DIFFUSION IN 13INARY SOLUTIONS
987
Diffusion in Binary Solutions’
by David W. McCall and Dean C. Douglass Bell Telephone Laboratories Inc., Murray Ha,New Jersey
(Received Auguat 1, 1986)
Experimental self-diffusion results are reported for the binary systems benzene-cyclohexane, acetone-chloroform, acetone-benzene, and acetone-water. Data were recorded as a function of concentration at 25” using the nmr spin-echo method. The data are discussed in connection with mutual diffusion results, previously published, with particular emphasis on the equations D = (b In al/b In z1)(QDZ xzD1) (Hartley-Crank and Darken) and D = Dl(b In all3 In s) (Bearman and Eyring). These equations are qualitatively but not quantitatively descriptive of the experimental data. A theoretical analysis is presented in which the mutual and self-diffusion coefficients are expressed in terms of integrals of molecular velocity correlation functions. This analysis approaches a molecular view of the three diffusion coefficients and gives some insight into the nature of the HartleyCrank relationship.
+
I. Introduction In this paper we discuss diffusion in binary liquid solutions of nonelectrolytes. Two groups of experiments can be distinguished-selfdiff usion and mutual diffusion. Self-diffusion is a measure of the mobility of the molecules. It occurs in chemically uniform systems and it, is made experimentally accessible by isotopic labeling or application of a magnetic field gradient in nmr spin-echo studies. In binary solutions there are two self-diffusion coefficients, one for each component. If (rt2) is the mean-square displacement of a molecule of component i in time T , the self-diffusion coefficient of component i is given by
Dg =
(T$)/~T
+ x~DI= D
D = Q(d2
(1)
Mutual diffusion studies, on the other hand, are concerned with the rate at which concentration gradients approach their equilibrium values (zero in miscible systems). In general, one can show that a single diffusion coefficient is sufficient in binary systems to describe this rate and we callthis the mutual diffusion coefficient, D . Intuitively we expect D to be closely related to D1 and Dz and, in fact, for thermodynamically ideal solutions, it has been claimed that28
Dl2 E x~DZ
the system must remain uniform. That is, if one component diffuses much more rapidly than the other, there must be a compensating bulk flow to keep the molecules from piling up at one end of the container. Thermodynamic considerations must enter the discussion of mutual diffusion because the driving force for mutual diffusion is the gradient of the chemical potential rather than the concentration gradient. It follows2bthat the mutual diffusion coefficient is pro~ , al and ct are the portional to (b In ai/b In c ~ ) T , where activity and concentration of component i. An equation that incorporates the mixture rule of eq 2 and the thermodynamic factor is
(2)
where the z values are mole fractions. Equation 2 is derived from the requirement that the pressure within
Q
= I1
+ XzDi)
+ (a In n / b In
(3) (4)
where y1 is the mole fraction activity coefficient. This equation can be regarded as a form of the HartleyCrank relation.a BearmanZb has given an illuminating discussion of the basis for this and other relations between D and D1 and D2. (1) Some of the data reported herein were previously published in a preliminary note (D. W. McCall and E. W. Anderson, J. Phys. Chem., 70, 601 (1966)). The preliminary data were not corrected for the isotope effect and the present paper supersedes the earlier note. (2) (a) J. E. Mayer and M. G. Mayer, “Statistical Mechanics,” John Wiley and Sons, New York, N. Y., 1940, p 30; (b) R. J. Bearman, J . Phys. Chem., 65, 1961 (1961). (3) G. 9. Hartley and J. Crank, Trans. Faradau Soc., 45, 801 (1949).
Volume 71, Number 4 March 1967
988
Equation 3 and other relations between mutual and self-diffusion parameters have often been discussed but rarely tested directly. Adequate data on activity and self-diffusionare unavailable for most systems. I n the present study we have determined the self-diffusion coefficients for four binary systems at 25”. Thermodynamic data exist for all four of the solutions studied. The systems are: benzene-cyclohexane, acetonebenzene, acetone-chloroform, and acetone-water. Mutual diffusion coefficients have been reported for the acetone solutions by Anderson, Hall, and Babb4 and Rodwin, Harpst, and Lyons5 have studied mutual diffusion in the benzene-cyclohexane solutions. Selfdiffusion results relevant to the latter system have been reported by Collins and Watts6 and Birkett and Lyons.’ A tracer study by Millss has appeared more recently in which self-diffusions coefficients for both components of the bensene-cyclohexane system were measured. The agreement between hlills’ tracer data and the nnir data reported herein is gratifying. Albright and MillsQprefer the term inlradifusion to selfdiffusion in the discussion of results for solutions. We use the term self-diffusion in discussing results determined by the nmr method whether the medium contains more than one kind of molecule or not.
