J . Phys. Chem. 1986, 90, 1509-1519
1509
calculation at the MRD-CI The results of the present study signal another triumph for high-quality a b initio calculations as a reliable approach to predicting the thermochemical properties of reaction intermediates. With the necessary blessing granted by experimental certification, some changes in the estimated thermochemistry of processes involving silylene are recommended. Related studies of substituted silylenes are in progress in our laboratory.
result strongly favors silylene chemistry, as opposed to silyl radical chemistry, as the primary decomposition reaction in silane pyrolysis. Using a similar approach, Hehre et aLt2have determined the proton affinity of dimethylsilylene to be 232 kcal-mol-]. The effect of methyl substitution (31 kcal.mol-') is close to that observed in comparing H,S to (CH3)*S(30.4 kcal-mol-I). Combining the known heat of formation25Aff?298(SiH2+)= 276.3 kcal-mol-' with the heat of formation of SiH2 derived in this study yields an adiabatic ionization potential of 8.99 eV for SiH,. This is in excellent agreement with the value 8.98 eV predicted by a b initio
Acknowledgment. We acknowledge the support of the National Science Foundation under Grant CHE-8407857.
(25) Ding, A.; Cassidy, R . A.; Cordis, L. S.: Lampe, F. W. J . Chem. Phys. 1985, 83. 3426-3432.
(26) Bruna, P. J., unpublished results cited in ref 25.
FEATURE ARTICLE Some Comments on Multicomponent Diffusion: Negative Main Term Diffusion Coefficients, Second Law Constraints, Solvent Choices, and Reference Frame Transformations Donald G. Miller,*'T2 Chemistry and Materials Sciences, Lawrence Livermore National Laboratory, Livermore, California 94550
Vincenzo Vitagliano,2 and Roberto Sartorio2 Dipartimento di Chimica, University of Naples, 801 34 Naples, Italy (Received: August 15, 1985; In Final Form: November 21, 1985)
We discuss some interesting aspects of multicomponent diffusion in liquids. These include the existence of a negative main term diffusion coefficient; the utility of taking different components as the solvent; the change-of-solvent transformation for Fick's law coefficients and Onsager coefficients; the validity of the Onsager reciprocal relations on changing solvents; and the calculation of partial molar volumes from diffusion data. For this, we first survey previous work on the irreversible thermodynamic basis for macroscopic diffusion; the importance of reference frames and transformations among them; and second law conditions on the volume-fixed diffusion coefficient matrix. Certain diffusion descriptions in other reference frames do not preserve these second law conditions.
I. Introduction Diffusion is one of the major transport processes in liquids. Such processes, especially in multicomponent systems, are increasingly recognized as significant in the earth sciences, biology, nuclear waste isolation, and chemical engineering. Unfortunately, multicomponent diffusion is more complicated than is often realized. For example, a ternary system (two solutes in a solvent) has four diffusion coefficients, not just two. The cross terms can be large and be positive or negative, thus having a substantial effect on flows of matter. The experimental investigation of liquid diffusion in binary systems began with Graham3 in 1850 and its theoretical description (1) Visiting Professor at the Facoltti di Scienze of the University of Naples, Spring 1983. Portions of this work were done under the auspices of the Office of Basic Energy Sciences (Geosciences), US.DOE at Lawrence Livermore National Laboratory, under Contract W-7405-ENG-48. (2) Portions of this work were supported by the Italian Minister0 della Pubblica lstruzione and the Italian C.N.R. (3) T.Graham, Trans. R. Soc. London, 140, 1 (1850).
0022-3654/86/2090-1509$01.50/0
began with Fick4 in 1855. However, the possibility of cross terms in multicomponent systems was not even suggested until 1932 by Onsager and F u o ~ s . ~The experimental verification of their existence by Gosting and collaborator^^^^ was as recent as 1955, a century after Fick. Precision experimental work did not begin until the late 194O's, following the design and construction of optical interferometers. Experimental matters are nicely described by Dunlop et aL8 and Tyrrell and Harris9 Among the complications of multicomponent liquid diffusion is the issue of reference frames. The one closest to most exper(4) A . E. Fick, Ann. Phys. (Poggendorf), 94, 59 (1855). (5) L. Onsager and R . Fuoss, J . Phys. Chem., 36, 2689 (1932) (6) R. L. Baldwin, P. J. Dunlop, and L. J . Gosting, J . Am. Chem. Soc., 77,5235 (1955). (7) P. J . Dunlop and L. J. Gosting, J . Am. Chem. Soc., 77, 5238 (1955). (8) P. J. Dunlop, B. J. Steel, and J. E. Lane in Physical Methods of Chemistry, Vol. 1, Part IV, A. Weissberger and B. Rossiter, Eds., Wiley, New York, 1972, Chapter IV, pp 205-349. (9) H. J. V. Tyrrell and K . R. Harris, Diffusion in Liquids. Butterworths, London, 1984.
t2 1986 American Chemical Society
1510 The Journal of Physical Chemistry, Vol. 90, No. 8 , 1986
iments is the volume-fixed reference frame,8,9but mass-fixed and solvent-fixed frames have useful theoretical properties. This question is significant because the numbers in the diffusion coefficient matrix depend on the choice of reference frame.I0-l2 Consequently to analyze different kinds of transport data measured on different reference frames, they must all be transformed to a common one. The study of reference frames and transformations among them was first explored by P r i g ~ g i n e ,then ' ~ by De Groot and his collaborator^,^^-^^ and by Kirkwood et aI.l7 The reference frame studies just noted, the unsuspected relations between diffusion coefficients resulting from the Onsager recipthe conditions on the diffusion rocal relations (ORR),14~t8-20 coefficient and the use of those conditions to solve various boundary value problem^^^-^^ have all been obtained in the context of linear irreversible thermodynamics. Irreversible t h e r m o d y n a r n i c ~ ~has ~ , ~thus ~ - ~ provided ~ the foundation for a complete macroscopic description of diffusion. In this article we call attention to some interesting additional aspects of multicomponent diffusion, especially in ternary systems. We shall deal with standard diffusion coefficients D, (based on gradients of concentration cj) and the Onsager diffusion coefficients L , of irreversible thermodynamics (based on gradients in terms of chemical potentials pj). Many results considered, while not new, are spread out in the literature. We discuss several major issues: (1) the conditions imposed by the second law of thermodynamics on the values of Dij and Lij, ( 2 ) the choice of a different component as the solvent and its impact on numerical values of D, and L,,, (3) the existence of a negatifie main term diffusion coefficient in a real system for a particular choice of solvent, (4) the D, and Lij transformation formulas for interconverting one choice of solvent to another in the same reference frame for some common reference frames, and (5) the calculation of partial molar volumes from diffusion data. From item (4), we find, not surprisingly, that the Onsager reciprocal relations remain valid. Many results will be given in general form, but most examples will be primarily for ternary systems. We shall be concerned only with linear diffusion processes. Unlike most Feature Articles, this one is partially a didactic survey of important known aspects of diffusion, partially a consideration of overlooked implications of alternative choices of the solvent constituent, partially an emphasis on recent unusual experimental data, and partially a presentation of new results. Therefore, the interested nonexpert is encouraged to first read the motivations, descriptions, and conclusions, rather than plow through all the mathematical details of the transformations. Those with a deeper interest can pursue the details later. 11. General Description of Diffusion Flows
Diffusion is the mixing phenomenon resulting from matter flows (IO) R. P. Wendt and L. J. Costing, J . Phys. Cheni., 63, 1287 (19.59). (11) D. E. Anderson, Reo. Mineral.,S, 211 (1981). ( 1 2) L. A. Woolf, D. G. Miller, and L. J. Gosting, J . A m . Chem. Sot., 84, 317 (1962). (1 3) I. Prigogine, Etude Thermodynamique des Phgnomenes Irrguersibles, Dunod, Paris, 1947. (14) S . R. DeGroot and P. Mazur, Non-Equilibrium Thermodynamics, Interscience, New York, 1962. ( 1 5 ) G. J . Hwyman, H. Holtan, P. Mazur, and S. R. DeGroot, Phpsica, 19, 1095 (1953). (16) G. Hooyman, Physica, 22, 751 (1956). (17) J . G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Costing, and G . Kegeles, J . Chem. Phys., 33, 1505 (1960). (18) D. G. Miller, Chem. Rev., 60, 15 (1960). (1 9) R. Haase, Thermodynamics of Irreversible Processes, AddisonWesley, Reading, MA, 1969. (20) J. Meixner and H. G. Reik, Thermodynamik der Irreaersiblen Pros. Fliigge, Ed., Springer-Verlag, Berlin, 1959, pp 413-523. zesse, Vol. "2, (21) J. S. Kirkaldy, D. Weichert, and Zia-U1-Haq, Can.J . Phys., 41. 2166 (1963). (22) H . T. Cullinan, Ind. Eng. Chem. Fundam., 4, 133 (1965). (23) H. L. Toor, AIChE J . , 10, 448 (1964). (24) H . L. Toor, AZChE J., 10, 460 (1964). (25) W. E. Stewart and R. Prober, Ind. Eng. Chem. Fundam., 3. 224 (1 964).
