J . Phys. Chem. 1994,98, 5565-5575
5565
Distinct Diffusion Coefficients in Binary Nonelectrolyte Mixtures: Frames of Reference R. Mills,' R. Malhotra, and L. A. Woolf Atomic and Molecular Physics Laboratories, Research School of Physical Sciences, Australian National University, Canberra 0200, Australia
Donald G. Miller Lawrence Livermore National Laboratory, University of California, Livermore, California 94550 Received: December 21, 1993; In Final Form: March 17, 1993"
Generalized transport coefficients are indispensable for the study of interactions between species of the same kind in binary solutions. Distinct diffusion coefficients are best suited for this purpose because of their finite limiting values and inherent symmetry. Equations are derived for the calculation of these coefficients from experimental data for the mass-, volume-, number-, and solvent-fixed frames of reference. Distinct diffusion coefficients in the 4 reference frames computed for 14 binary nonelectrolyte solutions show that the numberfixed frame is probably best suited for the comparison of the macroscopic properties of these solutions. A point of contention has been the frame of reference of the Friedman-Mills equation;' surprisingly, the mass-fixed frame describes its Dt values closely.
1. Introduction Generalized transport coefficients are becoming increasingly useful for examining the properties of electrolyte and nonelectrolyte solutions. This usefulness stems from the fact that certain of these coefficients can be related to specific interactions between pairs of molecules of the same kind. This specificity is not usually given explicitly by experimental transport coefficients such as self-diffusion and mutual diffusion coefficients. For nonelectrolyte binary mixtures, the generalized transport coefficients that have been used mostly in the past 2 decades have been velocity cross-correlation coefficients. These were first introduced by McCall, Douglass, and Frisch2 and their application later expanded by Hertz and c o - ~ o r k e r s .In ~ the present work we use a closely-related quantity termed the distinct diffusion coefficient which has recently been introduced and discussed by Friedman and c o - ~ o r k e r s . ~ The distinct diffusion coefficients Dd can be most simply defined in an Einstein-type formalism. ?he intradiffusion (selfdiffusion) coefficient of species i in a binary liquid mixture is defined by
0:= [ ( ( x i ( t )- x i ( 0 ) ) ' ) / 2 t ]
V-+ m, t
-
m
(1-1)
and the distinct diffusion coefficient for particles i and j by
Di = [ N ( ( x i ( t )- xi(o))(x,(t)- x j ( 0 ) ) ) / 2 t I V-
m,
t
-
m
(1-2)
where xi is a Cartesian coordinate of a particle of species i, (...) is an equilibrium ensemble average, Vis the total volume, t is the time, and N is the total number of particles. V-+ m specifies the thermodynamic limit; i and j are distinct particles and may be of the same or different species. In this formulation, the particle velocities, x, are in a coordinate system that is fixed in the laboratory." Thus, in descriptive terms, the distinct diffusion coefficient, Dlf, is a measure of the displacement of a particle i, due to the displacement of a particle j , where the particles are distinct even if the species are the same. Distinct diffusion coefficients can be evaluated from experimental data using expressions derived from linear response theory e Abstract
published in Aduance ACS Abstracts, June 1, 1994.
0022-365419412098-5565$04.50/0
as presented by Steele.5 These expressions, in their final form, use the velocity time correlation formalism instead of the displacement correlation functions given in eq 1-2. For binary nonelectrolyte solutions
(Lij)M= ' / , k T V J o m ( J i ( 0J)j ( t ) ) d t
(1-3)
where (Lij)Mis the Onsager phenomenological coefficient on the mass-fixed frame, k is the Boltzmann constant, Tis the absolute temperature, and (Ji(0) J j ( t ) ) is the ensemble average mass current cross-correlation function between particles i and j . The distinct diffusion coefficients (D$)Mcan be defined in terms of the above ensemble average by analogy with eq 1-2. Also the Onsager transport coefficients can be related to experimental quantities. Hence the (Dlf)Mcan be evaluated by the following equations (see section 2 ) :
+ x,M,)'B;r) i # j (1-4) (D;.)"= DVM;xj/(xi(xiMi+ xjMj)*B;r)- ((q),/~i) (1-5) (D$M = -DvMiMj/((x,M1
where Dv is the interdiffusion coefficient on the volume-fixed frame of reference, (4)j is the intradiffusion coefficient of i in a mixture of i and j , B; is the thermodynamic factor (a In ai/a In xi) with concentrations expressed in mole fractions xi, and Mi is the molecular weight of species i. Linear response theory' gives Onsager transport coefficients on the mass-fixed frame of reference, M, so that distinct diffusion coefficients are expressed initially on that frame. It will be noted that the interdiffusion coefficient Dv is measured on a volumefixed frame, hence the molecular weight conversion factors appearing in eqs 1-4 and 1-5. More generally, as Friedman and co-workers" have shown, the Onsager coefficients and D): are related on any internal reference frame R by the equation
n kT
[Lij]R= [a' /$i
d R + Xj[Dij] ]
where ni is the number density of species i and 6ij is the Kronecker delta. In the past, velocity cross-correlation coefficients on the massfixed frame have been the quantities that have been used to 0 1994 American Chemical Society
Mills et al.
5566 The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 TABLE 1: Binary Reference Frame Transforms reference frame
a!