11. Experimental Section The proton magnetic resonance spin-echo method has been used in this All data were obtained a t 25 f 1”. The absolute accuracy of the method has been estimated as somewhat better than 5% for liquids with strong resonance signals. In solutions dilute in protons, the accuracy is not as good. For example, in our spectrometer the 0.18 mole fraction of CHCL in (CD3),C0 solution had a signal-to-noise ratio of about 5 and the results are probably accurate to within 10%. The deuterated compounds were obtained from Merck Sharp and Dohme of Canada except for the dlz cyclohexane which was purchased from Volk Radiochemical Co. Isotopic purity was stated to be greater than 99% in each case. The solutions were examined by means of high-resolution nmr and it was found that no hydrogen-deuterium exchange occurred and that the resonance from the undesired compound was always negligible. The method involves the use of solutions for which one component is completely deuterated so that the other component can be studied separately. This procedure introduces some error, however, owing to the isotope effect on the self-diffusion rate. That is, deuteration of one component increases the visoosity and decreases the diffusion rate of the other component. The Journal of Physical Chemistry
DAVIDW. MCCALLAND DEANC. DOUGLASS
To correct for this effect, we have made self-diffusion measurements on the undeuterated solutions. The peak echo voltage is given by” 2
V ( 7 , G) = iC=Vl d 7 , 0 ) ~ X P [ - ~ ( T G ) ~ T ~ D (5)~ / ~ I where G is the magnetic field gradient and T is the echo , is the decay function for the ith comtime. V P ( 7 0) ponent in a homogeneous field, G = 0. For the substances investigated in this study, V f ( T ,0) = Vf(O, 0) for all T of interest. Vf(O, O)/V(O, 0) = p , is the fraction of protons in component i. p , = x,zf where zf is the number of protons for each i molecule. In the two-component systems studied here V(T, G ) / w - J 0)
=
PI exp(--KDl~~)4-p~ exp(--KDz~~) ( 6 )
A trial and error procedure has been adopted. V(T,G)/V(O, 0) is measured as a function of 7. The right-hand side of eq 6 is calculated for a given T using D1 and D, obtained from the deuterated solutions a t the same concentration. D1 and Dz are then increased a few per cent and the right-hand side is recalculated. This is continued until the calculated and observed V(T,G)/V(O, 0) agree. Then a new T is picked and further adjustments are made, if necessary. The corrections have not exceeded 10%. We would not claim that these isotope-eff ect corrections are very accurate, but the magnitudes seem to be reasonable. In analyzing our data, we have applied the same correction to both components a t a given concentration, Even though this may not appear to be entirely logical, the result is consistent with the data.
111. Results The data for the benzene-cyclohexane system are plotted in Figure 1 for comparison with the data of Rlills.8 The agreement is excellent by our standards of accuracy. Figure 2 shows our data in comparison with the mutual diffusion results of Rodwin, Harpst, and Lyons.6 The proper limiting behavior is observed. (4) D. K. Anderson, J. R. Hall, and A. L. Babh, J . Phys. Chem., 6 2 , 404 (1958). (5) L. Rodwin, J. A. Harpst, and P. A. Lyons, ibid., 69, 2783 (1965). (6) D.A. Collins and H. Watts, A u s t r a h n J . Chern., 17, 516 (1964). (7) J. D. Birkett and P. A. Lyons, J . Phys. Chem., 6 9 , 2782 (1965). (8) R. Mills, ibid., 6 9 , 3116 (1965). (9) J. G.Albright and R. Mills, ibid., 6 9 , 3120 (1965). (10) H.Y.Carr and E. M. Purcell, Phys. Rev., 94, 630 (1954). (11) D. C. Douglass and D. W. McCall, J . Phys. Cfiem., 6 2 , 1102 (1958).
989
DIFFUSION IN BINARY SOLUTIONS
5
lim D = D, xc +0
4
and lim D = Db
(7)12
zb 40
3
a n
The isotope corrections amounted to about 4% in the middle concentration range. Self-diffusion results for the acetone solutions are compared with mutual diffusion results of Anderson, Hall, and Babb,' in Figures 3-5. In each case the limiting behavior, eq 7, is satisfactory. The isotope correction was negligible for the acetonebenzene system. In the acetone-chloroform solutions the isotope cor-
c
2
o ACETONE BENZENE MUTUAL (ANDERSON, HALL AND BABB)
0
0
0.5 Xa
1.0
3.0i7T---7
Figure 3. Concentration dependence of mutual' and selfdiffusion coefficiente (cm*/sec) for the system acetone-benzene at 25".