Miller et al. caused by driving forces. In most applications, the driving forces are small enough that the flows and driving forces can be linearly connected. If the driving forces are concentration gradients, the linear relations are called Fick's law and the linear coefficients are called diffusion coefficients. If the driving forces are taken from irreversible thermodynamics, the linear Coefficients are called phenomenological coefficients, Onsager coefficients, or thermodynamic diffusion coefficients. Since flows are always measured with respect to some specific reference frame and since the numerical values of the linear coefficients depend on the reference frame, we are obliged to consider reference frame transformations. For Fick's law coefficients, only flow transformations are involved. However, the Onsager coefficient transformations are more complicated because the required invariance of the entropy production also makes the driving forces change with reference frame. Finally, we need to know the relations between Fick's law and Onsager coefficients to calculate the latter. The reason is that, whereas there are many kinds of concentration "meters", unfortunately there are no chemical potential "meters". With this motivation, we are ready to consider multicomponent systems. For an n-component system, diffusion flows in some arbitrary reference frame R can be described in terms of n - 1 independent (related to flows JF and n - 1 independent driving forces chemical potential gradients or concentration gradients).26 The flows on reference frame R are given by9,'4-17.19
where ci is the concentration (molarity) of component i in mol ern-), and ui and uR are the local velocities of i and the reference frame R, respectively, on any reference frame, including the apparatus frame. If ci were in the customary units mol dmW3or mol L-I, then factors of 1000 would appear in many equations below; for example, eq 3 would have a denominator of 1000.'* Any one component flow can be eliminated by the definition of reference frame:9~14-17~19
where a: is an appropriate weighting factor. Let n be arbitrarily chosen as the solvent and eliminated. Then one can write a generalized Fick's law for diffusion in terms of the independent remaining components (the solutes) as14.19 n- I
J? = -CD$acj/ax
( i = 1, ..., n - 1)
j= 1
(3)
which includes cross terms. For simplicity eq 3 is written in one dimension with x as the distance. The numerical values of D: depend both on the reference frame R'O." and on which component is labeled n and thus eliminated.27 We use molar flows (with HaaseI9) in mol cm-2 s-I rather than the mass-unit flows of DeGroot and Mazur.I4 Molarity gradients (or their density counterparts) are almost universally used by experimentalists in describing diffusion, rather than gradients of molality or of fractions such as mole, mass, or volume. With this choice, Dt will have the units cm2 SKI. In terms of the irreversible thermodynamics of isothermal diffusion, the dissipation TCTfor a system at mechanical equilibrium is given by9.I4.l9.20 n
Ta = x J p X i L 0
(4)
i= I
X , = -apI/8x (one dimension)
(i = 1, ..., n )
(5)
could be used, more (26) Although the dependent sets of Jp and transport coefficients are needed and more equations must be solved. It is therefore best to use the minimum number of coefficients sufficient to characterize the processes. (27) V. Vitagliano, R. Sartorio, S. Scala, and D. Spaduzzi, J . Solution Chem., 7. 605 (1978).
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986 1511
Feature Article where u is the "entropy production" (Le., the rate at which entropy is produced internally per unit volume) and T i s the temperature. Now Xn can be eliminated by using the Gibbs-Duhem equation (also valid at mechanical equilibrium)
- 1 components, with JR,Js, X, YR,and Ys being column vectors
and all others being square matrices, then the previous equations become
Tu = SOX = S R y R
(13)
ys = ARSYR
(15)
n
ZCJi
=0
i= 1
If J f is also eliminated by eq 2, then Tu becomes n-1
Tu =
ceXi
(7)
i=l
where are the independent flows and X I are the independent driving forces in the solvent-fixed reference frame,12,14s17 Le., a frame referred to J,. Using eq 2, one can transform flows on one reference frame to flows on any other by the procedure of Kirkwood et aL17 (there in terms of mass flows, here in terms of molar flows). Thus, if the two reference frames are R and S, then for all n flows: n
A is the transpose of A. For many processes, including diffusion, the independent JR and YRtaken from Tu can be linked14JsJ9~28 in matrix or subscript
where
forms by the linear expressions JR
JF = FL;?
J=
I
5 afck
( i = 1,
..., n )
If 8 is also eliminated by eq 2, then the transformation equation for the n - 1 independent flows is &ESJ~
(i = 1, ..., n - 1)
( i , j = 1, ..., n - 1)
(17)
LR?I = LRJ l
(18)
=
(19)
LR
]=I
(16)
(8)
k=l
Jp =
LRYR
Moreover, it has been shown by experiment that the ORR
c, C aFJs
JP=JS--
=
t R
are valid18,28*31,32 for proper choices of JR and YR,32Le., the LR matrix is symmetric.33 The actual numerical values in the L matrix depend on frame R (cf. ref 12). In addition, eq 3 is
JR = DRC
(9)
(20)
where A y is19 where the arrow indicates the matrix entry. It is easily shown by appropriate substitutions in eq 13-16 and 20 that the reference frame transformations for L and D
k=l
(10) and 6, is the Kronecker delta. However, Tu must remain invariant to a change of reference frame. Consequently, the driving forces Y, associated with each reference frame must also change.12,14.16,19,28 and are related by From eq 7 and 8-10,
are12,14,16.19
LR
=
ARSLSARS
(22)
n-I
yS = J=
1
( i = 1, ..., n - 1)
(11)
A consequence of e 10 is that the A? are the inverses (in the
matrix sense) of A S, I and are found simply by appropriate substitution in eq 10. Note the subscript reversal in eq 11. Certain reference frames are particularly useful. The mass-fixed and frame is used to derive the expressions for Tu (eq 4)14,19*20 can be related to velocity correlation coefficient^.^^ The solvent-fixed frame results from the most natural elimination of a dependent force and a dependent flow and is particularly apt for electrolyte solution^.^^^^^ The volume-fixed frame is appropriate when one end of an apparatus is closed off, and volume changes on mixing are small.17 In this situation, this frame coincides with the apparatus itself.I7 Almost all diffusion measurements are performed under these condition^.^.^ For the above reference frames, the values up are as follows: solvent fixed, all a: = 0 except a: = 1; volume-fixed, uy = V,where Vi is the partial molar volume; mass-fixed, af" = Mi where Mi is the molecular mass. Values of A? for one important conversion, solvent-fixed to volume-fixed, are easily found by substitution of the a, into eq 10
If we now switch to an obvious matrix notation in terms of n (28) D. G. Miller, J. Phys. Chem., 63, 570 (1959). (29) D. G. Miller, J.Phys. Chem., 85, 1137 (1981), and references therein. (30) D. G. Miller, J . Phys. Chem., 70, 2639 (1966).
