220
J. Phys. Chem. 1980, 84, 220-224
Application of the Velocity Cross-Correlation Method to Binary Nonelectrolyte Mixtures R. Mills" Diffusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra, A.C. T., Australia
and H. G. Hertz Institut fur Physikalishe Chemie und Elektrochemie II, der Universtitat Karlsruhe, Karlsruhe, West Germany (Received June 27, 1979)
Intradiffusion coefficients of good precision have been measured for the binary systems octamethylcyclotetrasiloxane (OMCTS)-CC14at 18,25,35,and 45 "C and acetone-water at 25 "C. Both NMR spin-echo and radiotracer techniques were used. These results have been combined with other data in the literature to calculate velocity cross-correlationcoefficientsfor several binary nonelectrolyte systems. For the OMCTS-CC14 systems the VCC's from measured quantities agree well with those calculated from simple momentum considerations. For the OMCTS-benzene and acetone-water systems however the VCC's for one of the components in each system are radically different. An attempt has been made to explain this behavior in terms of molecular rearrangements.
Introduction The relationship between the two intradiffusionl (or self-diffusion) coefficients and the mutual diffusion coefficient in a binary liquid mixture has been the subject of considerable discussion and controversy. Earlier Darken,2 Hartley and Crank: and Carman and Stein4proposed that the two types of coefficients could be related by equations which reduce to the general form DM = (xlDl + x2D;)a In al/a In x1 (1) where DM is the volume-fixed mutual diffusion coefficient, the Di*s were termed intrinsic diffusion coefficients, x, is the mole fraction, and a, is the activity of component i. It was later shown by Bearmad and Mills6 that these poorly defined intrinsic diffusion coefficients were an unnecessary complication and should be replaced in eq 1 by the measured intradiffusion coefficients (Dit). It has been found that, in general, eq 1 represents experimental diffusion data very poorly; the most that can be said is that the more closely a system approaches ideality, the better eq 1 is obeyed. In 1967 McCall and Douglass7 analyzed the binary diffusion case and, in particular, the relations between the diffusion coefficients of eq 1in terms of velocity correlation functions. They were able to show that eq 1 can only be true fortuitously, in fact only when the sum of three velocity cross-correlation coefficients (VCC's) is zero. The relation they derived between the mutual and intradiffusion coefficients by use of VCC's, however, affords the means of calculating the latter quantities. Further, these calculated VCC's can be used to obtain information on microscopic interactions in binary mixtures. In making such calculations, accurate sets of intradiffusion, mutual diffusion, and activity coefficients data are required. A t the time McCall and Douglass7 introduced this approach there were not many systems for which accurate data were available. In particular one set of VCC's for the system acetone-water studied by McCall and Douglass7appeared to show anomalous behavior. In recent years accurate mutual diffusion and activity data have been reported for a number of binary nonelectrolyte systems. We have therefore measured intradiffusion coefficients for these systems and examined the behavior of the VCC's which now become available. In particular, we have studied the carbon tetrachloride-octamethylcyclotetrasiloxane (OMCTS) system at 18, 25, 35, 0022-3654/80/2084-0220$0 1.OO/O
and 45 "C and the benzene-OMCTS system at 25 "C. We have also made new measurements of the intradiffusion coefficients in the acetonewater system to try and resolve the apparent anomaly there. The urea-water system has also been examined.
