assumptions made, particularly at the higher concentrations. If the differences are real, it would suggest that values of h for diffusion should be a little larger than those obtained from water activities. The experimental values in the middle range, cs = 0.5-2 moles/l., however, are open to question. The measurements of Irani and Adamson in this range are very scanty, and a recalculation of the experimental data in their Table I has yielded the bracketed values in Table I1 for cs = 1.0 and 1.5, which are appreciably higher than their values. More recent measurements by Sato, Hoshino, and MiyamotoZSaused a completely different experimental method which was most accurate in the middle range, but much less accurate at the extreme concentrations. Values taken from their curve are given in Table 11. At the extremes, both upper and lower, values of Irani and Adamson agree well, respectively, with those of English and and of Morris and G o ~ t i n g , *and ~ ~ must thus be accepted.
Discussion One of the most characteristic results of the Darken theory is the relationship between self-diff usion and interdiffusion coefficients in eq 5 for binary systems of nonelectrolytes and there are a number of nearly ideal systems which conform very closely to it, namely, (24) R. H. Robinson and R. H. Stokes, (1961).
J. Phys. Chem., 65, 1954
(25) (a) K. Sato, S. Hoshino, and K . Miyamoto, Kagaku Kogaku, 28,445 (1964): (b) A. C. English and M. Dole, J. Am. Chem. Soc., 72, 3261 (1950); (c) L. G. Gosting and J. M. Morris, ibid., 76, 3379 (1954).
SELF-DIFFUSION AND INTERDIFFUSION IN BINARY SYSTEMS
(a) benzene-brornobenzenez6 a t 20", (b) heptanecetane at 20°,27 (e) octane-decane a t 25 and 60°,28 (d) cyclohexane-carbon tetrachloride at 25°,29 (e) benzene-carbon tetrachloride at 25" ,13n30 and (f) benzene-diphenyl at 25°.81 Equations 18 and 35 are modified forms assumed by eq 5 when association polymers or compounds, respectively, are present. These require no departure from the basic Darken assumption that each diffusing species is characterized by its own intrinsic mobility. In the decidedly nonideal systems discussed in this paper, therefore, there is sufficient agreement between calculated and observed values of D12 to suggest that Darken's intrinsic mobilities have a wide validity in liquid systems. In the systems discussed, there is evidence from spectroscopy or from the behavior of the thermodynamic activities to enable the probable complexes to be identified and at least a rough estimate of their concentrations to be made. There are a number of other systems for which observed interdiffusion coefficients up to 10% larger than values calculated by eq 5 are found, e.g., (i) acetone-benzene at 25' ,21 (ii) nitromethane-benzene at 20",26 (iii) benzene-cyclohexane at 25°,21J2 (iv) benzene-normal heptane at 25°.Ya Since deviations from Raoult's law are positive, these could be explained satisfactorily by a small degree of association, but the possibility must also be considered that Darken's theory is not exact for all such systems. To appreciate this, some understanding of the relationship of Darken's theory to diffusive mechanism is necessary. The field in which eq 5 has been most successful is that of metallic solid solution^.^ In these, DA* and DB* can vary over orders of magnitude for the whole range of composition and the variation is independent, since the ratio DA*/DB*can have a wide range of values and the two coefficients usually have quite different energies of activation. In solid solutions, the independent mobilities can also be made visually evident by movement of inert markers (Kirkendall effect). That unmodified eq 5 is so widely applicable, even for very nonideal systems, is undoubtedly because of the fact that movement is by single atoms, so that complexes are excluded. The view taken by metallurgists is that the Darken theory is associated with a mechanism of activated jumps and is evidence for this mechanism. Thus, another possible mechanism such as place exchange leads to a completely different relationship between D A Band self-diffusion coefficients. Since each type of atom can be expected to require its own characteristic jump frequency and energy of activation for a jump, the independence of the intrinsic mo-
2571
bilities and hence of the self-diffusion coefficientsis to be expected. If Darken's theory can be applied to liquid systems, therefore, it seems probable that a mechanism of activated jumps plays at least a considerable role. Since liquid molecules are not on a lattice and are in continual motion, one must assume that a jump consists in escape from a cage of neighboring molecules to a new environment. This, however, is contrary to the most elaborate theories of liquid diffusion in vogue at present,84-86which are based upon a Brownian motion mechanism, i.e., liquids are treated essentially as dense gases. Now, in gaseous diffusion, an intrinsic diffusive mobility or intrinsic friction coeficient relative to the medium as a whole has no meaning, since the resistance to diffusion experienced by a component is the sum of a number of terms, one for each of the other components present. Each comprises a friction factor specific to the pair of components involved and their relative diffusive velocity. As a result, if, for example, component A of a binary mixture is labeled, the system is ternary and the self-diffusion coefficient DA* is not given by a simple mobility but by a sum of two terms, one for friction between labeled A with component B and one for friction between labeled and unlabeled A. The actual equation is