BENZENE-CYCLOHEXANE
-,
@
I
I
I
I
I
I
I
1
I
2.5
0
"0 2.0 CYCLOHEXANE
-
1.5
1.o
1
0
T
1
THIS STUDY
lo;lLL;
I
I
I
,
I
0.5
1.0
0
xb
0 b
ACETONE CHLOROFORM MUTUAL (ANDERSON. HALL AND BABB)
Figure 1. Comparison of nmr selfdiffusion results with the tracer data of Mills* for the system benzene-cyclohexane at 25". 0.5 Xa
0
-
BENZENE-25* CYCLOHEXANE C
-
n 0
CYCLOHEXANE KUTUAL (RODWIN, HARPST AND LYONS) 0.5
xb
Figure 2. Concentration dependence of mutual6 and self-diffusion coefficients (cmZ/sec) for the system benzene-cyclohexane at 25".
Figure 4. Concentration dependence of mutual' and selfdiffusion coefficients (cm*/sec) for the system acetone-chloroform at 25'.
rection was not appreciable at high acetone concentrations (20.8)and peaked a t about 10% near xs = 0.2. In the acetone-water system the isotope correction was 9% in the middle concentration range falling to 3% at xs = 0.3.
n 2
0
1.0
1.0
IV. Discussion As noted in the Introduction, a form of the Hartley~~~~
~
~
(12) We use the subscripta b, c, a, and w t o identify the components
benzene, cyclohexane, acetone, and water.
Volume 71, Number 4 March 1967
DAVIDW. MCCALLAND DEAND. DOUGLASS
990
I
I
L
I
I
4
-
4
Table I
ACETONE-WATER 25' C 4-
-
-
ACETONE 0 WATER e MUTUAL (ANDERSON, H A L L AND BASE) 0
-
-
3-
n
0
0
I
I
f
'
I
0.5
I
I
I
I
Zb
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
10%
106Do
1.88 2.11 2.26 2.34 2.41 2.44 2.44 2.42 2.38 2.31 2.25
1.42 1.58 1.76 1.87 1.97 2.03 2.09 2.15 2.21 2.14 2.09
Benzene-cyclohexane 2 5 O 1OWb0 1O'D D/Dh
1.88 2.05 2.17 2.20 2.23 2.24 2.24 2.24 2.24 2.16 2.09
1.88 1.87 1.84 1.80 1.80 1.80 1.81 1.85 1.91 1.97 2.09
1.00 0.91 0.85 0.82 0.81 0.80 0.81 0.83 0.85 0.91 1.00
Q
1.00 0.92 0.86 0.82 0.78 0.76 0.76 0.78 0.83 0.90 1.00
1.0
Xa
Figure 5. Concentration dependence of mutual' and selfdiffusion coefficients (cm*/sec) for the system scetone-water a t 25".
Table I1
1.5
ACETONE-CHLOROFORM
za
106D.
lo6&
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2.75 2.81 2.97 3.19 3.39 3.59 3.79 3.99 4.18 4.46 4.86
2.28 2.38 2.67 2.86 3.01 3.16 3.35 3.54 3.74 3.93 4.12
Acetone-benaene 2 5 O 1O6Dab lO6D D/D.b
2.75 2.77 2.90 3.09 3.24 3.38 3.52 3.68 3.83 3.98 4.12
2.75 2.58 2.56 2.61 2.70 2.81 2.97 3.16 3.42 3.72 4.12
1.00 0.93 0.88 0.85 0.83 0.83 0.84 0.86 0.89 0.94 1.00
Q
1.00 0.96 0.96 0.97 0.98 0.98 0.99 0.99 1.00 1.00 1.00
1.0
0.5
0.5
1.O
%a OR xb
Figure 6. Comparison of the thermodynamic factor Q = (3 In a l / b In x ~ ) ~ with , ~ ,the diffusion ratio D/(xlDt zzU1) a t 25".
+
Crank relation, eq 3, would indicate that D/Da should equal Q = (b In z l y l / b In x ~ ) = , ~An . empirical comparison of these quantities is given in Tables I, 11, 111, and IV and Figure 6. The thermodynamic factor The J o u r ~ dof Physical Chamistry
was computed from published data.13-16 Room-temperature activity data were available only for the acetone-chloroform system. It is obvious that DID12 and Q are qualitatively similar. Deviations from unity are in the same direction in each case. The acetone-water system exhibits the largest deviation in both quantities. The concentration at which Q and D/DH deviate most from unity seems to be the same: xs = 0.45 for acetonewater; xa = 0.40 for acetone-chloroform; and Z b = 0.55 for benzene-cyclohexane. For acetone-benzene the deviation of Q from unity probably lies within experimental error. Quantitative agreement, within the estimated 5% accuracy range, is observed only for the benzene-cyclohexane solutions. (13) (a) G. Scatchard, 8. E. Wood,and J. M. Mochel, J . Phye. Chem., and C. H. de Minjer, Rec. Frau.
