Oct. 20, 103.J
INTERACTING FLOWS : DIFFUSIONCOEFFICIENTS FROM MONENTS [CONTRIBUTIOK FROM
THE
DEPARTMENT O F CHEMISTRY, UNIVERSITY
,5235
O F WISCONSIS]
Interacting Flows in Liquid Diffusion : Equations for Evaluation of the Diffusion Coefficients from Moments of the Refractive Index Gradient Curves BY ROBERTL. BALDWIN, PETER J. DUNLOPAND LOUISJ. GOSTING RECEIVEDMARCH30, 1955 For systems of three or more components, flow equations of a more general form than Fick's first law may be required t o describe the process of diffusion. Generalized flow eqiations for liquid systems are proposed which are purely phenomenological and which follow from certain considerations of Onsager. Using these flow equations, expressions for the moments of refractive index gradient curves observed in free diffusion are derived in closed form and for a n y number of components. T h e four diffusion coefficients which appear in the flow equations for a three-component system a r e then expressed in terms of the second and fourth moments for two experiments. Providing t h a t both the specific refractive increments and the diffusion coefficients do not vary appreciably across the diffusing boundary ( a s assumed in t h e derivations), it is shown t h a t the refractive index gradient curves are symmetrical about the position of the initially sharp boundary.
In some liquid systems the flows produced by diffusion are not adequately represented by Fick's first law,' and more general flow equations2s3are required to describe experimental results within the error of measurement. Data for two such systems are presented in a companion article.4 The purpose of this paper is to show how the diffusion coefficients which appear in the following general flow equations can be evaluated from moments of the refractive index gradient curves observed in free diffusion experiments. Flow Equations.-We may divide liquid systems into three classes according to the complexity of the flow equations required to interpret diffusion experiments. In the first class are all systems of two components, wherein diffusion of either component is completely described by Fick's first law. 1.3 J , = --D(dc,/ax)l
(i = 0,l)
(1)
In this equation J ; is the flow5 of component i, and ( b C ; / d ~its ) ~ concentration gradient, a t position x and time t; the same diffusion coefficient, D , applies to both components. 2 , 3 , 6 A second class may be defined operationally as those systems of p 1 components (p 3 2) in which the flows of all but one are given separately by relations of the form Ji = --Di(dC,/dX)l (i = I, , . . , q ) (2) Equations of this type have long been used to describe diffusion in dilute solutions of two or more but satisfactory tests of their adequacy
+
(1) A. Fick, Pogg. A n n . , 94, 59 (1855). (2) 0. Lamm, A r k i v K e m i , MMineral., Geol., 18B, No. 2 (1944): J . P h y s . Colloid Chem., 6 1 , 1063 (1947). (3) L. Onsager, A n n . S . Y.A i n d . S c i , 46,241 (1945). (4) P. J. Dunlop and I,. J. Gosting. THIS JOURSAL. 77, 5238 (1955). (5) This and t h e following flow equations are written in terms of t h e x codrdinate only, rather than for three dimensions, because derivations in this paper are restricted t o t h e case of diffusion in one dimension in a cell of uniform cross-section. T h e flow is defined a s positive in t h e direction of increasing x, and i t 1s expressed as mass (or moles) of component i crossing unit area per second when Ci is t h e mass (or moles) of component i per unit volume of solution. I t is supposed t h a t no volume change occurs on mixing, so t h a t this coordinate system may be taken as fixed relative t o t h e cell. This condition is met in experimental work by making every concentration difference within t h e cell sufficiently small t h a t t h e partial molal volume of each component can he taken a s constant throughout t h e cell. Correspondingly, taking small concentration differences also decreases variation of t h e diffusion coefficients. (6) G . S. Hartley a n d J. Crank, T r a n s . F a r a d a y SOL., 46, 801 (1949). (7) 0. Lamm, X o u a Acto Regiae SOL.Sci. Upsaliensis, Series IV, 10, No. 6 (1937).