DR
=Mc,
ARO
YZ
examine phenomenological properties of solutions, but a large difference in mass between the two constituents may distort such interpretations.6 Although it will still be necessary to tabulate the (D$)Mfor use primarily with theories based on statistical mechanics, it would seem that other frames of reference could be more suitable for some approaches. For example, in theories of transport in electrolyte solutions (such as the AltenbergerZhong-Friedman theory') where the solvent is regarded as a background material, the D$ are converted to a solvent-fixed frame of reference. For approaches in which the D$ are correlated qualitatively with solution properties which can be deduced from equilibrium thermodynamic data, such as dimerization, compound formation, and aggregation of constituents, other reference frames may be more useful. The question of the relation between cross-correlation coefficients and binary nonelectrolyte solution properties has also been discussed by Mills and co-workers,8 Hertz and Leiter,3b Bender and Pecora? and Weingartner.Io However, all these papers have been in terms of thef;, coefficients (D; = xh,)and it is possible that the D$ with their finite limiting values and inherent symmetry (i.e. (@JR = (D$R, where i # j ) could be more informative. Kat011 has used the distinct diffusion coefficient formalism, but his study and all the work mentioned above have focused on the mass-fixed frame of reference alone and with only a limited number of systems being studied in each case. In this work, then, equations for binary nonelectrolyte solutions which related the D; to experimental quantities are obtained for the solvent-fixed, mass-fixed, volume-fixed, and number-fixed frames of reference. In addition to the general equations for these frames, special attention is given to deriving equations for obtaining the limiting D$ for a trace species in a preponderant partner since these are not given by the general equations. In the next section, derivations of these several equations will be presented; in the following one, the results of calculations of D i in various frames of reference for 14 binary systems will be discussed. In this paper only representative tables and figures will be shown, but a tabulation of numerical data and sources of data for all 14 systems will be available as a separate publication.
(4)"
2. Derivation of and Limiting Equations for Binary Nonelectrolyte Mixtures It is usually simpler to analyze the diffusion properties of binary systems in terms of the single independent interdiffusion (Le. mutual) coefficient DRor the "diffusion Onsager coefficient" LR. The superscript R indicates the reference frame of interest, and the numerical values of DR and LR for a given system depend on
ARN
ficients are usually measured on the volume-fixed reference frame, DR must be obtained from them by reference frame transformations. Since one purpose of this paper is to compare the D$ on different reference frames, it is useful to begin with transformations of the independent DR and the independent LR. The solute is denoted by 1, and the solvent, by 2; superscript 0 denotes the solvent-fixed reference frame, V the volume-fixed, M the massfixed, and N the number-fixed. The notation used is from Miller et a1.,12 one of many papers dealing with reference frame transformations. To minimize the number of equations, that paper should be read in conjunction with this one. All equations are in mole quantities. Equations 10,22,23, and 32 of ref 12 arevalid for any number of components and can be given in scalar form for two components as follows:
L~ = ( A ~ ) ~ L ~
dz = A M Y
(2-3)
where ARSis the transform matrix, ASRis its inverse, p i is the chemical potential of component i, ci is the volume concentration of i, and a: is the weighting factor associated with component i in reference frame R. Because these matrices have only a single entry, the transpose is the same as the original matrix and the inverse matrix is simply the reciprocal of the original. Table 1 contains the weighting factors a:, CaFck = pR and transform coefficients of the binary case for the four reference frames of interest. Here 6 is the partial molar volume of i, Mi the molecular weight of i, p the density, c the total concentration cl + c2, wi the weight fraction of i, xi the mole fraction of i, and 4i = c i 6 the volume fraction of i (actually the "partial molal volume fraction" of i). It is particularly convenient to consider LO and g0 because the expressions are simpler in terms of concentration gradients.12 Thus eq 2-3 yields
R. However, as noted earlier, the "distinct diffusion coefficients" (D$)R are useful for comparison with statistical mechanical calculations. Fortunately the (0;)" can be calculated from the dependent diffusion Onsager coefficients L t , using the macroscopic form of eq 1-6:
where R is the gas constant, T i s the absolute temperature, and y1 is the solute activity coefficient on the molarity scale. It can be shown13 that
DO (1
(1 -6A)
+ c1 a In yl/t3c,)
--
DV
(1
+ m t3 In -yl/am)
(2-6)
so that The dependent LE can in turn be obtained from the independent LR. Finally LR can be obtained from DR plus activity coefficient derivatives. Since experimental values of interdiffusion coef-
LO =
Dvcl
kT(1
+ m t3 In -yl/t3m)
(2-7)
Reference Frames for Distinct Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 5567
where m is the solute molarity and y1 is the solute activity coefficient on the molality scale. For nonelectrolytes, the mole fraction scale is most convenient, and it can be shown from the results of Tyrrell and Harris14 that
with the Onsager reciprocal relations relating the L;
LF2 = L;l
(2-22)
where JF are flows on any reference frame R and Xi = -&/axi is the driving force associated with the solvent-fixed reference frame in terms of independent forces and flows.12 In addition, it can be shown12 that where
(2-10)
Bf = 1 + c1 a In y1/dcl
(2-1 1)
+ w1 d In ul/dwl
(2-12)
BY
=1
+ a:LFl = o ~ F L F+~UFLF~= o
(2-23)
aFLrl
q = 1 + x1 a lnfl/dxl BY = 1 + m a In yJdm
(2-9)
(2-24)
Equations 2-22 to 2-24 lead to aR
(i+j-2)
L!=(--$)
(2-25)
Lpl
These equations and the Gibbs-Duhem equation
where f l is the mole fraction and u1 the weight fraction activity coefficients of the solute, respectively. Consequently from eq 2-8 we can write LO as
c,x1
+ c&,
=0
(2-26)
enableX2to beeliminated from eq 2-20. The resulting expression for JY is in terms of X1 and that very factor ARo(Table 1) which transforms X1 into the appropriate independent force T.l2 Comparison of the expressions immediately shows that where in anticipation of later results, we define LF1 = LR (2-14)
This expression has Lo in terms of the experimentally measured diffusion coefficient Dvand the mole fraction activity coefficient derivative q. Expressions for LRin terms of Lo are obtained using eq 2-2 and the expression for ARO
ARo =
c2a;
claP + c2aF
(2-27)
Consequently the dependent LF can be related to experimental quantities through eqs 2-25 and 2-27. This in turn allows (0;)” to be obtained from LF and the experimental quantity (with eq 1-6A rewritten for (D;)R):
or
R
(2-15)
= Y2
(2-29)
where# (different from y i ) ,the “transform fraction”, is directly related to the expected concentration fractions associated with the corresponding reference frames. The expression for LR, equivalent to eq 2-2, is R 2 0 LR = (y2) L
It is clear that (Dt2)R= (D;,)” because of eq 2-22. If the first term on the right hand side of eq 2-29 is denoted by QF,then by eqs 2-27, 2-25, 2-16, and 2-14
(2-16)
The explicit expressions for LR are also given for future use in terms of the experimental quantity Z L V = -‘p;xlclZ =-
v2
xlx2Dvc(-)2
AT6
RT
cxiy.