43, 119 (1939); (b) W. Reinders Chim., 59,369 (1940).
(14) H.Rock and W. Schroder, 2. Physik. C h m . (Frankfurt), 11, 41 (1957). (15) D. F. Othmer and R. F. Benenati, I d . Eng. Chem., 37, 299 (1946). (16) R. York and R. C. Holmes, ibid., 34,345 (1942).
DIFFUSION IN BINARY SOLUTIONS
991
'
t
Table III
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
,
-
'-
Acetone-chloroform 2 5 O 28
,
10KDB
10KD6
106Dm
1ObD
D/Dm
Q
2.35 2.41 2.54 2.72 2.86 3.02 3.23 3.49 3.80 4.20 4.90
2.58 2.67 2.83 2.89 2.91 3.05 3.21 3.42 3.60 3.62 3.62
2.35 2.45 2.60 2.77 2.88 3.04 3.22 3.44 3.64 3.68 3.62
2.35 2.67 2.97 3.18 3.37 3.44 3.47 3.52 3.56 3.60 3.62
1.00 1.09 1.14 1.15 1.17 1.13 1.08 1.02 0.98 0.98 1.00
1.00 1.20 1.43 1.58 1.59 1.54 1.46 1.36 1.21 1.10 1.00
D/Daw
Q
Table IV zg
1O*Dg
0.0
1.28
2.45
1.28
1.28
1.00
1.00
0.7 0.8 0.9 1.0
3.13 3.50 3.99 4.94
2.74 3.23 4.24 -5.5
2.86 3.27 4.22 -5.5
1.60 2.32 3.39 -5.5
0.56 0.71 0.80 1.00
0.35 0.56 0.76 1.00
104ow
106DBw
1O*D
I
I
i
I
I
I
-
BENZENE- CYCLOHEXANE 25' C
RODWIN, HARPST AND LYONS
0
n n 2;
2
00
Y
0
-
1-
0
1.o
0.5
0
xb
Figure 7. Comparison of mutual diffusion data6 with predictions based on eq 8 for the system benzene-cyclohexane a t 25'. The solid curves are calculated results, eq 8.
-
ACETONE-BENZENE 25OC
-
0
L
2-
-
ANDERSON, H A L L A N 0 BABB
0
-
1-
Bearman2b has proposed that rate-process theory1' will yield a relation between D and D1 or D2. The relation he suggests is D = D1 (b In f~cl/dIn C
= Q1D1
I ) ~ , ~
(8)
where f is the activity coefficient appropriate to the molar concentration, c. Note that this thermodynamic factor is not the same as the mole fraction thermodynamic factor, eq 4 Le., Q1 # &. Further, Q1 # Q2. The factors are related by &I
=
&(a In ~ 'In b~
d
~
,
p
(9)
Solution densities were taken from the "International Critical Tables."18 We have evaluated Q1 and Q2 for the solutions studied. These results and the comparisons relevant to eq 8 are shown in Table V. Figures 7-10 exhibit the results graphically. The solid lines are the computed mutual diffusion coefficients and the circles are
t 0
I
'
I
'
I
'
I
'
I
0.5
0
1.0
Xa
Figure 8. Comparison of mutual diffusion data4 with predictions based on eq 8 for the system acetone-bensene a t 25'. The solid curves are calculated results, eq 8.