Theoretical Description McCall and Douglass7 have given formulae with which to calculate VCC's but, as the definitions, notation, and units differ from those used here, we outline the main equations. The development is similar to that given by Hertzs for a salt-water system, except that the concentrations are converted to mole fraction units. The mutual diffusion coefficient, (D12),expressing proportionality between experimental flows and concentration gradients for components 1 and 2 and the theoretical phenomenological coefficients (al2)expressing proportionality between flows and chemical potential gradients are related by the equation
a In al (clM1 + c2M2)2 D12 = - M 1 2 M 2 2 ~ 1V2RT~ 2 a In c1012 I
(14
where ci, Mi, Vi, and ai are the concentration in mol/L, molecular weight, partial molar volume, and activity of component i, respectively (i = 1, 2 ) . From linear response theory we have
where V is the total volume of the system and ( Jl(0).J2(t) ) is the ensemble average of the mass current cross-correlation function between particles 1 and 2. From eq l a and 2 we can derive D12
=
where No is Avogadro's number. We now define velocity correlation coefficients as (4)
0 1980 American Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 2, 7980 221
Diffusiori in Binary Nonelectrolyte Mixtures
so that eq 3 can be written as (clMl f czM2)2- a In al D12 = VZ----fl2 MlM2C2 a In c1
X1 0
0.2
0.4
0.6
(6)
Because concentrations for miscible binary liquid systems are generally expressed in terms of mole fractions we write (xlM1 + x2Mz)2a In al D'12 = fi2 (7) M1M2x2 a In x i where x, is the mole fraction of component i. One can thus obtain f12 from either eq 6 or eq 7 . To calculate the fii)s one uses the relationship obtained by Fitt,sg Q12 = a21 = --a11= 4 2 2 (8) to obtain altlernative VCC's of the form
where the two terms within the square brackets are the integrals over the velocity autocorrelation function (identifiable with the self- or intradiffusion coefficient Dlt) and the velocity cross-correlation coefficient fll for component 1. Use of eq 5 and translating to mole fractions gives
OMCTS -CARBON TETRACHLORIDE ( 25°C ) Figure 1. Standard and experimental VCC's for the system OMCTSCCl., at 25 "C.
thermodynamic data (Albright and Mills1). and similarly for f Z 2 .
Experimental Section Two techniques were used to measure self- or intradiffusion coefficients in the binary liquid systems. Field gradient pulsed NMR was used to measure the intradiffusion of OM[CTS in carbon tetrachloride-OMCTS mixtures at 18, 25, 35, and 45 "C. As described by Harris, Mills, Hack, and WebsterlO our NMR machine is calibrated over a range of temperature by using values of Dt for water from the data of Milldl and for benzene from the data of Collings and Mills.12 The g or field gradient strength values obtaincsd in this calibration agreed to within &0.5%. Intradiffusion coefficients for acetone in D20 at 25 "C were also measured by the NMR method. The remainder of the diffusion coefficients reported below were measured by radiotracer methods with diaphragm cells. The measuring techniques are described in the monograph by Mills and W00lf.l~ CC1, labeled with 14C was used to measure intradiffusion coefficients in OMCTS mixlures at 18 and 25 "C. Acetone labeled with 14C and tritiated water were used for the acetone-water system. Results and Calculations The intradiffusion coefficients measured in this work and the calculated fLlo and fLI values are reported in Tables I-IV (availabde as supplementary material). Sources of additional data used in the calculations for each system are as follows: OMCTS-CC4, mutual diffusion coefficients and activity coefficient data (Marshl4J6)); OMCTS-C6F&, mutual diffusion coefficients, OMCTS (Barrie, Dawson, and Sheppardi6), C6H6 (Mi11sI7);acetone-H20, mliitual diffusion coefficients (Anderson, Hall and Babb18),thermodynamic factor, a In a,/a In x , (McCall and Douglass7); urea-H,O, mutual diffusion coefficients, (Gosting and Akeleylg), intradiffusion coefficients and
Discussion The f,,and fb," for the OMCTS-CC14 system at 25 "C are plotted against mole fraction in Figure 1. The most striking feature is the almost exact agreement between the experimental VCC's, f,,, fL,, and the calculated standard values, fb:, In order to give an interpretation of the observed VCC's it is worth recalling the way the formulae for the standard VCC's are derived. These formulae are
f,:.