(31) D. G. Miller in Transport Phenomena in Fluids, H. Hanley, Ed., Marcel Dekker, New York, 1969, Chapter 11 (a reprinting of ref 18 plus new material). (32) D. G. Miller in Foundations of Continuum Thermodynamics, J. J. Delgado Domingos, M. N. R. Nina, and J. H. Whitelaw, Eds., Macmillan, London, 1974, Chapter 10 (see also E. A. Mason, Chapter 11). (33) Since the ORR will be essential in the following, certain repeated criticisms need a response. These criticisms, most sharply put by Tr~esdell,-'~ are primarily based on the fact that the vector flows of matter (and heat as well) used here and elsewhere to describe diffusion are not the flows used by Onsager in his original statistical mechanical justification of the ORR.35 It is true that an arbitrary splitting of Tu into J and Y does not ensure symmetry, and that there exist transformations which leave Tu invariant but which do not preserve symmetry. Consequently, a "proper choice" of J is important (given J,, Y , follows automatically for each term of T u ) . One such "proper choice" is the matrix of molar flows on the mass-fixed reference frame, and it will be seen shortly that the ORR are preserved by reference frame transformations among the mass-, volume-, solvent-fixed frames. This "proper choice" of J M has been amply justified by classical, quantum, or relativistic statistical mechanics and kinetic theory. That literature is too large to fully cite here, but ref 14 and DeGroot et al. contain some of it. The critics seem to persistently ignore this extensive literature. Moreover, it is equally valid to take a purely macroscopic view and take as hypotheses that J M is a "proper choice" and the ORR are correct, similar to the hypotheses of the first and second laws of equilibrium thermodynamics. The validity of the ORR then becomes a matter for experimental test. However, experimental tests do confirm the ORR for diffusion with our vector flows, as well as for other p h e n ~ m e n a . ' * ~This ~ ' , ~issue ~ and other unwarranted criticisms have been discussed elsewhere in more detail and with more reference^.^^.^'.^^ (34) C. Truesdell, Rational Thermodynamics, 1st ed, McGraw-Hill, New York, 1969; 2nd ed, Springer-Verlag, New York, 1984. ( 3 5 ) L. Onsager, Phys. Reu., 37, 405 (1931); 38, 2265 (1931). (36) S. R.DeGroot, W. A. van Leeuwen, and C. G. Van Weert, Relatiuistic Kinetic Theory. Principles and Applications, North Holland, Amsterdam, 1980. (37) Reference 30, footnote 18. (38) J. Meixner, Adu. Mol. Relaxation Processes, 5 , 319 (1973).
1512
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986
Moreover, simply by comparing eq 22 with its transpose and using eq 19, one can easily see that if LR is symmetric, so is Ls and conversely. Therefore the O R R are preserved by these reference frame transformations. The DR can be related to the LR,since the general transformed driving force matrix YR can be related to the chemical potential gradient matrix X (X is simply Yo). But X can be related to a chemical potential deriuatiue matrix p and, in turn, to the column matrix of concentration gradients C . Thus
x = /Lc
-
where p
(24)
(i, j = 1, ..., n - 1)
(F,,= all,/&,)
(25)
The matrix p is not symmetric,28 but it can be related to the symmetric one pm by p
where pm
M
-
-
=
= +,/dm,)
(/L:
(26)
pmM
(i, j = 1, ..., n - 1)
( i , j = 1. ..., n
(ml, = am,/&,)
-
1)
(28)
(29)
Therefore YR
= A@RX= A@RpC=
;jORpmAOV
/ ,M,
The key issue in these considerations is that the eigenvalues of the diffusion matrix must be real and positive in order that solutions of the diffusion equation be real and nonperiodically decaying. We are concerned here only with conditions on the D matrix. The original a ~ t h o r s ~ ' - used ~ ~ ,a~variety ~ - ~ ~of notations, reference frames, driving forces, concentration gradients, and proof methods, as well as dependent and independent flow equations. Consequently, no simple direct comparison is possible. We shall survey these results in terms of independent quantitites, i.e., n 1 flows, n - 1 forces, and (n - 1) X ( n - 1) L and D matrices, where component n is chosen as the solvent. From the second law of thermodynamics, the entropy production (dissipation) T a must be positive and, using eq 16 and 19, can be written for an arbitrary reference frame satisfying eq 2 as T~ = j R y R = q R E R y R =
Q = APA
Lf:
(30)
LRYR
= LRAORpC = DRC
(31)
=
(32)
Consequently, in general DR
=
LRAOR
If R is the volume-fixed frame, eq 32 appears more symmetric: DV
=
LVAOV
cc mAOV/C,M,
(33)
If R is the solvent-fixed frame, the D expression has fewer terms, is the unit matrix, and thus since A@@ Do = L@pmAov / c,M,
(34)
111. Second Law Conditions on L, pm,and D
We now turn to what can be learned about diffusion from exploiting the power of irreversible thermodynamics and the second law. Reference frames will again be important, particularly the volume-fixed one. The properties of the diffusion coefficient matrix and the most powerful methods of solution were almost completely described between 1963 and 1965 in a group of landmark papers by Kirkaldy et al.,*' Cullinan,22Toor.23324and Stewart and Prober.25 These papers provided the connection between the second law of thermodynamics and the diffusion m a t r i ~ ;proved ~ ' ~ that ~ ~ this ~ ~ matrix ~ showed how the multicomponent could be diffusion equations could be decoupled (Le., brought to binary forms) by transforming DV to diagonal form and using the same transformation matrix to get "combine concentration^";^'-^^ and showed how the solutions of these binary forms in terms of these combine concentrations and diagonal elements of DV could be transformed back to solutions in terms of the original concentrations.2'-25 These ideas had been stated without proof by Onsager39in 1945. Important earlier, related, or partial results were given by Kirkaldy,4@i4'Sundelof and S o d e r ~ i Cussler .~~ and L i g h t f ~ o t S, ~~~h o n e r tand , ~ ~Gupta and C ~ o p e r . ~ ' (39) L. Onsager, Ann. N . Y . Acad. Sci., 46, 241 (1945).
(36)
> 0 , L52 > 0, LP,L?* - LF2L5]> 0
47 > 0, =
(35)
is also positive definite, whether or not A is.21,46 Therefore, by eq 22, Ls is positive definite if LR is. This also follows directly for frame S in terms of Tu from eq 35. It can also be shown that pm must be positive definite for a multicomponent system to be stable with respect to phase equil i b r i ~ m . ~ This ' is not necessarily the case for p.28s47 The conditions that a matrix P be positive definite are that (1) det (P) be positive, ( 2 ) all its leading principal minors be positive, and (3) all the Pij be positive. For a ternary system this yields
and from eq 20 JR
>
~ R L R ~ R0
Since this is a quadratic form and LR is symmetric, L R must be positive definite. Moreover if any matrix P is positive definite, then
(27)
Here m, is the molality of constituent i in mol (g of solvent)-', rather than the customary mol (kg of solvent)-'. This avoids factors of 1000 later, e.g., eq 63. However, by comparing A: (eq 12) with the expression for m , (cf. eq 28, ref 28), we have M = Ao'/c,M,
Miller et al.
$2
> 0.