of eq 8. It is seen that eq 8, like eq 3, gives a quantitative description of the data only for the system benzenecyclohexane. The present data do not provide a basis for choosing between eq 3 and 8. In making these comparisons it is necessary to do considerable analysis on the thermodynamic data. We have carried this through in a primitive way and we have no doubt that superior analyses are possible. A thermodynamic identity QlV2
=
Q2Vl
(10)
Volume 71, Number 4 March 1967
DAVIDW. MCCALLAND DEANC. DOUGLASS
992
Table V -Benecne-cyclohexan-
-Acetone-benzen-
-Acetonechloroform-
106 Zb
Q
&b
QbDb
1WD
za
Q
Qa
106 &sDa
0.1 0.3 0.5 0.7 0.9
0.92 0.82 0.76 0.78 0.90
0.91 0.78 0.69 0.68 0.74
1.91 1.83 1.69 1.64 1.71
1.87 1.80 1.80 1.79 1.98
0.1 0.3 0.5 0.7 0.9
0.96 0.97 0.98 0.99 1.00
0.94 0.92 0.89 0.87 0.83
2.64 2.94 3.20 3.47 3.71
20
C2
&o
Qo&
losD
zb
Q
0.1 0.3 0.5
0.90 0.78 0.76 0.82 0.92
0.92 0.82 0.84 0.94 1.07
1.98 1.76 1.71 1.76 1.69
1.98 1.79 1.80 1.80 1.87
0.1 0.3 0.5 0.7 0.9
1.00 0.99 0.98 0.97 0.96
106
0.7 0.9
-Acetonewate106
105D
Zs
Q
Qs
2.58 2.61 2.81 3.16 3.72
0.1 0.3 0.5 0.7 0.9
1.20 1.58 1.54 1.36 1.10
1.21 1.61 1.63 1.44 1.22
1.02 1.05 1.09 1.12 1.15
QbDb
4.02 3.71 3.45 3.20 2.73
2.67 3.18 3.44 3.52 3.60
Q
Za
3.72 3.16 2.81 2.61 2.58
Zo
0.1 0.3 0.5 0.7 0.9
Q
0.1 0.3 0.5 0.7 0.9
0.57 0.24 0.14 0.35 0.76
0.73 0.48 0.38 1.21 3.48
0.71 0.65 0.81 3.79 13.9
Zw
Q
Qw
QwDw
1.10 1.36 1.54 1.58 1.20
1.09 1.32 1.46 1.52 1.07
3.95 4.52 4.45 4.40 2.85
I
I
3.60 3.52 3.44 3.18 2.67
I
0.1 0.3 0.5 0.7 0.9
1
I
0.76 0.35 0.14 0.24 0.57
I
0.70 0.27 0.09 0.11 0.19
I
l
3.0 0.74 0.20 0.18 0.33
-
ANDERSON, HALL AND BABE
3.4 1.60 0.85 0.62 0.82
0
-I
I
4 4 -1
105D
I
ACETONE- WATER 25OC
-
0
0.82 0.62 0.85 1.60 3.39
105 106D
QoDo
Qo
1060
Qs
105 105D
5 ACETONE-CHLOROFORM 25OC
-
2.91 4.39 4.92 5.03 5.13
105 Qb
105 10aD
3n
-
o
2-
0
ANDERSON, HALL AND BABE
0
I
I
I
I
I
I
I
I
I
0.5
1.0
Xa 0.5
0
Figure 10. Comparison of mutual diffusion data4with predictions based on eq 8 for the system acetone-water at 25". The solid curves are calculated results, eq 8.
I .o
Xa
Figure 9. Comparison of mutual diffusion data' with predictions based on eq 8 for the system acetonechloroform at 25'. The solid curves are calculated results, eq 8.
where the S, values are partial molar volumes provides a consistency check on the Qi factors. We have computed the 'lf' and find that eq lo is Obeyed adequately, Table VI. This check does not bear on the accuracy of Q, however, as Q cancels in the ratio Qi/Q2, by eq 9. BY eq 8 D = DiQi D2Q2 (11) and, using eq 12
Di/Dz
=
QdQi= V*/Vi
The ,Journal of Physical Chemistry
(12)
Thus we have included the ratio D1/Dz in Table VI. Equation 12 holds fairly well for benzene-cyclohexane and acetone-benzene solutions, holds roughly for acetontrchloroform solutions, and is not obeyed for acetone-water solutions. BearmanIgand i\/Iills2° have discussed this equation previously. We should keep in mind that eq 8 does not follow from eq 12 even though the reverse does follow. A rather different point of view can be stated as follows. Acetone-chloroform solutions exhibit negative deviations from Raoult's law and the thermo(19) R. J. Bearman, J. Chem. Phys., 3 2 , 1308 (1960). (20) R. Mills, J. Phys. Chem., 67, 600 (1963).