>’
L M = w,~x,c,Z -=x1x2Dvc( M2
AT6
RT
(2-18)
cxiMi
x ~ x l c l Z xlx2Dvc
LN=.-=RT
kTq
+
JF = LFIX, LFG2
Analysis of the cases of eq 2-30 yields the following compact expressions for QF: (2-31)
Q; = (-l)(i+j)(l-yF)(1 -$)Z
(2-19)
The relation of the four dependent L; to the single independent LR will now be obtained. At mechanical equilibrium, the dependent forms are
J p = L.p1Xl+ L $ Y 2
(2-30)
(2-17)
(2-20) (2-21)
This in turn yields a compact expression for (D$)R:
(-l)(i+j)(1 - yR)( 1 - yj”)DV (0;)” = -XlX26
(2-32) Xi
Since B; = (valid for the mole/fraction scale), B; can be replaced by in eq 2-32. The explicit expressions for (D$)Rhave been collected in Table 2 for the convenience of future workers and are obtained using the quantities in Table 1. Note the unsymmetrical forms for the
5568 The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 TABLE 2
Expressions for (0;)" in Terms of Experimental Quantities
reference frame solvent-fixed (0)
(#I)"
(@2)"
Dv
fl
x1x2q
XI
x1x2q
mass-fixed (M)
number-fixed (N)
(1 -w1)'Dv
(1 - xI)'Dv XIX24
--DS XI
fl
(1 - xl)Dv
x1
x14
--=
- bl] + [a1 - kl]
(1-x2)'DV - -@ =
--
x1x2q
x1
( ~ 9 ~
+
V, = ( V J 0 h I x l 0
x2
+ h,x12+ h3X13+ h4xI4
(2-36)
Vl = (vl)o+ 2h,x1 + (3h3 - h,)X,2 +
-bl
2(2h4 - h3)x13- 3h4xI4 (2-37)
2M _$] + 2
V2 = (Vz)o- h,x12- 2h3x13- 3h4x14 - (@lo
- (@lo
solvent-fixed reference frame compared to those for the other reference frames. Finally, the limiting behavior of these expressions as X I goes to zero is examined: at infinite dilution, 6 = 1 and x2 = 1. In addition, with the definition of component 1 as the solute, Dv (D;)o as XI 0, where the subscript 0 refers to the value at infinite dilution. Using the definitions of the various kinds of fractions, it is found that (D;,)" and (D:,)" have the finite values given in Table 3 . These depend only on infinite dilution values of quantities, regardless of the concentration dependence of Dv, q,and BT. On the other hand, there is a possible singularity in (Dtl)Ras xI goes to zero because XI is in the denominator of each term. However, finite values are obtained with a Taylor series analysis using the following reasonable assumptions for the concentration dependences of Dv, q,and B; for nonelectrolytes:
-
+ a l x l + a 2 x I 2+ a3x13+ a4xI4 (2-33) = (Df)o + k l x , + k,xI2 + k 3 x I 3+ k4x14 (2-34)
Dv = (DY), D:
x24
It is then easy to show that
[a1 - kll
-
(1 -x2)Dv _-@
- (@)o
[a1 - kll
(D?)O[l
x2
described by the form
(fl)o[
massfixed (M)
(4,"
XI
x1x24
TABLE 3: Infinite Dilution Values of
(@)o[l
= 0
(1 -41YDV --fl
volume-fixed (V)
solventfixed (0) volumefixed (V)
Mills et al.
For the volume-fixed case, the partial molal volumes are also functions of the composition, so the Taylor series expansion is morecomplicated. It is reasonable toassume that indilutesolution the volume per mole V, of a binary nonelectrolyte solution is
(2-38)
and that
The Taylor series expansion in terms of x1 depends on having w1 in terms of xi; namely,
41and
XIVl
41
= XIVl + X,V,
--
XlVl
+x1(1VI- X 1 ) V z
(2-40)
The infinite dilution expressions for (Dd)", denoted as (D;)!, are given in Table 3 . Unlike (D12) d # and (D&)R, the limiting values of (D;JR do depend on the nature of the concentration dependence of DV, 4,and q.The volume-fixed case also depends on the concentration dependence of VI and V2. The values of (Of1)" in dilute solutions are given in order x12,as given in Chart 1. If the partial molal volumes are constant, Le., h2 = h3 = hq = 0, then eq 2-43 has the same form as the mass-fixed result in Chart 2, with Mi replacing (V~)O. Note the minus sign in front of the in the zero-order term of eq 2-45. This equation also results from setting M I = M2 in the mass-fixed eq 2-44.