v,")
so that f2: = f l ~ ( D J x , M 2 / D ~ x 2 MIn l ) the . present system Mi = 296.62 and M 2 = 153.82. The argument is as follows. The mean value of all molecular velocities in the system is zero. Now assume that at t = 0 any molecule selected a t random has an instantaneous velocity ul(i) (i = 1, 2). Then, since the mean velocity is zero, there must be movement of the remaining molecules in the opposite direction. The magnitude of the corresponding mean drift velocity u' as a consequence of the law of linear conservation of momentum is determined by the ratio of the mass of the molecule observed to the mean molecular mass of the remainder of the systeim:
where
M = xlMl + x2Mz Thus at t = 0 we have ( ul(i)(0)U'(O)) = -(Mi/'&$)((di))')
(14)
222
The Journal of Physical Chemistty, Vol. 84, No. 2,
1980
Mills and Hertz
Now we say that the contribution to the correlation product on the left-hand side of eq 14 which stems from the species i or j is simply given by the mole fraction of this species
X, 0
t
j=iorj# i with ub)' = Ulb)
+ u p + ugh) + .,.
Next we assume that the velocity cross correlation function ( U ~ ( ~ ) ( O ) U ( Jdecays ~ ( ~ ) ) essentially in the same manner as the velocity self-correlation function of species i. This means that the normalized time integrals are equal
Dl+
EM
(16)
-4.0
v
f:i
0
f22
0
e 2
Thus we obtain the result -5 0
xjMj -
Dit
i tf j or i = j ( l l a )
XlMl+ XzMz For the system OMCTS-CC1, at 18, 25,35, and 45 OC it appears that the above model is verified by experiment very satisfactorily. There is another aspect of the excellent agreement between the experimental and predicted VCC's which should be discussed. This is the fact that eq 7 and 10, which lead from the primary observed quantity D,, to the "experimental" VCC's, involve two coordinate system transformations. (1) The transformation from the center-of-volume determined coordinate system (equal to the cell fixed reference frame) to the local center-of-mass fixed coordinate system. This transformation is of course necessary because in the theoretical prediction of the VCC's the conservation of linear momentum enters directly. (2) The analytical field expressed in terms of the partial mass densities as a function of position is transformed to a field which describes the chemical potential as a function of the position. This is the origin of the factor a In al/a In x1 in eq 7 and 10. Of course, the VCC is a quantity which always must exist in a molecular system since the velocity correlation functions refer to the system in the equilibrium state. It is an interesting point at this stage to enquire whether the effective Hamiltonian connected with these correlation functions is in fact the gradient of the chemical potential or instead the concentration gradient multiplied by RT. For the OMCTS-CC1, system at 25 "C the reported datal5 show that a In al/a In xl for this system is about 1.1at x1 = 0.5. The VCC ratios at this mole fraction are fl:/fll = 1.04, f2i/fiz = 1.05, and f l ~ / f l = z 0.9. If the concentration gradient were more appropriate than the chemical potential gradient then fL:/fi, should be equal to a In al/a In xl in all cases. This is not the case and on average it is found that the gradient of the chemical potential gives a better representation of the effective Hamiltonian function. Next we turn to the systems OMCTS-benzene (Figure 2) and acetone-water (Figure 3). In both of these systems, the VCC calculated from experiment for the heavier component, (l),agree fairly well with the predicted standard behavior of fl:. In contrast to this, the different species correlation coefficient, f 1 2 , shows a deviation from the standard function flzo which is positive for the range x1 2 0.5 and negative for x1 5 0.5, the effect being much
OMCTS
- BENZENE (25OC)
Figure 2. Standard and experimental VCC's for the system OMCTSCBHBat 25 OC.