DIFFUSION IN BINARY SOLUTIONS
993
must provide for breaking down the water structure and the formation of a water-acetone complex
Table VI -Benzen?y$ohexane-
-Acetone-Jen:en-
Xb
DdDc
vc/vb
Qc/Qb
%a
0.1 0.3 0.5 0.7 0.9
1.32 1.28 1.20 1.13 1.08
1.21 1.22 1.22 1.23 1.23
1.26 1.21 1.22 1.21 1.18
0.1 0.3 0.5 0.7 0.9
Za
0.1 0.3 0.5 0.7 0.9
-AcetonehloroformDa/Dc Vc/Va Qc/Qa
0.91 0.94 1.00 1.03 1.25
1.11 1.11 1.11 1.11 1.11
1.12 1.07 1.12 1.10 1.13
De./&
1.16 1.12 1.13 1.12 1.15
vb/vs
1.22 1.22 1.22 1.22 1.22
1.22 1.22 1.22 1.21 1.23
yAceton_e-w_ater--XB
0.1 0.3 0.5 0.7 0.9
Da/Dw
Vw/Va
0.62 0.27 0.81 0.24 0.99 0.23 1.09 0.22 ~ 1 . 0 -0.16
(HzO),
Qb/Qs
Qw/Qa
0.26 0.23 0.24 0.22 0.20
dynamic behavior has been discussed extensively21 in connection with various models for deviations from ideality. Although there are several ways of expressing the ideas, they all involve association or compound formation of the type
This ties in directly with the empirical fact that D. = Do a t x. = 0.5. I n the context of the diffusion experiments the distinction between association and compound formation is not important; i.e., the rate of exchange can be quite large by chemical standards and still be small compared with the diffusion jump rate. At concentrations other than equimolar, the interpretation of the measured self-diffusion coefficients is complicated owing to the introduction of an associated entity. D, will be some sort of a weighted average of the self-diffusion coefficients of the free acetone (present in solution) and the complex. Similarly, Do will be an average. Each of these coefficients will be concentration dependent. The interpretation of the mutual diffusion coefficient could be complicatedas well. Stokes has recently22published a paper dealing with the effects of dimerization equilibria on diffusion. He discusses the problem of the relation between dimer and monomer mobilities in terms of a modified StokesEinstein hydrodynamic theory.23 Acetone-water solutions exhibit large positive deviations from Raoult’s law that are usually attributed to association of one of the components.21 Of course, water is a strongly associated liquid and the thermodynamic behavior is of the expected form. Experimentally we observe that D, = D, at x. = 0.5 and this is somewhat more surprising than the similar result for the acetone-chloroform system. A successful model
+ n(CH3)2CO
n[(CH3)2CO. . H 2 0 ]
This is not an unreasonable suggestion, but we do not wish to imply that the data lead directly to such a model. It is possible that the equality of the selfdiffusion coefficients is merely a coincidence. I n any case, interpretations are complicated owing to our ignorance of the diffusion coefficients for the various components of the model.
V. Theory To clarify the relationship between mutual and selfdiffusion experiments, it is useful to make a comparison in the language of irreversible thermodynamic^.^^ The approach is to write down linear relations between particle fluxes, ji, and chemical potential gradients, V p r . The diffusion coefficients are then derived in terms of the linear (Onsager) coefficients, Q i p For a binary solution
-jl =
+
= VD1’Vnl
(13) where nl is the concentration, Le., number per cms. At constant temperature and pressure the GibbsDuhem relation nlVp1 nJp2 = 0 (14) Q11Vp1
5212Vl.12
+
places a constraint on the variables so that we find D1’
=
[a11
- (nl/n2)Ql2](b~l/bnl)(l/V)(15)
and D2’ = [-(nz/nl)91
+ 92l(bp~cz/bnd(1/V)
(16)
for the diffusion coefficients of components 1 and 2, respectively. For a mass-fixed reference frame in mechanical equilibrium it can be shown that
ml2Ql1=
-m1mzQ12
=
-m1m25221
= mt2&
(17)
Thus D1’
= (Q11m/n2m~V) (bwlbnd
(18)
where m = nlml -k
n2m2
(19)
The mf values are molecular masses. (21) J. H. Hildebrand and R. L. Scott, “The Solubility of Nonelectrolytes,” 3rd ed, Reinhold Publishing Corp., New York, N. Y . . 1960, Chapter 11. (22) R.H. Stokes, J. Phys. Chem., 69,4012 (1965). (23) F. Perrin, J . Phys. Radium, 7, l(1936). (24) D.D.Fitts, “Nonequilibrium Thermodynamics,” McGraw-Hill Book Co., Inc., New York, N. Y., 1962.
Volume 71,Number 4 March 1967
994
DAVIDW. MCCALLAND DEANC. DOUGLASS
I n general, D1’ and D2’ are not equal in this frame of reference. However, using the thermodynamic identity n1fl2(b~l/bnl) = n2WWbn2)
(20)
mSflaa
D =
= (mih/rn2)D1’ =
D
(mfll/ml)D2’
(22)
which is the same for the two components. This corresponds to the experimental mutualdiffusion coefficient commonly encountered. For self-diffusion, D = D1’ = D2’. As in the case of single-component systems, the diffusion coefficients may be expressed in terms of velocity correlation functions via the results of Kirkwood and According to these results the linear coefficients may be expressed as
n,,
=
( 1 / 3 k T ) ~ ( J , 0 J j ( 0 ) ) ds
(23)
where the pointed brackets indicate an average over an equilibrium ensemble.