Reference Frames for Distinct Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 5569
CHART 2 mass-fixed
number-fixed
A more compact form for eqs 2-42 through 2-45 for all four reference frames up to terms in xf is given in Appendix I. The above expressions permit the calculation of (Dt,)R in dilute solution from fits to experimental data at higher concentrations. If the other end of the mole fraction range is of interest, the same equations apply with solute-solvent subscripts interchanged.
3. Discussion 3.1. Thermodynamic Factor I$' = dln a,/dln xi. In eqs 1-4 and 1-5, % can have a profound influence on the calculated D$ values. B;" can range in value from above unity to near zero for various binary systems reflecting the deviations from ideality (B;x = 1). It is useful to try to account for the behavior of this factor over the mole fraction range to illustrate points to be made later in the Discussion. In Figure 1, B;" has been plotted as a function of mole fraction for some of the systems studied here. By definition B: equals unity for pure liquids. For a binary system which behaves ideally B: remains unity over the whole mole fraction range. Near-ideal systems such as cyclooctane/cyclopentane have B: values quite close to unity over the whole concentration range. In moderately nonideal binary systems such as those with benzene as one component, the l37 decrease to about 0.5 a t their lowest points. However, in very nonideal systems such as water with acetone or acetonitrile as the other constituent, the % are much lower and in the latter case decrease close to zero. An exception is the acetone/chloroform system, where BY is equal to or greater than unity over the whole range and reaches a maximum value of 1.4. The latter system shows negativedeviations from Raoult's law, and this can be correlated to its %values being greater than unity. Conversely, systems which have positive deviations from Raoult's law have B; values between unity and zero. This
-
1 .o 0.8
ss
0.6
0.4 0.2
0.0 0
0.2
0.4
0.6
0.8
1
Xl
Figure 1. Thermodynamicfactor (g) versus mole fraction component (1) at 25 O C ; 0, acetone (l)/chloroform (2); A, cyclooctane (1)/ cyclopentane(2); 0 ,cyclohexane (l)/benzene (2);., acetone (l)/water (2); 0,acetonitrile (l)/water (2).
variation in behavior, which can be correlated with different types of compound formation, will be discussed later. The influence of the thermodynamic factor has also been discussed by Czworniak, Andersen, and P e c ~ r a , Hertz '~ and Leiter,3b and Weingartner.10 3.2. Comparison of Frames of Reference for In section 2, equations were formulated with which the D i can be calculated on several frames of reference. In the past, only the mass-fixed frame has been used for nonelectrolytes. Examination of all figures and tables for the 14 binary systems shows the following general features: (a) In most cases, the D; in the mass- and volume-fixed frames follow each other closely. This might be expected as mass and volume are somewhat related; see, e.g., Figures 2 and 3.
5570 The Journal of Physical Chemistry, Vol. 98, No. 21, I994
Mills et al. of D; values in any frame of reference must therefore be relative to change across the concentration range. Taking feature b into account, we select the number-fixed frame for discussing the concentration and temperature dependence of the D;. 3.3. Standard Equations. Relations between the (D;)Mwere given by Friedman and Mills' in the form
Oa5
-1.5 -2.0
The equations are derived using the Onsager reciprocal relations. The corresponding equations for the number-fixed frame are
- 2 . 5 L ' 0
I
0.2
"
"
0.4
0.6
"
'
0.8
I
+ X,(D;,)~ + x ~ ( D : , ) ~= -Dl
1
X ' ( D ; , ) ~x ~ ( D ~= ,-0: )~
Xl
Figure 2. (Di2)Rversus mole fraction (1) for the OMCTS (l)/benzene (2) system at 25 OC: 0, mass-fixed; 0, volume-fixed; and 0 , number-
fixed frames.
(3-4)
(3-5)
Relations between the (D$)Mas given in eqs 3-2 and 3-3 have been used by Friedman and Mills' with an additional relation for the Di to give standard coefficients, which are then assumed to apply to ideal solutions. One such additional relation can be obtained by use of a version of the Hartley-Crank equation which is applicable only to ideal solutions
Dv = xlD; + xzD;
(3-6)
If eq 3-6 is incorporated into eq 1-4 and adapted to the numberfixed frame, one obtains (0Df2)N= -(x,Dl -3.9
0
0.2
0.6
0.4
0.8
1
+ x2D:)
(3-7)
Equation 3-7 can be combined with eqs 3-4 and 3-5 to give
Xl
(oD:l)N = - ~ 2 ( D f- DS,) -
Figure 3. (Dy2)Rversus mole fraction (1) for the cyclooctane (1)/ cyclopentane(2) system at 25 OC: 0 ,mass-fixed; 0 , volume-fixed; and 0,number-fixed frames.