PEA42 - PL;"Z/L;
0
(37) (38)
Now let us consider DV on the volume-fixed frame. From eq 33, DV can be written in terms of Q = Aov,,,lIlAov (39) as D~ = L~Q/C,M,
(40)
where Q is positive definite by the theorem of eq 36, and of course c,M, is positive. Therefore, DV is the product of the two positive definite matrices Lv and Q. Such a product need not be either symmetric or positive definite, as can be seen by counter examples. Indeed, DV is typically nonsymmetric. However, det (Dv) is positive because the determinant of a product is the product of the determinants. Therefore, for a ternary DylDy2- Dy2Dy1> 0
(41)
Much more can be derived from the fact that DVis the product of two real, symmetric, positive definite matrices, Lv and Q. First, each of Lv and Q must have all real and positive eigenvalue^.^^^^^ Second, from this, their product LVQcan be shown to have real, positive eigenvalues as even though this product (and thus Dv) need not be symmetric or positive definite. But real, positive eigenvalues are the conditions that solutions of the diffusion (40) J. S . Kirkaldy, Can. J . Phys., 36, 899 (1958). (41) J . S . Kirkaldy, Can. J . Phys., 37, 30 (1959). (42) L . - 0 . Sundelof and I . Sodervi, A r k . Kemi, 21, 143 (1963). (43) E. L. Cussler and E. N . Lightfoot, AZChE J . , 9, 702 (1963). (44) H. J. Schonert, Z . Phys. Chem. (Frankfurt am Main), 42, 247 (1964). (45) P. K. Gupta and A. R. Cooper, Physica, 54, 39 (1971). (46) D. C. Murdoch, Linear Algebra f o r Undergraduates, Wiley, New York, 1957. (47) I . Prigogine and R. Defay, Chemical Thermodynamics, Longmans, Green and Co., London, 1954. (48) G. A. Korn and T. M. Korn, Mathematical Handbook f o r Scientists and Engineers, 2nd ed, McGraw-Hill, New York, 1968, p 420. (49) G. Strang, Linear Algebra and Ifs Applicafions, 2nd ed, Academic Press, New York, 1980, p 250.
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986 1513
Feature Article equation be real and decay nonperiodically.21 This eigenvalue condition also leads to the p r o 0 f 2 ~that ~ ~ ~transformations exist which can diagonalize DV,and thus validates the powerful solution and Stewart and Prober.25 methods of K i r k a l d ~ , Toor,23924 ~' Real, positive eigenvalues are also equivalent to conditions on the diffusion coefficients.2' If X i are the eigenvalues ( i = 1, ..., r ) , ( r = n - 1) then the coefficients wi for the characteristic polynomial written with -A substituted for X (-A),
+ wI(-X)'-l + WZ(-X)'~
+ ... + W , = 0
(42)
can be given in two ways: first, as sums and products of the roots X i , and, second, from the results of expanding the determinant of the eigenvalue matrix by its diagonal elements." The determinant expansion leads to
wi = T , ( D ~ )
(i = 1,
..., r )
(43)
where Ti, the generalized trace of order i, is the sum of all principal minors of Dv of order i. The usual trace, TI,is sum of the diagonal elements, and T , is the determinant of DV. If the roots X i are all real and positive, their sums and products are also. Therefore all wi must be positive. Conversely, if all wi are positive, then the characteristic polynomial with (+A) will have all terms alternating in sign. If the roots can also be shown to be real, then all Xi will also be positive by Descarte's rule of signs.50 These are necessary and sufficient conditions on the DVmatrix. For the three-component system ( r = 2), these conditions arez1
TI = Dyl
+ DY2 = XI + Xz > 0
T2 = det (Dv) = DTID&- DyzDY, = X,X,
>0
(45)
+
where eq 46 is simply the discriminant condition that the quadratic characteristic polynomial have two real roots. Analogously for the four-component system ( r = 3), the conditions are45
+ DYz + Dy3 = XI + X2 + X3 > 0 (47) T2 = (DylDYz- D y 2 D ~ ,+ ) (DyZDy3 - Dy3DyJ + (Dy,Dh - D y 3 D ~ ,= ) XIXI + 4 x 3 + X 3 X 1 > 0 (48) TI = DTl
X1X2X3
(TI
TI
=
>0
(491
(DYl XI =
+ 04;)- [(D';;+ DYz)2- 4 det (DV)]'/' TI - [ TI2 - 4T2] 2
(DY, + 04;)+ [(DTl + DyJ' - 4 det (Dv)]'/2 X2 = 2 TI [ TIZ- 4T2] 2
+
(50) J . V . Uspensky, Theory of Equations, McGraw-Hill, New 1948, p 124.
(51)
XI
(53)
n
det (AoR) = det
(AoR) = CciaF/cnat i=O
(54)
Thus, for the solvent-fixed frame, det (Aoo)= 1, which is obviously positive. For the mass-fixed frame, CciMiis just the density d , and hence det (AoM)= d/c,M,,
(55)
which is positive. For the volume-fixed frame det (Aov) = l/cnr,,
(56)
However, there is no guarantee that det (Aov) is positive or even that A y are positive. The reason is that there exist systems which have components with negative partial molar volumes. If such a component is chosen as a solvent, the determinant is negative and, depending on the concentrations, A:v may be negative. Since E. can even go through 0, the determinant can approach infinity. But Aov appears in eq 32 for every reference frame; therefore, even if the determinants of LR, AoR,and pmare positive, one cannot guarantee that det (DR) 2 0. Only when AoR = Aov does this problem disappear, i.e. only for DV is the determinant guaranteed to be positive or zero when molarity gradients are used as driving forces. We have emphasized molarity gradients in Fick's laws because this is the customary experimental use.10-12,17 However, a little reflection on the solvent-fixedcase suggests that molality gradients may give an interesting result. Thus we write
m
-
Jo = Dzm
(57)
(mjl = -am,/ax)
(58)
and the elements of D: do not have the units cm2 s - I . X can be written simply as X = pmm
Therefore
(59)
and from
Jo =
Dim = LOX = Lopmm
(60)
we have D z = Lop"'
2
1 =-
It is especially important to note that eq 41 does not necessarily carry over to other reference frames when DR is defined by molarity gradients (eq 20). To see this, consider DR on an arbitrary reference frame as given by eq 32. In this case, multipliers of pm are not transposes of each other. Therefore, for det (DR) to be positive or zero, the determinants of AoR and Aov must both have the same sign. In general, det (AoR)is easily found from eq 10 to be
where Equation 50 is again the discriminant condition that the cubic characteristic polynomial have three real roots. Unlike Lv and pm, DV is not symmetric or positive definite. Therefore we cannot prove that all Dp are positive, but only that their sum is.21 Although many experimental systems and model theories have all Dp positive, recent experimentsz7show ternary systems with one negative Dp. However, we will see in section VI that the sum of Dii is positive, as required by eq 44. It is useful to give the eigenvalues for a ternary system, since they are easily made explicit:
+ [TI2- 4T21"' 2 T2
(44)
(TI2- 4T2) = (DYl + DYz)2- 4 det (Dv) = (DY2 - DTI)' 4Dy2DYl = (XI - X2)' 2 0 (46)
T3 = det (Dv) =
Experimentalists almost universally use the solutions of the diffusion equations given by Fujita and Gosting (three-c~mponent)~' and Kim ( f o u r - c ~ m p o n e n t ) .Their ~ ~ ~ ~"eigenvalues" ~ ui are the reciprocals of Xi (ui in ref 2 1 , Di in ref 22-24). For example, for a ternary
(61)
The use of eq 34 and 23 in eq 61 yields D: in terms of Do, and in turn DV on the molarity basis: ) - '( c , M ~ ) A ~ ~ D ~ ( A(62) ~~)-~ DL = ( C , M , ) D ~ ( A ~ ~ =
(52) York,
Clearly, Dk is obtained from DV by a similarity transformation which is then multiplied through by cnM,,. Consequently, after multiplication by c,,M,,, its eigenvalues, as well as the invariants (51) H. Fujita and L. H . Gosting, J . Am. Chem. Soc., 78, 1099 (1956). (52) H . Kim, J . Phys. Chem., 70, 562 (1966). (53) H . Kim, J . Phys. Chem., 73, 1716 (1969).