where vta’(s) is the velocity of the kth molecule of type i relative to the center of mass. For convenience we define the following molecular velocity correlation functions and their integrals. flaa
= ~ r0 ( ~ l a ’ ( s ) v l a ’ (ds 0))
(25) (26)
and f12
= ~ (0 ~ I ~ ‘ ( S ) V ~ ~ ’ds (O))
(27)
where via' is the velocity of molecule one of type a, etc. I n terms of these functions the linear coeficients are and Q12
=
(nmfd3kT)
Using the conservation of momentum we obtain The Journal of P h y e h l Chemhtry
(29)
0
- ( r n 2 ~ 2 / 3 m m V )In ( b al/d In nl)f12 D1 = f1aa/3
(21)
Following Fitts, we define a mutualdiffusion coefficient
=
(30) and a similar relation for component two. From straightforward algebraic combinations of the above equations we obtain the desired relations
it follows that (Dl’lD2’) = (mzfll/m1$
+ n m f I a g+ nzm&
(31) (32)
D2 = f 2 a a / 3
(33) The thermodynamic factor (b In al/b In nl) enters simply from the assumption that the fluxes are proportional to the chemical potential gradients. VPl = (bPl/bnl)Vnl p1 = 1.11’
+ kT In a1
(drddnl) = (kT/nd(b In al/b In nl)
(34)
In the self-diffusion experiment the thermodynamic factor is unity. Owing to the restrictions imposed by the conservation of momentum, only three of the five correlation functions defined above are independent. The three experimental diffusion coefficients are then sufficient to evaluate the integrals of all five correlation functions but one does not obtain a functional relationship among the three diffusion coefficients. Such a general relationship requires information beyond the phenomenological equations. By way of illustration, observe that one may rearrange the foregoing equations to form the identity D
- Q(z2D1 + d ? z )
+
= [ S C Z V Q / ~ ( S4 (flap
+
f2ap
1X
- 2fd
(35)
where V is the volume of the system. The right-hand side of this equation is clearly the error made by using the Hartley-Crank relation. In order that the HartleyCrank relation apply over the entire concentration range, there should be a relation among the cross correlations such that the right-hand side of eq 35 is small compared to D. A related result has been In~ dilute solutions the correcobtained by T ~ r r e 1 1 . ~ tion is small and eq 35 should apply, but only in a limited sense is it a relationship among the three diffusion coefficients. This result may be extended somewhat if one observes that the cross correlations in the error term may (25) J. G.Kirkwood, Rend. Scuoka Intern. Fia. Enrico Fermi, 10,200 (1960). (26) R. W.Zwanzig, Ann. Rev. Phys. C h m . , 16, 67 (1965). (27) 9. A. Rice and P. Gray, “Statistical Mechanics of Simple Liquids,” Interscience Publishers, Inc., New York, N . Y., 1965. (28) H.J. V. Tyrrell, J. Chem. Soc., 1599 (1963).
DIFFUSION IN BINARY SOLUTIONS
-
995
=
(%a’(O)%Z’(s)) BENZENE-CYCLOHEXANE
-
2S°C
(kT/m,)(1
1
t
- (s2/2m3c$gf*92uffdrff+ cJgijV%jdrij + - . 1
(38)
and K(S)
+
= (s2P/2m)[(l/mi2)SgiiV2uiidrii
(l/mz2).l’gd72uzzdm(2/mlm?).l’gd‘Urzdri2 1
+ - ..
(39) With the exception of K , the correlation functions have a maximum at zero time and subsequently decay. The coefficient of s2 is the mean-square force on a molecule and is therefore positive in all except the expression for K. The function K is zero at zero time and may increase in magnitude before decaying. In many cases, then, the integral contained in K is smaller than the integrals of the other correlation functions and the Hartley-Crank relation may apply reasonably well over the entire concentration range. The integral of the I
t -5
S
-
1
0
,
,
,
,
,
,
,
,
,
ACETONE-BENZENE 25’C
1.o
Ob XC
Figure 11. Experimental velocity correlation function integrals for the system benzene-cyclohexane at 2 5 O .
be combined into a single correlation function which has a clear, though esoteric, physical interpretation. Equation 35 becomes
-Q(d1
+
ZlD2)
r t
=
+
(c~czVQ/~(CIc ~ ) [0K ( s )ds (36)
t
where K(S)
=
([km-
VZa(0)
1[%9(s)
- 2128(9) 1
0
is the cross-correlation of the relative velocity of one pair of unlike molecules with the relative velocity of a distinct pair of unlike molecules. The behavior of K&) is unlike that of the other correlation functions. This is readily apparent if one expands the various functions as power series in time.
1 1
t
K(S)
(QU’(O)U,B’(S))
=:
- (kT/m){ 1 - (s2P/2mfm,J.gt,v22ut,drf,
+
. .1
(37)
-5
1 1 1 1 1 1 1 1 1 1 )
0
0.5 Xa
1.0
Figure 12. Experimental velocity correlation function integrals for the system aceton&ensene at 25”.