(b) For most systems, D$ values for the number-fixed frame show the least variation across the mole fraction range. This is particularly marked where the mass difference between components is large; see, e.g., Figures 2 and 3. (c) The Dt2 curves for the 3 frames intersect one another; see, e.g., Figure 3. This crossover effect is due to the changing velocities of the frames of reference relative to laboratory coordinates across a concentration range. For example the crossover concentration for (Df2)Mand (DfJN can be obtained by setting equal their appropriate equations in Table 2. For those reference frames it is given by
Similarly crossover points can be calculated for the other two cases. (d) Values of the limiting D i on the solvent-fixed frame (see Table 3) are (with one or two exceptions) all positive and reach high positivevalues with increasing concentration. There is some conceptual difficulty in interpreting data on this frame in a binary nonelectrolyte system, so they will not be further discussed. As can be seen from Figures 2 and 3, the D$ values differ according to the frame of reference used. In the exceptional case where the masses or molar volumes of the two components are equal, the respective reference frames would, of course, coincide with the number-fixed frame. The fact that absolute values of the D; should differ was predicted by Raineri and Friedman4 in the context of mblecular theory. A qualitative interpretation
= -xl(Di - 0;) - Ds,
(3-8) (3-9)
These standard coefficients give good agreement with the (D;)" of near-ideal systems such as cyclooctane/cyclopentane, as is shown in Figure 5 , discussed below. When applied tononideal systems, there is considerable divergence (see Figure 6) which serves to give a qualitative measure of nonideality. However, for the latter systems, the OD; calculated by eqs 3-8 and 3-9, while anchored at one end to thenegativevalue of oneof thecomponents' self-diffusion coefficient, usually differ widely from that component's limiting trace value. To rectify this, we have adapted, for the number-fixed frame, the procedure of Friedman and Mills' in which experimental values for the limiting trace coefficients are built into the standard equations. It should be noted that, later in this paper, two other standard equations for the D$ based on the Friedman-Mills equation' (FM) will be briefly discussed. Friedman and Mills' proposed an approach to estimate limiting trace values while Bender and P e ~ o r ain, ~their extension of the F M equation to the whole mole fraction range, gave a further standard equation. However, both of these standards will be shown below to be applicable only to the mass-fixed frame. A more satisfactory standard adaptable to any frame of reference and preferably derived on a statistical mechanical basis, awaits development. 3.4. General Interpretationof the Concentration Dependence of Before attempting such an interpretation, a brief discussion on the properties of correlation functions is needed. If the velocities of two particles are completely random over time, then their velocity cross-correlation integral will equal zero. If,
4-
Reference Frames for Distinct Diffusion Coefficients
t
1
0
i
The Journal of Physical Chemistry, Vol. 98, NO. 21, 1994 5571
-0.4
2*5
I
2.0
2
5 1.5
1.o
O.5b
'
'
0.2
'
'
0.4
'
I
0.6
'
I
0.8
'
1
-""F 0.2
-2.0 0
0.4
Xl
however, two particles have an attractive force between them, sufficient for their velocities to be positively correlated, then the time integral will have an excess of positive elements. Dimerization, compound formation, or loose aggregates of particles may result from these attractive forces. Conversely, if two particles, on the average, repel one another, then their timecorrelation integral will have excess negative elements. Now, all D i (where i # j ) are negative by definition (see eq 1-4). Also the D; in pure i are equal to the negative value of the self-diffusion coefficient, as can be seen from eq 1-5. With increasing mole fraction (increasing j ) , self-association or aggregation will be evidenced by decreasing negativity of the D: and in some cases positive values. Conversely, particle repulsion will give the 0:. increasing negativity across a concentration range. On the phenomenological level, Kate" has suggested that the "kinetic" diffusion coefficient (I) can be used to measure selfassociation in nonelectrolyte mixtures. H e gives the equations
f
1
XI
Figure 4. L /I0 versus mole fraction (1) for the systems acetone (1)/ chloroform (2), A; cyclooctane (l)/cyclopentane (2), 0 ;OMCTS (1)/ benzene (2), 0;acetone (l)/water (2), all at 25 O C and acetone (1)/ water (2), 0 at 5 O C .
f
0.8
0.6
+ xlx2(Df,+ Di2 - 2Dy2)
Figure 5. versus mole fraction (1) for the system cyclooctane (l)/cyclopentane (2) at 25 O C : -, experimentally derived values; - -, standard equation.
-151
0
'
I
0.2
'
I
0.4
'
"
I
0.6
'
0.8
1
1
x1
Figure 6. versus mole fraction (1) for the system acetonitrile (l)/water (2) at 25 O C : -, experimentallyderived values; - -,standard equation.
(3-10)
where
Io= x2Di + x l D i
+
(3-1 1)
If Ofl Di2 > 2Dt,, it can be deduced that molecules of the same kind tend to diffuse together, whereas if Dtl Di2 < 2Dt2, there is the opposite tendency. Thus L/Locan be used as an index of self-association in the mixture. Examination of the individual D:l and D;2 curves for a particular mixture can indicate which component is associating more than the other. It should be noted that L / L0 is independent of the frame of reference. The independence of the expression, Ofl + D;2 2Dt2 was discovered, initially, when manipulating the experimental data. This invariance to the frame of reference by the above expression can also be shown by substituting for the (Of.)"the equations given in Table 2. It is of interest that Trullas and PadroI6 derived equations for velocity correlation functions on a microscopic basis which are independent of the frame of reference and are similar in form to the above expression (e.g., see their eq 16). In Figure 4, we show plots of L/& for several systems, and these will be discussed later in this section. Figures 5-7 are representative of near-ideal and nonideal systems. The D; in these figures are all on the number-fixed frame and theN superscript is omitted in the following paragraphs.