1514
The Journal ofphysical Chemistry, Vol. 90, No. 8,1986
Miller et al.
IV. Different Choices for Solvent Having finished our survey of older general results, in this and presubsequent sections, we consider some newer ideas and results reference multipler gradient weight involving the choice of solvent species. Once again we will need frame LYR GR factor a; transformations, this time from choice to choice. A. Motivation. In this section, we refer to volume-fixed DB, volume-fixed 1 -ac,iax r;; solvent-fixed c,M, -am,/ax 6,Il unless specifically stated otherwise. It is customary to choose the mass-fixed d -( 1i M , ) ( a w , i a x ) M, constituent in largest amount as the solvent (the others are solutes). mole-fixed C -a,v,iax 1 However, in principle, the choice of solvent is completely arbitrary because actual experimental diffusion data do not depend on this “ d is the density, C the total concentiation in mol ~ m - w~ ,, the choice, only the interpretation and numbers do. For example, in weight fraction, and N , the mole fraction. This table is a n extension of a ternary Rayleigh interferometry e ~ p e r i m e n t the , ~ ~total ~~~~~ Cullinan’s Table I.22 T h e gradient GR is given for one dimension Note that 23‘ = DV. number of fringes, the fringe numbers and their respective positions, and the time are independent of the solvent choice. Similarly, in a ternary Gouy interferometry e ~ p e r i m e n t ,the ~ ~ total ~~’~ of eq 43, are the same as for DV. We have just shown that this number of fringes, the fringe positions, the time, and the area Qo invariance is not necessarily retained for molarity gradients on under the s1 graph do not depend on this choice. the solvent-fixed or other reference frames. However, to calculate the DE, the solutes have to be assigned If we write eq 57 and eq 20 respectively as because their concentrations appear directly in the expressions relating the DE to experimental quantities. Moreover, the reJo = (c,,M,,)Born (63) fractive index increments used in these expressions also depend on the choice of solutes. Thus, a different choice of solutes will Jv = DVC (64) change the numerical values of D; and refractive index increments, then the units, eigenvalues,and invariants of Boand DVare exactly but not the actual flows of matter in the experiment.66 the same, and the diffusion matrices provide “similar” solutions With the customary choice of solvent, all reported main term to the diffusion equation^.^^-^^ We note that identical eigenvalues coefficients DB have been positive so far, thereby automatically do not imply identical numbers in the two matrices, except for satisfying eq 44. However, it is extremely interesting that, with a binary system. a minor constituent chosen as solvent, there exist systems with This idea, suggested by De Groot and Mazurl4*Isfor binary negative Dz.27 An example will be discussed in section V. systems with different reference frame^,^ was very elegantly exThe choice of a minor constituent as the solvent is not just of tended by CullinanZZto multicomponent systems. For each theoretical interest. For example, consider a high molecular weight reference frame, there is a premultiplier like c,M, and a different polyelectrolyte serving as a rigid membrane through which water driving force, but the diffusion matrices BRare all related to DV and a salt can pass. This system can be modeled by using a series by similarity transformations. Consequently, the DR matrix for of ternary systems which begins with the constituent monomer each reference frame will have the same units, eigenvalues A,, and unit, and continues with larger and larger molecular weight invariants T, as DV,provided the appropriate driving forces are polymers. To correlate the flows of water and salt through the used. Table I shows some examples. Interestingly DV is the one polymer, it is clear that the polymer should be taken as the solvent most commonly measured by experimental arrangements, has the in each member of this ~eries.~’-~O Since the ultimate extrapolation simplest premultiplier (= l ) , and uses the classic concentration is to a nonmoving membrane, it is obviously desirable to transform gradients of Fick‘s law. the experimental volume-fixed quantities to the solvent-fixed (Le,, We conclude this section with some qualifications and additions. a polymer-fixed) reference frame. Such an extrapolation may For the diagonalization procedures of K i r k a l d ~ ,Toor,23s24 ~’ and be needed, for example, because optical methods cannot be used Stewart and ProberZSto work, the boundary conditions for all within a solid membrane. solutes must be of the same class and the D , must be ~ o n s t a n t * ~ , * ~ Let us now turn to transformations from one solvent choice to (i.e., in practice, the Acl must be sufficiently small). Some disanother within a given reference frame. For the volume-, mass-, cussion of concentration-dependent D , can be found, for example, and number-fixed frames, the reference velocity is independent in Kirkaldy et a1.54Js and Gupta and Cooper.45 Wc have also of the solvent choice. Hence, the flows also have the same values, implicitly assumed that all eigenvalues are distinct. The case of independent of the solvent. However, solvent-fixed flows will multiple eigenvalues has been discussed most generally by T 0 0 r ~ ~ change as the choice of solvent changes, and it will be necessary and for special cases by Kirkaldy et a1.40~ss3s6 and Sundelof and and to distinguish the flows of constituent 1, for example, by S o d e r ~ i There . ~ ~ are cases near critical mixing points where det (& for solvent choices 2 and 3, respectively. On the other hand, ’ (Dv) = 0. These have been discussed by S ~ n d e l o f , ~Kirkaldy values of the driving forces X i for the solvent-fixed frame do not and P ~ r d y and , ~ ~Vitagliano et aL2’ Finally, the diffusion of depend on the choice of solvent, although this choice determines labeled species (isotope, self, or tracer diffusion) can be treated which set of (n - 1) of them will be used. However, the driving by considering the labeled species as additional components with forces for other reference frames change with that choice. The special properties, and examining the expanded diffusion matrix. diffusion coefficients will need to be distinguished in all cases, This case, first studied by Albright and has been most e.g., ( D t ) kor DF for solvent k . elegantly analyzed by Schonert.61.62 He has also shown63that the eigenvalues of the expanded matrix are the eigenvalues of the (64) J . M. Creeth, J . Am. Chem. Soc., 77, 6428 (1955). (65) D. G . Miller, J . Solution Chem., 10, 831 (1981). original matrix without labels, plus the tracer diffusion coefficients (66) If the components are A, B, and C, then any one of these can be the D,* of all the components including the solvent: all D,* must also solvent. If A and B form a compound, the solution could also be considered be real and positive. to be made up of (AB, A, C) or (AB, B, C), and AB could be taken as a TABLE I: Mole Type Diffusion Descriptions JR = aRDRGR Whose DRHave Matrix Similarity to Each Other”
(fi)2
(54) J. S. Kirkaldy, J. E. Lane, and G. R. Mason, Can. J . Phys., 41, 2174 (1963). (55) J . S. Kirkaldy, Adu. Mater. Res., 4, 5 5 (1970). (56) J . S. Kirkaldy, R. J . Brigham, and D. H. Weichert, Acta Metall., 13, 907 (1965). (57) L.-0. Sundelof, Ark. Kemi, 20, 369 (1963). ( 5 8 ) J . S. Kirkaldy and G . R. Purdy, Can. J . Phys., 47, 865 (1969). (59) J. G . Albright and R. Mills, J . Phys. Chem., 69. 3120 (1965) (60) J . G . Albright, J . Phys. Chem., 72, 1 1 (1968). (61) H. Schonert, Ber. Bunsenges. Phys. Chem., 82, 726 (1978). (62) H. Schonert, Z . Phys. Chem. (Frankfurt a m Main), 113, 165 (1968). (63) H. Schonert, Z . Phys. Chem. (Frankfurt a m Main), 137, 125 (1983).
solvent or solute. This case is under consideration by the authors. These various choices depend only on the materials. In principle one could also use the ‘combine concentrations” as components and use one of them as a solvent. However, this concept is less useful, because the “combine concentrations” depend on the concentrations and diffusion coefficients as well as the materials. Thus such solvent and solute components would be different for each composition, unlike components A, B, and C. (67) R. Paterson, Pontil. Acad. Sci. Scr. Varia, 40, 517 (1976). (68) V . Vitagliano, R. Sartorio, P. Padulano, R. Laurentino, and 0 . Ortona, Gazz. Chim. I f a l , , 109, 427 (1979). (69) R. Sartorio, V. Vitagliano, L. Costantino, and 0. Ortona, J . Solution Chem.. 11, 875 (1982). (70) V . Vitagliano and R. Sartorio, manuscript in preparation.