Volume 71,Number 4 March 1867
DAVID w. MCCALLAND DEANc. DOUGLASS
996
t t
;
- 5 L
0.5
1.0
xa
Figure 13. ]Experimental velocity correlation function integrals for the system acetone-chloroform a t 25’.
-5
0
0.5
1.o
Xa
Figure 14. Experimental velocity correlation function integrals for the system acetone-water at 25’.
f values are not great but the trends in some systems are interesting. Among the cross correlations only fWaB in the water-acetone system has a positive excursion. In solutions dilute in acetone fwa8 is negative. Somewhat loosely speaking this is a consequence of the conservation of momentum. For the pure water system with zero total momentum and with a mole cule with specified momentum, pl, the remaining molecules must, on the average, have --pi/(N - 1) as their momentum. This arrangement must persist D = (lcY/3(~1 ~ z ) ) & [ ~ p / ~ i ~ S g ~ A ~ ~(40) ~~~irzI”’ in such a way that the integrated function is negative. This situation would obtain over the entire concenAs mentioned before, the integrals of the five correlation functions may be obtained from the three diftration range for solutions composed of similar mole cules, e.g., the benzene-cyclohexane system. As the fusion coefficients without recourse to assumptions concentration of acetone increases, the restriction imbeyond those inherent in nonequilibrium thermoposed by the conservation of momentum becomes less dynamics. These integraIs have been determined by dominating since the molecule with specified momenuse of eq 30 and 36 for the systems studied. These tum may distribute its momentum to both water and derived quantities are listed in Table VI1 and plotted acetone molecules as dictated by the differing interin Figures 11-14. molecular forces. The correlation function must be The concentration dependences exhibited by the
autocorrelation function of the relative velocities may be obtained directly from the two self-diffusion coefficients. One might also note that methods employing expansions like eq 37, previously used for the calculation of self-diffusion coefficients, apply with similar assumptions to the calculation of the mutual diffusion. For example, if one assumes that the functional form of the velocity correlation function is gaussian, one finds
+
The JOUTW~ 01 Physical Chem@try
997
DIFFUSION IN BINARY SOLUTIONS
Table VI1 zo
0 0.2 0.4 0.6 0.8 1.0
-Benzene-cyclohexane lo'foao 1 0 % ~ ~ 1o'Nfbo
6.27 6.63 6.27 5.92 5.28 4.26
-0.60 -0.67 -0.71 -0.69 -0.65 -0.57
Acetone-watm IO%aa 1O"Nf.w
z*
0 0.2 0.4 0.6 0.8 1.0
6.75 7.14 7.32 7.24 6.79 5.64
3.84 3.24 5.11 7.95 10.5 14.8
7.36 4.54 5.74 7.33 9.70
-0.19 -0.19 -0.28 -0.31 -0.26
7
lo'Nfou@
-0.59 -0.56 -0.56 -0.53
lO$Nfap@
-0.17 -0.35 -0.59 -0.79
lo*Nfbao
-0.65 -0.68 -0.72 -0.71
IO"fWU@
-0.13 -0.03 +O.ll +0.29 -0.24
zero at zero time but at some later time it must be positive in order that the integral fwag be positive. This implies that a moving water molecule tends to impart its momentum to other water molecules so as to make them move in the same direction in spite of the opposing influence of the conservation of momentum. A corresponding trend may be observed infa@. One should be careful, however, in drawing detailed conclusions about the reasons for this effect inasmuch
7
Acetone-benzene 1o'Nfsb
2s
lo%za
0 0.2 0.4 0.6 0.8 1.0
8.25 8.90 10.2 11.4 12.5 14.6
9.03 10.1 11.2 12.4
zs
104/apa
Acetone-chloroform l 0 ~ ~ o u u lO*NIao
lO*NfSlrs
0 0.2 0.4 0.6 0.8 1.0
7.05 7.63 8.59 9.70 11.40 14.7
7.74 8.50 8.7 9.6 10.8 10.9
-0.52 -0.47 -0.47 -0.60 -1.08
losfbaa
6.84
8.00
-0.46 -0.52 -0.65 -0.81 -0.93 -1.04
-0.23 -0.25 -0.33 -0.47 -0.77 -1.67
1o'Nfap@
-0.48 -0.53 -0.64 -0.83 -1.07
1O'Nfh@
-0.61 -0.78 -0.99 -1.26 -2.02
lO'Nf~U@
-0.63 -0.82 -1.06 -1.58 -2.83
as one may visualize several mechanisms to produce this observation. A deeper insight must come from the details of the correlation functions themselves rather than just their integrals. Acknowledgment. We are indebted to W. P. Slichter and J. L. Lundberg for helpful comments and suggestions during the preparation of this manuscript. D. R. Falcone and E. W. Anderson assisted in some of the experiments.
Volume 71, Number 4 March 1967