+
-2oL 0
"
0.2
"
0.4
"
0.6
'
I
0.8
'
1
1
Xl
Figure 7. versus mole fraction (1) for the system acetonitrile (l)/water (2) at 25 OC: 0,experimentallyderived values; - -,standard equation. Figure 5 shows the D t for the near-ideal system cyclopentane (l)/cyclooctane ( 2 ) at 25 O C . Its near ideality is judged by the closeness to unity of the thermodynamic factor (B;x) (see Figure 1). As can be seen, the standard and D t curves are very close together and progress monotonically between the values for the limiting concentrations. The opposite extreme is provided by the Dfl and Dt2 curves for acetonitrile (l)/water ( 2 ) a t 25 "C in Figures 6 and 7. In Figure 6, the Ofl curve deviates positively from the standard almost over
5572 The Journal of Physical Chemistry, Vol. 98, No. 21, 1994
t
i
Mills et al. 1 - *O* ’ -
t
i
-1.5
-2.0
8 4
3
v
U
-21
0
’
I
0.2
’
0.4
”
0.6
”
0.8
’
-2.5
1
1
Figure 8. (D&)”/L$ versus mole fraction (1) for the system acetone (l)/water (2): -, 5 O C ; .-,25 OC.
the complete concentration range with a high peak at X I = -0.3. The Of2curve (Figure 7 ) for the same system deviates negatively from the standard with a minimum at X I = -0.3. The D:2 curve shows a smaller maximum a t XI = -0.3 and then increases to a high positive value for the limiting value of water in pure acetonitrile. The same pattern is shown a t 5 ‘C with the peak values increased and a shift to XI = -0.4. As mentioned in an earlier paper,6athere is a phase separation in the acetonitrile/water system at -1 OC and 0.38 mole fraction of acetonitrile, and the suggestion has been made from thermodynamic evidence that there are microheterogeneous regions near theconsulate point. It is noteworthy also that the thermodynamic factor for this system approaches zero around the same concentration (see Figure 1). Thus the general features of the 0;.curves for the acetonitrile/water system both a t 5 and 25 O C ! are in accord with the thermodynamic interpretation. The high positive D:2 limiting values for water in acetonitrile also indicate clustering of water molecules as their concentration decreases. Some of the above features are shown in the acetone/water and methanol/water systems but to a lesser degree. A comparison forJ,’s U j = DY./xj) on a mass-fixed frame of the above three water mixtures has been made by Mills and Woolf.*b In all systems except for acetone (l)/chloroform ( 2 ) , the experimental Of2 values are more negative than the standards. The exception indicates compound formation between acetone and chloroform, and this is confirmed by thermodynamic studies. Also the L/Loindices (see Figure 4) are below unity, which again is indicative of compound formation. 3.4.1. Temperature Dependence of D t . The three mixtures with water as a common component ( 2 ) have been measured at two temperatures, 5 and 25 O C . Of the components, water has the greater tendency to form hydrogen bonds and is generally considered to have more structure, the lower the temperature. Consequently if the D;2 for water (with a given other component) are compared at the two temperatures, the one at 5 ‘C should show the greater positivity. To obtain the relative increase, the Di2 have been divided by the water self-diffusion coefficients D; at each temperature. Such a comparison for the acetone (1)/ water ( 2 ) system is given in Figure 8, and the anticipated trend is clearly shown. The same behavior is shown in the -C/& plots in Figure 4 with a higher peak for the data a t 5 O C . Similar trends are shown for methanol and acetonitrile. 3.4.2. Comparison of D t for Isothermal Systems with a Common Component. In this study there are five binary systems where benzene is a common component (2) and three where it is water. Figure 9 shows the D:2 for benzene with five different partners. In the systems with benzene and either cyclohexane, chlorobenzene, or OMCTS, the D:2 values become more positive
.;
Xl
Xl
Figure 9. (&)” versus mole fraction (1) for benzene (2) with OMCTS (I), 0;chlorobenzene (l), 0;cyclohexane (l), hexane (l), V; heptane
(I), 0 .
a
n
&a
-1
U
-2
-31
0
..
’
I
0.2
’
1
0.4
’ X!
I
0.6
’
.;
1
0.8
’
1
1
Figure 10. (@2)Mversusmole fraction (1) for benzene (2) with OMCTS (l), 0;chlorobenzene (l), 0;cyclohexane (l), hexane (l), V; heptane (1),
as the benzene mole fraction increases. For hexane and heptane, however, the D:2 curves become more negative with increasing chain length. An explanation for the behavior in the first three cases could be some form of self-association of benzene in these solutions. There appears to be no thermodynamic evidence to support self-association. However, Monte Carlo studies by Evans and Watts” on the intermolecular potential between benzene molecules in pure benzene give some support to self-association. In the vapor phase, their calculations show a favored configuration of T-shaped pairs with benzene planes perpendicular to one another. This kind of configuration may persist in the liquid phase. Moreover the radial distribution function in pure liquid benzene shows a small shoulder a t -4 A which is attributed to the planar stacking of pairs of molecules. There is also experimental evidence for structure in liquid benzene from the neutron diffraction studies of Bartsch et al.,I8 where the statement is made that “models assuming a preferred orientation describe the neutron experiments better than theones with more randomly oriented molecules”. The -C/& indices are also indicative of a small amount of self-association in the benzene mixtures of this section (see Figure 4 for the OMCTS/benzene index). Let us assume then that some self-association is present in the mixtures which is dependent on the nature of the other component and it affects the (D:2)Ncurves shown in Figure 9. In Figure 10 the same curves are plotted using the mass-fixed frame. It will be observed that the general pattern is different but in particular the (Di2)Mcurve for OMCTS/benzene is now well above the others and even becomes positive a t high OMCTS mole fraction. Such behavior would indicate a high degree of self-association which
Reference Frames for Distinct Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 5513
is not given by the other evidence above. Undoubtedly it is caused by the high mass ratio in this system (3.8:l)and shows that a degree of distortion is introduced by use of the mass-fixed frame, as predicted in ref 6. It seems that the use of the number-fixed frame gives a more credible picture of benzene behavior in the various solutions than does the mass-fixed frame. For water in the three systems with methanol, acetone, or acetonitrile as partners, there is a progression in D:2 {water = 2 ) which can be related to the water clustering in a changing response to the hydrogen bonding and polarities of the three partners. This case has already been discussed by Mills and Woolfsb for the mass-fixed frame; again, some distortion will be introduced when that frame is used.