1 DA*
-
+-
NA NB (DA*)o D A B
where ( D A * ) ~ is the self-diffusion coefficient for pure A.87*38Similarly, in liquids, if mobilities or frictional coefficientsare based upon relative velocities, as in the theories of Lammas and Laity,4O two terms are re-
(26) L. Miller and P. C. Carman, Trans. Faraday SOC.,55, 1831 (1959). (27) L. Miller and P. C. Carman, ibid., 58, 1529 (1962). (28) A. L. van Geet and A. W. Adamson, J . Phys. Chem., 68, 238 (1964). (29) M. V. Kulkarni, G. F. Allen, and P. A. Lyons, ibid., 69, 2491 (1965). (30) C.S. Caldwell and A. L. Babb, ibid., 60, 51 (1956). (31) R. Mills, ibid., 67, 600 (1963). (32) (a) D.W.McCall and E. W. Anderson, ibid., 70, 601 (1966); (b) I. Kamal and E. McLaughlin, Trans. Faraday Soc., 62, 1762 (1966). (33) A. de Korte, Dissertation Abstr., 26, 1932 (1965). (34) J. G.Kirkwood, J . Chem. Phys., 14, 180 (1946). (35)R. J. Bearman and J. C. Kirkwood, ibid., 28, 136 (1958). (36) R. J. Bearman, ibid., 32, 1308 (1960). (37) L. Miller and P. C. Carman, Trans. Faraday Soc., 57, 2143 (1961). (38) L. Miller and P. C. Carman, ibid., 60, 33 (1964). (39) 0.Lamm, Acta Chem. S a n d . , 8 , 1120 (1954). (40) R. W.Laity, J . Phys. Chem., 6 3 , 80 (1959).
Volume 7 1 , Number 8 July 1967
P. C . CARMAN
2572
quired for self-diffusion in binary systems, even if no complexes are formed. If the friction coefficient between labeled and unlabeled A is assumed independent of composition, Lamm has shown that an equation closely analogous to eq 42 results, namely
-1- --
+----
Darken’s theory, not to Bearman’s. The range for (b) is 40%’ which would make it impossible even to approximate eq 5 on Bearman’s theory. The further point, which is significant for this paper, is that modified eq 18 and 35,obtained when complexes are formed, cannot be derived from friction coeficients based upon relative velocities. A conclusion which does seem inescapable, however, is that a Brownian motion mechanism must play a t least some role in liquid systems. To this extent, the Darken type of equation and probably any equation based upon a simple premise cannot be expected to give a highly accurate correlation with experimental data in most systems. This may well be the case for systems i-iv.
C A V A NB b l n u (43) DA* (DA*)o DABb In N A where V Ais the molar volume of pure A.283a7J9,40 Obviously eq 43 is a completely different relationship to eq 5, but other equations can be derived on this theory by assuming other relations for the friction coefficients. Bearman has shown that the Lamm-Laity type of theory4I is based upon a Brownian motion mechanism by a statistical mechanical approach and pointed out that, under certain conditions, the ratio Appendix A for the two friction coefficients can be determined by Derivation of Eq 6. If the partial molar volumes of the two components and is therefore effectively independent of composition. J A = -cAqAvpA Under these circumstances, eq 5 is 0btained.3~8~~ J B = -cBqBVPB = CAqBVPA A great deal of confusion has arisen from this, since it has been interpreted as a proof that the Darken and then the volume mean diffusive velocity, Bearman theories lead to the same r e ~ u l t , ~ ldespite ,~l by the contradictory nature of the basic assumptions. vov = J A V A JBVB It must be remembered that the restrictions imposed
210’
is given
+
in Bearman’s case are very highly specialized and that, under other circumstances, eq 43 is an equally valid deduction from the same theory. They require that A and B molecules be of much the same size and shape and approximately the same intermolecular potential. The various friction coefficients then bear a k e d ratio to one another, independent of composition, with the result that DA* and DB* are not independent but bear a fixed ratio t o one another which should be in the inverse ratio of the molar volumes. Furthermore, Di*q, where q is the viscosity of the mixture, should also be independent of composition If we turn to the systems listed at the beginning of this discussion for which eq 5 is valid, we find hardly any of Bearman’s requirements fulfilled. Only in (d) and (e) is Di*q even approximately constant for one of the components over the range of composition and in (b) the variation can exceed 100%. I n (c), D A * / D B * is reasonably constant and is fairly near to VB/VA, so that it does conform t o Bearman’s assumptions in this respect. I n (f), DA*/DB* is constant, but quite different from Vg/VA. I n the others, DA*/DB*shows a sufficient range of variation to suggest that DA* and DB* are in fact independent and therefore conform to
and the diffusive flux relative to uovl JA’, is given by
=
- ~0’) = J A ( ~- CAVA)- CAJBVB V B ( C B JA CAJB)
=
-cArB(cBqA
JAv =
+ CAqB)VpA
However J~~ =
-DABVCA
and
b In C A --
41342
The Journal of Physical Chemistry
CA(VA
b In N A
VB
+
NAVA N B V B
where
Therefore
(41) R. J. Bearman, J . Phys. Chem., 65, 1961 (1961). (42) J. K.Horrocks and E. McLaughlin, Trans. Faraday Soc., 58, 1357 (1962).