\ i.0
-5.0
A
fll
1
I
-
I
ACETONE WATER (25°C)
Figure 3. Standard and experimental VCC's for the system acetone-water at 25 'C.
stronger for the aqueous system than for the OMCTSC6H6mixture. Finally, the lighter particle Vcc, fz2, in both cases shows a positive deviation from the predicted fzzo. In the case of water, fzz becomes even positive and numerically larger than the self-diffusion coefficient of acetone at this composition. We may state that the anomaly reported by McCall and Douglass7 for the system (CH3)zCO-H20 not only is confirmed, but in our representation appears to be much more pronounced. It may be noted that the a In al/a In x1 factor varies greatly for this system and that its minimum value coincides with the maximum value of fzz. If we now attempt to interpret this quite remarkable effect in f i z , we are immediately led to the conclusion that
The Journal of Physical Chemistry, Vol. 84, No. 2, 1980
Diffusion in Binary Nonelectrolyte Mixtures
0
223
C, (mol. I-')
n
1
0
2
3
I
-1.0 -
0 0
-2.0
-
'4
b
-50
02 Flgure 4. Representation of molecular rearrangements in an acetone-water system in an initial state (a) and (b) in a state in which a molecule of type 2 is trapped.
it cannot be explained by the postulation of a particular local mole fraction, by an effective dynamic mass of the molecule different from the real one, or by a drastic change of the correlation time compared with the standard assumption. Any of these assumptions would not account for the change in sign of the observed VCC with respect to the predicted one from eq 11. Thus there must be a more complicated mechanism. We start from the statement that bath systems show a positive deviation from Raoult's law. Again, the effect is much stronger for the aqueous system than for the nonaqueous one (see Table 11). A positive deviation from Raoult's law, if translated to a molecular picture, is connected with 11and 22 association, in othler words, we have microscopic heterogeneity in such a way that the local mole fraction of kind i around a molecule of species i is larger than the mean or macroscopic mole fraction %,. In Figure 4a we give a schematic representation of such an instantaneous molecular configuration. If the mixture has a mole fraction x, = 0.5, then, it is seen that the first coordination sphere of both species i contains three molecules of species i and one molecule of the other component, thus we have x,(loc) = 0.75 instead of 0.5. Now let us assume that at a given instant the left-hand quartet of molecule 1 distorts somewhat so as to form a hole or a channel which opens to the right. This fact causes a very high velocity u1@)(0)of a particle of kind 2 in that direction to fill the hole or channel. As 2 moves to the left, in order to keep the momentum conservation satisfied, the molecules of type 1 move to the right and during this process the hole collapses. The situation is depicted in Figure 5b. A.lso, due to the conservation of momentum, other molecules 2 have moved from left to right when u ~ ( ~ ) ( Owas ) large and directed from right to left. We see that eventually one molecule 2 is in a trapped situation and other molecules 2 are accumulated in excess. Now it is clear that there will be a fairly slow flow of molecules 2 from the right to the left, which means that the time integral Som( I Y ~ ~ ~ ~ ( )Odt) will U ~ take ~ ~ ~a ~comparatively ~ ) large positive value because a long-persisting positive velocity correlation is produced which ends when the coor-
'
UREA -WATER (25OC)
1
Flgure 5. Standard and experimental VCC's for the system urea-water at 25 O C .
dination number 3 is reattained. The effect must not be symmetric, in that if one type of particle is trapped as a consequence of a high instantaneous velocity, this need not be true for the other particle as well. The larger particles OMCTS or acetone, obviously, do not undergo this trapping mechanism. Of course, this interpretation can only be very tentative and qualitative. Further experiments will have to show whether positive deviations from Raoult's law are always connected with a positive velocity correlation effect for one component. Then it may also be shown whether the "oscillation" of the observed fiz around the standard curve f l z o is a typical effect which occurs regularly. Whatever the true correlation of properties may be, it is striking that for the system urea-water (see Figure 5 ) we also have negative and positive deviations of f l z and fll from the curves of fi2and fit, respectively. It is perhaps significant that both Stokesz0 and Levay,21 from analysis of the self-diffusion data of Albright and Mills' for this system, have postulated that urea-urea association is present. In both these studies mutual diffusion data which describe a collective dynamic process have not been used. Supplementary Material Auailable: Four tables giving our measured intradiffusion coefficients and calculated fi? and f i j values for all systems are available as supplementary material (5 pages). Ordering information is available on any current masthead page. Copies of the tables can also be obtained from the authors.