4. Hydrodynamic Equations Recently Friedman and Mills’ presented a semiempirical equation based essentially on hydrodynamic interactions for D$ in binary nonelectrolyte mixtures. There is some question as to the frame of reference which is applicable to this equation. It is an adaptation of one given by Altenberger and Friedman's for electrolyte solutions which is based on a solvent-fixed frame. However, the solvent-fixed frame in an electrolyte solution is difficult to equate with one in a binary nonelectrolyte solution. In addition to hydrodynamic factors, the F M equation contains an equilibrium molecular quantity, the pair-correlation function. It reads
DY2 = ( 2 / 3 ~ ) k B T p J o m rhI2(r) d r + ...
(3-12)
where 7) is the solvent viscosity, k g is the Boltzman constant, T is the temperature, p = N/Vis the total number of molecules per unit volume, and h12(r) + 1 = glz(r) is the radial distribution function. With the aim of obtaining a reference equation for measuring deviations from ideality, Friedman and Mills’ made simplifying assumptions to eq 3-12. An essential step in this process is to estimate limiting values of the D; for a trace component 1 in a virtually pure solvent 2. These calculated values can then be compared with the experimental ones for near-ideal and nonideal systems. It is of renewed interest here because such a comparison should also shed light on the frame of reference applicable to eq 3- 12. In brief, two approaches were adopted by Friedman and Mills.’ The first was termed the “structural” approximation, and the second, the “thermodynamic” approximation. In a later paper4b they showed that the structural approximation was the better of the two approaches from the point ofview of a reference equation. This approach will be the only one described here. It incorporates the simplification
h,2(r)= -1
=0 where
u12 is
< uI2 if cI2< r
if r
(3-13)
an effective diameter which can be expressed by
(3-14) The final equation, which involves only the kinetic elements of the diffusion equation, is
D!, = -27rD,3(1/Dt where
+ 1/Di)2
(3-15)
TABLE 4 Experimental (Reference Frame Dependent) and Calculated Limiting Values of Distinct Diffusion Coefficients in Binary Nonelectrolyte Systems in the xz = 1 Limit at 25 OC solute (1) cyclooctane cyclopentane benzene chlorobenzene benzene cyclohexane benzene OMCTS benzene n-heptane benzene carbon tetrachloride OMCTS acetone chloroform methanol water acetone water acetonitrile water
eq 3-15
M
cyclopentane cyclooctane chlorobenzene benzene cyclohexane benzene OMCTS benzene n-hexane benzene n-heptane OMCTS
-4.1 -0.17 -0.71 -1.9 -0.57 -1.9 -0.04 -6.1 -2.1 -1.5 -1.2 -0.07
-4.3 -0.09 -0.77 -2.6 -1.3
-1.5 -1.1 -1.92 -0.73 -1.5 -1.1 -1.74 0.1 1 -1.5 -4.3 2.1 -2.1 -3.0 -1.6 -0.37 -1.8 -3.5 -0.26 -1.04
-3.4 -0.21 -1.45 -1.33 -0.98 -0.74 0.06 -3.6 -0.03 -3.2 -0.52 0.08
carbon tetrachloride chloroform acetone water methanol water acetone water acetonitrile
-4.3
-1.3
0.0
-3.2
-2.2 -5.8 -5.1 -5.0 -7.5 -2.8 -4.5 -2.1
-2.6 -13.6 -2.4 0.70 -2.1 15.4 -9.9 26.8
-5.0 -5.9 0.07 -1.2 3.6 8.1 -5.7 20.1
-4.5 -6.6 -3.2 1.6 -3.1 16.6 -1 1.2 28.3
solvent(2)
-
N
V
Limiting values of Dtl ( X I 0 ) calculated from eq 3-15 are compared with experimental values in three frames of reference and are shown in Table 4. It will be seen from Table 4 that for near-ideal systems and moderately ideal systems (particularly those with benzene as a component) that the D$ from eq 3-15 give the best agreement with those on a mass-fixed frame. For very nonideal systems, such as those with water as a component, eq 3-15 is sometimes satisfactory for the nonaqueous component but quite inadequate for water. Disagreement with the latter systems is of course expected because of the approximations involved. The important point is that for near-ideal systems the reference equation agrees surprisingly well with experimental values on a mass-fixed frame. Bender and Pecora9 extended the scope of applicability of the FM equation by using the Ornstein-Zernike approximation for gi,(r), thus eliminating the restriction given by eq 3-13. Then they expressed gij in terms of the Kirkwood-Buff parameters from equilibrium theory. The parameters G,,are defined by
G , = K 4 r ? [ g , ( r ) - 11 dr
(3-16)
Bender and Pecora’s modification of the FM eq 3-12 is (converted fromJj to D$ notation):
D$ = 2 k T / 3 ? ) R - ( ’ / , ) + ci:((Gij + ( 4 / 3 ) ? r u t ~ ) / 8 ~ u J ] (3-17) where 9 is now the solution viscosity (cf. eq 3- 12) and Pthe molar volume. The uij were calculated from eq 3-14. With eq 3-17, Bender and Pecora9 calculated Jj G, = D$/xj) coefficients for a number of near-ideal and nonideal systems and in general found good qualitative agreement with experimentally derived coefficients on a mass-fixed frame. As mentioned earlier, Bender and Pecora also derived a reference equation for the ideal mixture case based on thermodynamic considerations. Weingartner’o later demonstrated a close connection betweenhj coefficients and Kirkwood-Buff integrals for the system methanol/carbon tetrachloride. He also commented on the frame of reference problem for eq 3-12, since its origin was in electrolyte theory where a solvent-fixed frame of reference is used.