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986 1515
Feature Article Except as noted below, the results of the next two sections are new. We assume the original solvent is constituent 3 and the new one is constituent 2. Other choices are obtained simply by appropriate relettering. Results will be given for n-component systems in the Appendix, as well as alternate derivations based on transformation matrices. B. Transformations of Dt. We sketch the argument first for a ternary on an arbitrary reference frame (not solvent-fixed). Consider 3 as the solvent. Then
JY = -(apJ?
c2’ = -(PjCl’
+ aFJF)/aF + v3c3’)/72
(67) (68)
where c/ is defined by eq 21 and where eq 68 is the Gibbs-Duhem equation for partial molar volumes. By substituting eq 65 and 66 into 67 and then substituting eq 68 both into the result and into eq 65, one obtains expressions which can be compared term by term with the direct expression in terms of solvent 2: J? =
(@1)2cl’ + (D?3)2c3’
(69)
The results for all reference frames satisfying eq 2 (except solvent-fixed) are
C . Transformations of L t . The argument for a ternary on an arbitrary reference frame (not solvent-fixed) is more complicated are for L t than for D; because the independent driving forces not defined for the solvent. Therefore there is no simple relation A “direct” proof requires exlike eq 6 connecting all the panding the ? for each solvent choice into its appropriate X I by means of the transformation AoR for that solvent choice. Each set in turn is then brought to a uniform set of n - 1 independent X I terms by means of eq 6. If A denotes the original solvent n and B the new solvent m, then J i is obtained from the others and Jf using eq 2. Appropriate expressions of eq 17 are substituted into the JR expressions for flow set B and ( ?)B evaluated in terms of the common set of X I terms. The same is done again in terms of the flows of set A, and the ( ?)A are evaluated in terms of the common set of X I . The two equal expressions for each flow (Jy)Bhave combinations of (Lt)+as coefficients of the XI on one side and other combinations of (Ll,)Bas coefficients of the X I on the other side. For each X , in each flow, those coefficients must be equal. Therefore, there are just enough equations to solve for (L;)B in terms of ( L ; ) , and conversely. The algebra for this “direct” path is lengthy, even for a three-component system, hence only the results are given below:
I;”
e.
Equations 7 1-74 generalize the equations of Vitagliano et al.,27 who first obtained this type of transformation only for the voland cancellation in eq ume-fixed frame. (For this case a: = 73 and 74 yields the results of ref 27.) We will see later that, with molarity gradients, this solvent-solvent transformation is a similarity transformation only for DV. For the solvent-fixed reference frame, we can write
vi,
(89)
(4)3= (@l)3cI’ + (@2)3c2’ (4)*and (4)2 from eq 8, recalling that
We now find except for the solvent for which a: = 1. Then
(76)
a: = 0
(77)
(812
=
c3
c3
(8)3- -(4)3 = ---(4)3 C2 C2
(79)
If we now substitute eq 75 and 76 into 77 and 79 and eliminate c2’ with eq 68, we get expressions which can be compared term by term with
For the solvent-fixed reference frame, the solution principle is the same as given above for D,. Equations 77-79 are used, a common set of X i are employed, and appropriate coefficients are set equal to one another. The algebra, however, is much simpler because the forces are already in terms of Xi.The results are
1516 The Journal of Physical Chemistry, Vol. 90, iVo. 8, 198h
From eq 87 and 88 or 91 and 92 it is clear that, if the Onsager reciprocal relations are valid for one solvent choice, they will be valid with any other in any arbitrary reference frame defined by eq 2. V. Experimental Values of Negative D: As noted in section 111, thermodynamic conditions do not require
Dp to be positive, whereas their sum T , must be (eq 44). Contrary to popular belief, there are systems with a negative D,,for a particular choice of the solvent component.27 One system is the water (I)-chloroform (2)-acetic acid (3), which shows a negative chloroform main term diffusion coefficient over a wide range of concentration, provided water is chosen as solvent component (see ref 27, Table IV). In ref 27, experimental data were treated by assuming acetic acid as the solvent component. A new set of D: with water as solvent were then computed by using the set of eq 71-74 for DV,assuming the partial molar volumes could be approximated by pure component molar volumes. As noted earlier, the DZ can be computed directly from the experimental data in data set 5 given in Table V of ref 27, using in turn acetic acid, water, and chloroform as solvents. The results are in Table 11. Since the resulting D,"k do not involve any assumptions on the VI, their closeness to those given in Table I V of ref 27 indicates that pure component volumes are a good approximation to VI. What is important is that Dx Dik is positive as required by eq 44, as is the determinant in eq 45. Note also that these invariants T I and T 2 are quite constant. Although the negative Dil may appear as some sort of artifact, this is not the case. The solvent component choice is made arbitrarily by introducing into the calculations the desired pair of Aci with the set of experimental diffusion data, excluding the Ac, of the component taken as the solvent. The accuracy of the results is directly related to the accuracy of the measured Ac,, which is why taking the most abundant component as solvent is generally preferred.
+
VI. Determination of from Volume-Fixed Diffusion Measurements As noted previously, diffusion experiments are independent of the solvent choice. but the numbers in the diffusion coefficient matrix do depend on this choice. For each choice, the input to calculate D,J consists of the experimental diffusion data together with the concentration differences Aci of the solutes. It is interesting to note that the partial molar volume of the components at their average concentrations can be obtained from any pair of experimental (D:)! sets. Let us take for simplicity a ternary system on the volume-fixed reference frame. In general, if cI and c2 are the solute concentrations, then the solvent concentration c3 can be obtained from clMl
+ czM2 + c,Mj = d
(94)
where d is the density. To calculate ,le3 and the average c j for the various solution pairs, c I . c2, and d must be known for each top and bottom solution. From eq 71 and 72 and the equation c1V1+ c2Vz
+ c3V3= 1
(95)
there are three equations from which the three v, can be determined. In these equations, the ci are the average concentrations of the two or more experiments. Suppose now, the diffusion coefficients are determined by using ,lel and Ac2. Then (DYl),, (DY2),,(DYl),, and (Dy2)3are obtained. If Acl and Ac3 are used. (Dyl)2,(Dy3)z,(Dyl)2,and (Dy3)2are obtained. Solving eq 71, 72, and 95 for V! yields
Miller et al. TABLE 11: Diffusion Coefficients for the System Water (1)Chloroform (2)-Acetic Acid (3) at 25 OC for Different Choices of Solvent Component' (Data from Ref 27) acetic acid water chloroform component i acetic acid water component k chloroform water chloroform solvent j acetic acid 105(D9, 0.307 -0.616 1.012 --0.266 0.370 - - I ,088 IO S ( D i ) , -0.264 0.340 1.162 lO'(D:!), 0.229 0.938 1.813 lO'(DYi), 1.197 1.241 IO'T, 1.245 0.147 0.162 I O'"T, 0.162 0.139 0.148 10% 0.148 1.058 1.093 I0'X2 1.097 O F , = 6.94. (;z = 4.90, F3 = 8.57. The results of (D,vk)s(solvent acetic acid) given here a r e slightly different from those previously published2' (although within the experimental error), because of a slight improvement in the computer program used to compute the Ds. Units are r , in mol dm-? and ( D s ) , in ern's-'.