References and Notes (1) For definition of "intradiffusion" see J. G. Albright and R. Mills, J . Phys. Chem., 69,3120 (1965). (2) L. S.Darken, Trans. Am. rnst. Min. Metall. €nu., 175,184 (1948). (3) G. S. Hartley and J. Crank, Trans. Faraday S&., 45,801 (1949). (4) P. C. Carman and L. H. Steln, Trans. Faraday Soc., 52, 619 (1!356). (5) R. J. Bearman, J. Phys. Chern., 65, 1961 (1961). (6) R. Mills, J. Phys. Chem., 67,600 (1963). (7) D. W. McCall and D. C. Douglass, J . Phys. Chem., 71,987 (1967). (8) H. G. Hertz, Ber. Bunsenges. fhys. Chem., 81,656 (1977). (9) D. D. Fitts, "Non-equilibrium Thermodynarnlcs", McGraw-Hill, New York, 1962. (IO) K. R. Harris, R. Mills, P. J. Back, and D. S. Webster, J. Magn. Reson., 29,473 (1978). (11) R. Mills, J . fhys. Chern., 77,685 (1973). (12) A. F. Collings and R. Mills, Trans. Faraday Soc.,66, 2761 (1!370).
224
J. Phys. Chem. 1980, 84, 224-226
(13) R. Mills and L. A. Woolf, “The Diaphragm Cell”, ANU Press, Canberra, 1968. (14) K. N. Marsh, Trans. Faraday Soc., 64, 894 (1968). (15) K. N. Marsh, Trans. Faraday Soc., 64, 883 (1968). (16) J. A. Barrie, R. B. Dawson, and R. N. Sheppard, J . Chern. Soc., Faraday Trans. 1 , 74, 490 (1978).
(17) R. Mills, Trans. Faraday Soc., 67, 1654 (1971). (18) D. K. Anderson, J. R. Hall, and A. L. Babb, J . Phys. Chem., 62, 404 (1958). (19) L. J. Gosting and D. F. Akeley, J. Am. Chem. Soc., 74, 2058 (1952). (20) R. H. Stokes, J . Phys. Chem., 69, 4012 (1965). (21) B. Levay, J. Phys. Chem., 77, 2118 (1973).
COMMUNICATIONS TO THE EDITOR Reactions of the Xe(3P,) and Kr(3P,) Resonance States wlth Halogen Donor Molecules
Xe(3Pl)
Sir: With the advent of rare gas halide lasers has come interest in the formation, quenching, electronic state transfer, and vibrational relaxation processes of the rare gas halide B and C states. Although quantitative measurements of rate constants and branching fractions for the reactions of the metastable (3P2o) rare gas atoms with halogen donors have been made,l-k little information is available regarding the reactions of the (3P1)and (IPl) resonance states with halogen containing molecules, Existing measurements6-10establish that the quenching rate constants for the resonance states should be comparable to or slightly larger than those for the metastable states. However, there are fewlo measurements of the branching fractions for rare gas halide formation or product state distributions from reactions of the resonance states. In this work we report the formation of KrX* and XeX* (X = F, C1, Br, and I) from the reaction of the (3P1)resonance states with halogen donors. The low pressure XeX and KrX emission spectra from these reactions is compared to that from the (3P2)metastable atom reactions. Relative rate constants for formation of XeCl(B,C) are estimated for a series of chlorinated reagents. From the pressure dependence of the XeCl and XeF B/C ratio, transfer rate constants are assigned for He, Ne, and Ar bath gases. The experimental apparatus is essentially the same as that used by othersl1J2 for observing excitation transfer reactions of the resonance states. A microwave powered resonance lamp was interfaced via a MgF, window to a square stainless-steel reaction cell; the rare gas halide emission was observed through a quartz window at right angles to the light from the resonance lamp. The quartz discharge lamp, which has a 5-7-cm extension arm for cooling, was attached to the cell via an O-ring seal. The extension was cooled with liquid Ar and with liquid N2for the Xe and Kr lamps, respectively. The spectra from the lamps, examined with a vacuum monochromator, consisted predominantly of the (3P1)resonance lines (147.0 nm for Xe, 123.6 nm for Kr). Previously, this apparatus was used for the photolysis of XeF2.13 The rare gas halide emission was observed with the same computer interfaced monochromator used for study of the metastable atom reaction~.