5574
Mills et al.
The Journal of Physical Chemistry, Vol. 98, No. 21, 1994
whole concentration range. This is an interesting result in that the FM equation is derived on a Smolukowski level. This rpises the possibility that an equation of this form might be derivable on a more fundamental level such as the Born-Oppenheimer 0ne.19 Acknowledgment. We thank Professor H. Weingartner for helpful comments. Portions of the work by D.G.M. were done under the auspices of the U S . Department of Energy, Office of Basic Sciences (Geosciences),at Lawrence Livermore Laboratory under Contract No. W-7905-ENG-48. Appendix I - 4 . 5 r . I " ' 0 0.2 0.4
0.6
'
I
0.8
'
1
x1
Figure 11. (D;'JR versus mole fraction (1) for cyclooctane (1)/ cyclopentane (2) at25 O C : 0,mass-fixed; 0,volume-fixed;and 0,numberfixed frames. - -, Bender and Pecora, eq 17.
An examination of terms up to x14for the Taylor series expansion using eqs 2-33 through 2-36 showed that a more compact, recursive form can be found covering all four reference frames. Let the Taylor series expansion be written as n
Then i- 1
pi =
+ C b ( i - j p j+~ (1 - P)'[1-( i + PI +f j=O
i- 1
a
h
8=
v
qi =
-2
['(,+I)
- k[i+1,1 + Z'(i-jpj j=O
-4
i
where, for i = 0, the sum terms vanish @e., aj = b, = 0 for j = 0, -1). Thef are additional terms involving the h, and the ratio rR,which appear only for the volume-fixed reference frame for all i > 0. No obvious patterns could be observed for these additional terms, so the explicit values for i = 1, ..., 4 are given below. The expressions for ( D f l ) Rfor all four reference frames can be fitted into the above pattern with the following assignments of rR, hi, and f :
ro = 0, hi = 0,
8=0
number-fixed reference frame: rN = 1, hi = 0,
f =0
solvent-fixed reference frame:
mass-fixed reference frame:
rM = M l / M 2 , hi = 0, jy = 0 volume-fixed reference frame:
rv = ( V J ~ / ( V ~hj) ~ z,0,
f' given a s follows
8' = 4 h 2 / ( ~ 2 ) 0
.c=2(1
- rv)(9rv - 4)h2
+ 2(7rv - 4)h, - 8h4 +- 8h: (VJ;
(V2)O
fl= 2( 1 - rV)2(14rV - 5)h2+2( 1 - r V ) (12rV- 5)h, (V2)O
+ 2(9rv - 5)h4+
22h2h, - 2( 13rV- 9)h:
(V,),Z References and Notes (1) Equation 2.4 in: Friedman, H.L.;Mills, R.J . Solution Chem. 1986,
IS, 69.
Reference Frames for Distinct Diffusion Coefficients (2) (a) McCall, D. W.; Douglass, D. C. J . Phys. Chem. 1967,71,987. (b) Douglass, D. C.; Frisch, H. L. J . Phys. Chem. 1969,73, 3039. (3) (a) Mills, R.; Hertz, H. G. J . Phys. Chem. 1980,84,220. (b) Hertz, H. G.; Leiter, G. Z.Phys. Chem. (Munich) 1982,133, 45. (4) (a) Friedman, H. L.; Mills, R. J . Solurion Chem. 1981,10,395. (b) Mills, R.; Friedman, H. L. J . Solution Chem. 1987,16, 927. (c) Friedman, H. L.; Raineri, F. 0.;Wood, M. D. Chem. Scr. 1989,29A,49. (d) Raineri, F. 0.;Friedman, H. L. J. Chem. Phys. 1989,91,5643. ( 5 ) Steele, W. A. In Transport Phenomena in Fluids;Hanky, H. J. M., Ed., Marcel Dekker: New York, 1969; pp 230-249. (6) Miller, D. G. J . Phys. Chem. 1981,85, 1137. (7) (a) Altenberger, A. R.; Friedman, H. L. J . Chem. Phys. 1983,78, 4162. (b) Zhong, E. C.; Friedman, H. L. J . Phys. Chem. 1988,92,1685. (c) Wu, Y. C.; Koch, W. F.; Zhong, E. C.; Friedman, H. L. J . Phys. Chem. 1988, 92,1692. (8) (a) Easteal, A. J.; Woolf, L. A.; Mills, R. Z.Phys. Chem.(Munich) 1987,155, 69. (b) Mills, R.; Woolf, L. A. J . Mol. Liq. 1992,52, 115.
The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 5515 (9) (10) (11) (1 2) 1509. (13) (14)
Bender, T. M.; Pecora, R. J . Phys. Chem. 1989,93,2614. Weinglrtner, H. Eer. Bunsen-Ges Phys. Chem. 1990, 94,358. Kato, T. J. Phys. Chem. 1985,89,5750. Miller, D. G.; Vitagliano, V.; Sartorio, R. J . Phys. Chem. 1986,90,
Miller, D. G. J . Phys. Chem. 1966,70, 2639. Tyrrell, H. J. V.; Harris, K. R. Diffusion in Liquids; Butterworth: London, 1984. (15) Czworniak, K. J.; Andersen, H. C.; Pecora, R. Chem. Phys. 1975, 11, 451. (16) Trullas, J.; Padro, J. A. J. Chem. Phys. 1993, 99,3983. (17) Evans, D. J.; Watts, R. 0. Mol. Phys. 1976,32,93. (18) Bartsch, E.; Bertagnolli, H.; Schulz, G.; Chieux, P. Eer. Bunsen-Ges. Phys. Chem. 1985,89, 147. (19) Friedman, H. L. Private communication.