TABLE 111: Partial Molar Volumes of Water, Chloroform, and Acetic Acid Computed from Diffusion Coefficient Data of Table I1 by tisine Ea 96-99
c, cm3 mol-'
water
chloroform
acetic acid
16.9 17.4 18.5
80.0 80.7 78.7
57.2 56.4 56.7
acetic acid-chloroform water-acetic acid chloroform-water
18.07
84.64
57.52
molar volumes of pure components from ref 27
pair of solvents chosen
where [
1 = C l [ ( D ~ I -) , (DYl)21 + c2(DY*,, - c,(DY3)2
(99)
Relations between other pairs are obtained by relabeling subscripts. If these VI expressions are substituted into the remaining eq 73 and 74, then eq 74 yields the equality of the traces and eq 73 the equality of the determinants of DY and DY. The equality of these invariants suggests that the solvent-to-solvent transformation is a similarity transformation, as we will indeed see in the Appendix. For the three choices of solvent three DY matrices can be calculated from ex erimental data and a pair of AcJ,j # i . Substituting the Di$elements in eq 96-99 for each of the three (DY, DY) pairs yields three separate calculations of Vi, which should be in good agreement with each other. Table I11 shows the partial molar volumes computed from the data of Table I1 using, in turn, all three possible pairs of solvents. These data are in reasonably good agreement, especially since the original experiments were not performed with the aim of obtaining the VI data, and hence uncertainty is present in the acetic acid Ac3 values. We note that if Ac, are small enough, and the densities of each solution are known, then the density data can be fitted to the equation d = d(cI,i.,)
+ H ~ ( c I- ? I ) + Hz(c2 - C,)
(100)
from which the Vi at t i and ?, can be calculated directly." Here t,are the average concentrations. Since the concentration and density data necessary to calculate the DB sets are already sufficient to obtain 6 from eq 100, the indirect calculation of from transport data actually contains no new information. However, the comparison of the direct and indirect values of 8 is a powerful test of experimental consistency. Such a comparison can be made by solvent transformations and V! calculations using the data of Kim,72obtained from a diffusionieter of substantially higher precision73than that for Table
E.
( 7 1 ) P. J . Dunlop and L. J . Gosting, J . Phys. Chem., 63, 86 (1959). ( 7 2 ) H . Kim, J . Chem. Eng. Data, 27, 255 (1982). (73) L. J. Gosting, H. Kim, M . A . Loewenstein, G. Reinfelds, and R . Revzin, Rec. Sri Insfrum.. 44, 1602 ( 1 9 7 3 ) .
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986 1517
Feature Article
TABLE IV: Diffusion Coefficients of NaCI-KCI-H,O at 25 OC for Different Solvent Choices (Data from Ref 72) all 4 expt best 3 component i component k component j (solvent)
NaCl KCI
NaCl
KCI
HI0
H20
KCI
1o5(~:), 10s(D$, IO5(D, 1 105(Dfkj, I 05 T ,
1.4459 0.0877 0.0892 1.6624 3.1083 2.3958 1.4142 1.6941
1.3878 -0.0502 0.1617 1.7204 3.1082 2.3957 1.4144 1.6938
NaCl 1.5286 -0.0769 -0.2455 1.5799 3.1085 2.3962 1.4146 I .6939
10IoT2 lOSX,
1 05X2
HI0
NaCl KCI HI0 1.4643 0.0844 0.0681 1.6662 3.1305 2.4341 1.439 1 1.6914
NaCl
KCI
H20
H2O
KCI 1.4077 -0.0485 0.1832 1.7228 3.1305 2.4341 1.4391 1.6914
NaCl 1.5646 -0.0584 -0.2731 1.5659 3.1305 2.4341 1.4391 1.6914
“For this system C, = 1.9, E2 = 0.1, E3 = 53.14066. Kim’s values of DZ for H 2 0as solvent differ slightly, but a t e well within reported experimental error. Differences a r e probably due to different computer programs for processing the raw data. Units are c, in mol dm-’, (Di),in cm2 s-’. TABLE V: Partial Molar Volumes of NaCI-KCI-H,O from Diffusion Data of Table IV all 4 expt
best 3
NaCl
KCI
H20
NaCl
KC1
H20
solvent pair
20.85 20.89 20.74
31.47 31.34 31.48
18.013 18.012 18.017
21.02 21.00 21.03
31.34 31.33 31.34
18.008
18.008 18.007
H2O-KC1 NaCI-H20 NaCI-KCI
20.91
31.37
18.011
20.91
31.37
18.011
eq100
from one solvent to another in a given reference frame are given in general index form for both Dl and L t , with illustrations for four-component systems. A. Matrix Transformations f o r One Solvent to Another. Without loss of generality we can let the old solvent be n and the new one be m = n - 1 in an n-component system, and let subscripts on matrices refer to the solvent choice. We also take a fourcomponent system for illustration. The general transformation eq 14, when written to take (Jf‘, JF, JY) to ( J r , JF, J f ) , is
“ F r o m eq 96-99. The bottom row are Vi from eq 100 fitted to experimental densities in ref 72.
11 data. The c3 were calculated from tabulated c I ,c2,and d for each top and bottom solution. From the Ac, pairs and the raw diffusion data, the three sets of (D$)j were calculated from the four reported experiments and are given in columns 2-4 of Table IV. Note the much greater constancy of the invariants T I and T 2 from these more precise data. Values of from these three sets of (Dik), are given in columns 1-3 of Table V. As can be seen, the Vi from eq 96-99 are in excellent agreement with each other and with those obtained from a least-squares fitting of eq 100 to densities. The maximum deviation is less than 0.2 cm3. An additional illuminating comparison of overall consistency is possible with Kim’s data,72because one of his four experiments is slightly off the average Ci of the other three. If the best three are used to calculate (Dik),,columns 5-7 of Table IV are obtained. These differ from the results of all four experiments within the estimated error. However, when these (Dik),are used with eq 96-99, the internal agreement of the E. (columns 4-6 of Table V) is within 0.03 cm3 and differs from eq 100 values by a maximum of 0.12 cm3. Moreover, the invariants of these three sets are now constant to five significant figures. Clearly the three best experiments are more consistent. This three-experiment set is probably more accurate as well, as its calculated Q, values (not shown here) are closer to the experimental values than those from the four-experiment set. However even the slightly lower consistency of the 4-set still indicates a very high standard of experimental precision.
vi
Acknowledgment. D. G . Miller is very grateful for the hospitality of the Dipartimento di Chimica during his stay in Naples in the Spring of 1983. Appendix. Solvent-to-Solvent Transformations for Four or M o r e Component Systems
There has been some work on solutions of the diffusion equation for multicomponent systems with different boundary conditions,’1.23.24,40,41.43.45,51-53,56,74.75 Specific results exist for the theoretical analysis of four-component system^,^^^^^ but very little experimental work has been rep~rted,’~-~* and the results are not very precise. Owing to increasing interest, explicit transformations (74) H. Fujita, J . Phys. Chem., 63, 242 (1959).