~-* Each halide donor was photolyzed in the absence of rare gas to check for a photodissociation process which would give an interfering fluorescence. The cell then was evacuated and filled with mixtures of rare gas and donor. The following equations describe the system for Xe:
-
Xe(PP,)
Xe(’So) + hv(147 nm)
+ - + + -
Xe(3P1) + RX
Publication costs assisted by the U.S.Department of Energy
Xe(lSo) + hv(147 nm)
-
(1)
0022-3654/80/2084-0224$01.00/0
Xe(’P1)
RX
Xe(3Pl)
Xe
RX
(2)
+R
(34
other products
(3b)
XeX*
Xe(3P2)+ RX
(34
Searches were made for rare-gas halide emission with rare gas/RX ratios of 0.1-10 at total pressures ranging from 0.03 to 1torr. The relative RgX emission intensities from the Rg(3P1) + RX reactions closely resembled those observed from the metastable reactions, i.e., a halogen donor with high XeX branching fractions for Xe(3P2)also give intense XeX emission with Xe(3Pl). For X = F, the most intense emission was obtained from the reaction of F2, followed by CF30F and NF3. For X = C1, C12was the best donor, followed by SC12and COCl2. On the basis of these observations and the values of the quenching rate constantslJ+10for the resonance state, which are slightly larger than those of the metastable atoms, reaction 3c is thought to be unimportant. This was qua1itat)ively confirmed by failure to observe any atomic absorption from Xe(3P2)by using the source lamp previously used to monitor the metastable states.l The 3P1concentration in the resonance cell and the 3P2concentration in the flowing afterglow are estimated to be approximately equal. Since the KrCl and XeCl emission intensities from C12 in the two experiments were comparable, by implication the branching fractions for the resonance states are similar to those of the 3P2atom reactions, which are 1.0 and 0.9 (Clz* = 0.1) for Xe(3P2) and Kr(3P2),respectively. Experimental conditions could be adjusted so that the XeCl emission intensities were first order in [RCl]. For these conditions the relative intensities are proportional to the rate constants for XeCl formation; the values for C12,SC12, C0C12, and HC1 were 1.0,0.2,0.1, and 0.01. Judging by the similarity of C12*emission from reaction of Xe(3P1)and Xe(3P2)with C12,and the reduced rate constants for XeCl formation for larger donor molecules, we feel reaction 3b is important, just as for the (3Pz) atom reaction^.^ The high RgX branching fractions from diatomic halogens with Xe(3P1) and Kr(3P,) contrasts sharply with the claimlo of a low ArF branching fraction for A@P2)with F,. The ArF branching fraction for Ar(3P2) with F2 is reduced, relative to Xe(3P2),but is still large (0.53).4 The low pressure spectra of XeCl and KrCl from the (3Pz)and (3P1)reactions are shown in Figure 1. Two features distinguish between the emission of the (3P1)and (3P2)atom reactions: (i) In general the B-X spectrum from the 3P1reactions extends to shorter wavelength than does the spectrum from the 3P2atom reactions. (ii) The C-A spectrum from the 3P1reaction also shows, as revealed by the position of oscillations and redmost band, a generally 0 1980 American Chemical Society