(75) E. L. Cussler, Multicomponent Diffusion, Elsevier, Amsterdam, 1976.
where (JY)k indicates the flow of i for solvent choice k . The J , without superscripts and an Lij without superscripts (see below) have the same range of indexes as solvent choice 4 does and are used for consistency in indexes. At the end, J 3 , L33,etc. can be replaced by (J,“),, (Lf4)3,etc. For reference frames other than solvent-fixed, consideration of eq 2 shows that AR is of the form AR=
(i: 0
3 0
1
where
ai = -aR/aF
(‘43)
For the solvent-fixed reference frame, consideration of eq 8 (or the extension of eq 77-79) leads to a matrix of the form (A4
Here
ai = -c,/c3
( i # 3)
a, = -c4/c3
(‘45) (‘46)
Transforms for the driving forces Y R of irreversible thermodynamics must keep TCIinvariant. They are given by eq 15, and are the transposes of eq A2 and A4 for nonsolvent-fixed and solvent-fixed frames, respectively. Consequently, the transformation of the Onsager coefficients LE follows from eq 22, i.e.
~p
=L =
ARL,R;IR
(‘47)
where A R can now be either eq A2 or A4. Since eq A7 has the same form as eq 22, if LF is symmetric, so is L?; i.e. the Onsager reciprocal relations are preserved in all solvent-to-solvent transformations. However, since eq A7 is not a similarity transfor(76) G. P. Rai and H. T. Cullinan, J . Chem. Eng. Data, 18, 213 (1973). (77) M. Tanigaki, K. Kondo, M. Harada, and W. Eguchi, J . Phys. Chem., 87, 586 (1983). (78) M. Tanigaki, S. Machida, M . Harada, and W. Eguchi, J . Chem. Erin. Jpn., 16, 257 (1983).
1518 The Journal of Physical Chemistry, Vol. 90, No. 8. 1986
Miller et al.
mation, the traces and eigenvalues of the L matrices are not preserved. However, it is easy to verify for three components that the determinants satisfy the equations det (LF)
--
det (L:)
--
(A81
a:
a:
c: det (Lq) = c: det (L!)
('49)
using eq 86-89 and 90-93, respectively. For diffusion coefficient transforms, we need a matrix B which takes the "driving forces" (cl', c2/, c4') into (c,', cz', c3'): namely
c,=
=
c
[tiir].
(I;)
= pij,.f;:)=
(A10)
c3*
- 4
c3
BC,
The quantity ci* is used to maintain consistency in indexing. Equation 68 is generalized to an n-component system as
xF',c:
=O
( i = 1. ..., n )
(All)
(i # 3)
(A131
V3
(A141
I
From this, B is given by
where
bj = -F',./V3 b3 = -V4/
The diffusion coefficient transformation is obtained as follows:
Jp = DpC3
('415)
J%= DfC4
(A161
Jp = ARJf = ARD4C4= ARD4BC,
(A171
Therefore
and
It is instructive to multiply out the D and L transformations for a four-component system. (The general forms can be obtained by induction.) Alternatively a summation analysis can also be carried out with the range of indexes for solvent n, using the unsuperscripted J and L mentioned above. At the end, indexes relative to solvent n can be restored. Equation A1 9 can be proved analogously for n-component systems. Consequently, the solvent-to-solvent transformation in the volume-fixed reference frame for n components is a similarity transformation, preserving eigenvalues and traces of all orders. Since Cul1inan's2*matrices with his driving forces and premultipliers are related to DV by similarity transformations, (e.g., Bo of eq 63 and Table I), solvent-to-solvent transformations with them will also be similarity transformations. Note that for binary systems, the DR matrix has only one entry and thus "similarity" means identity. Consequently, either solvent choice has the same value of D and all reference frames have the same D,with due regard to appropriate driving forces and p r e m ~ l t i p l i e r s . ~ ~ ' ~ We now turn to explicit results, and in all cases give them in terms of new solvent m and old solvent n. Although a single equation is given for all cases, it is more illuminating to break up the various terms for cases ( i j # n,m), (ip:i # n,m), ( n j ; j # n,m), and (n,n). B. Change ofSolvent for Dt (Molarity Gradients). ( 1 ) All but Soluent-Fixed
5
(D!), = (DE), - -(D&),
( i # m,n)
(A30)
Vm
DF = ARDfB
('418)
where R can be any of our reference frames. It is interesting to note that if R is the volume-fixed reference frame, substitution and multiplying out yields
A ~ = BI
V"
(DiR,), = --(D&),
vnl
( i z m,n)
(A3 1)
('419)
where I is the unit matrix. Therefore B = (Av)-]
('420) ('433)
and hence
DY = AVD:(AV)-'
(A211
Since this is a similarity transformation, the eigenvalues A, (or uI) and the traces of all orders for Dy and DY are the same. See Tables I1 and IV. These results are also easily verified for three components by using eq 71-74. These results are not true for the other reference frames when concentration gradients are the driving forces, as can be verified from eq 71-74 or 82-85. However, they will be true for the Boof eq 63 with molality gradient driving forces, and for the other cases shown in Table I. All the previous results remain valid for n-component systems if, in the matrix equations, solvent 3 is replaced by m and solvent 4 is replaced by n. We can also write Af: and B , in general form as A; = 6,(1 - 6m,)
+ 6,,la,
(A221
all frames but solvent-fixed a, = -aF/a,"
(A23)
or in a single equation
If R is the volume-fixed frame, there will be some cancellation. ( 2 ) Soluent-Fixed
1519
J . Phys. Chem. 1986, 90, 1519-1524 cn (OH,), = --(Dkj),,
C,
5
( 2 ) Solvent-Fixed
+ -(Dkm),,
(j # m,n)
(A37)
Cmvm
Cm
(j # m,n)
(A47)
or in one equation
.... . .
k # n I+n
('443)
anL
1
(L;), = cicj (1 - Sni) x
or in one equation
6ni(l - 6 n j ) af:
6dnj
k+n
af(LC),
+ R2 ( a n )k # n
z a F a f ( L B ) , (A44)
/#n
These explicit equations also show that the ORR are preserved in changing solvents.
SPECTROSCOPY AND STRUCTURE Binary Overtone and Combination Band Intensities of Methyl Fluoride Shigeo Kondo,* Yoshinori Koga, and Taisuke Nakanaga National Chemical Laboratory f o r Industry, Tsukuba Research Center, Yatabe, Ibaraki 305, Japan (Received: May 30, 1985)
Integrated intensities of the binary overtone and combination bands of CH3F and CD3F have been calculated by using the experimental first derivatives and theoretical second derivatives of the dipole moment as well as the recently reported cubic force constants. They were compared with the experimental values obtained with the pressure broadening technique. A marked agreement was obtained between the calculated and measured intensities. Many terms of the second derivatives of the dipole moment have been determined from the observed band intensities by subtracting the contribution of the mechanical anharmonicity. These values have been compared with those obtained from analysis of the effective dipole moment in the excited vibrational states.
Introduction General expressions have kn developed for the band intensities of two quantum transitions by Secroun, Barbe, and Jouve,] and by Yao and Overend.2 According to these expressions, analysis (1) Secroun, C.; Barbe, A.; Jouve, P. J . Mol. Spectrosc. 1973, 45, I.
0022-3654/86/2090-1519$01.50/0
of the overtone and combination band intensities requires extremely accurate information O n the anharmonic force Constants (2) (a) Yao, S. J.; Overend, J. Spectrochim. Acta, Part A 1976, 32, 1059. (b) Overend, J. J . Chem. Phys. 1976.61, 2878. (c) Overend, J. "Vibrational Intensities in Infrared and Raman Spectroscopy"; Person, W. B., Zerbi, G., Eds.; Elsevier: Amsterdam, 1982; p 203.
0 1986 American Chemical Society