were difficult and seem to have awaited the development of modern interferometric techniques.12-21 Recent studies22'23 with a Gouy diffusiometer have shown that in certain dilute aqueous solutions these relations are adequate to describe the Gouy fringe positions (and hence the refractive index gradient curve) for free diffusion with an accuracy of about 0.0270 of the maximum fringe displacement. In those cases, equation 2 was used to describe each solute flow, leaving the flow of solvent, component 0, u n s p e ~ i f i e d . ~It~ can be seen later that equation 2 is only a special case of equation 3 in which each cross-term diffusion coefficient is zero (Dij = 0 for i # j ) ; when somewhat greater experimental accuracy becomes available, many systems which can now be considered in this second class must be moved to class three and described by equations 3 . In the third class we consider the general case, that is, systems of three or more components in which the flows interact: the flow of each solute de. pends not only on its concentration gradient but also on other concentration gradients. Onsager formulated a description of this case by expressing the flow of each component as the sum of every concentration gradient multiplied by a diffusion coefficient (equation 3 of ref. 3 ) , thus introducing (q diffusion coefficients for a system of p 1 components. He then showed t h a t only p? diffusion coefficients are necessary to describe the
+
+
(8) N. Gralen, Kolloid-Z., 96, 188 (1941) ; Dissertation, Uppsala. 194$. (9) 0. Quensel, Dissertation, Uppsala, 1942. (10) H . h'eurath, Chein. Revs., SO, 357 (1942). (11) E . M. Bevilacqua, E . B. Bevilacqua, M. M .Bender and J . W . Williams, A n n . .V. Y . A c o d . Sci., 46,309 (1945). (12) L. G. Longsworth, THISJ O U R N A L , 69,25lC (1947). (13) G. Kegeles and L. J. Gosting, ib'd., 69,2510 (1447). (14) C . A. Coulson, J. T. Cox, A. G . Ogston and J. St. L. Philpot, Proc. R o y . SOL.( L o n d o n ) , A191, 382 (1948). (15) A. G . Ogston, ibid., 8196, 272 (1919). (16) P. A. Charlwood, .I. P h y s . Chein., S7, 125 (1953). (17) E. Calvet, Compt. l e n d . , 220, 597 (1915); 221, 403 (19t.5); Rev. opt., as, 35 (i950). (18) J . S t . L. Philpot and G . H. Cook, Research, 1, 234 (1918). (19) H. Svensson, A d a Chem. S c a n d . , 3 , 1170 (1949); 4,359 (1950); S , 72 (1951). (20) L . G. Longsworth. THISJOURNAL, 74, 4155 (1952). (21) G. Scheihling, J. c h i i n p k y s . , 47,689 (1950); 48,559 (1951). 76, 3085 (22) D. F. Akeley and L. J. Gosting, T H I S JOURNAL, (1953). (23) P. J. Dunlop, ibid., 77,2991 (1955). (21) T h e solvent flow, J3. is specified indirectly since t h e partial molal volume, vi, of each component is assumed constant (ref. 5 ) . Therefore, Jo is given b y equation 3a.
R.L. BALDWIN, P. J. DUNLOP AND L. J. GOSTING
5236
f l ~ w s . ~ For J ~ convenience, because solute rather than solvent concentrations are usually measured, we choose to omit t.he concentration gradient of the solvent (or major component), and also its flow equat i ~ n , ? “ from ? ~ the set of flow relationships. The equation for the flow of any solute, i, may then be written J, =
-
c 9
D i , ( d C , / d ~ ) f (i = 1,. . .,Q)
(3)
j = 1
This set of flow equations defines y 2 diffusion coefficients for a system of q 1 components, as required by Onsager’s theory; however, these coefficients differ from those defined by Onsager.3 They are related to certain combinations of his (y l ) ?coefficients and the various partial molal volumes. In the accompanying article the coefficients, Dij, of equation 3 are evaluated for two systems; equations for obtaining them from even moments of the refractive index gradient curve in free diffusion will now be derived. A Relation between the Time Derivative of the rth Moment and the Flows.-The rth moment of the refractive index gradient curve, ( b n / b ~versus )~ x,is defined by
Vol. i 7
where ACL =
[(GIB
- (C,)al
(9)
and for convenience we define solute fractions the basis of refractive index by = R,AC,/An
011
(10)
Differentiation of equation 3 with respect to s and substitution of the result into equation 1 yields, after rearrangementz6
+
+
m I= ( l / A n )
Jl
x‘(dn/ax),dx
(I =
0,1,2,. . . ) (4)
+
+
P ~
i
(
-~ Ci) i +.
,
.
(5)
i = 1
Here n~ = n(C1,. . ., Cq) is the value of n when the concentration of each solute is the mean concentration
ci
+ (Ci)B]/2
= [( ci).4
(6)
Solutions X and B are, respectively, the upper and lower solutions used t o form the initially sharp boundary a t time t = 0. In general, each differential refractive increment (7) Ri = Ildn(c1,. . . , C g ) / d C i I ~ . ~ . r j # i l , ,,,.,, ~ - ~ c p = cp depends on the temperature, T , pressure, P , and the nature and concentrations of each of the q solutes present. It follows from equation 5 that the total change in refractive index across the boundary may be written 4
R~AC,
An = i = l
(25) This was done hy means of t h e conditions
y?
L z J $= 0
(3a)
i= 0 9 vi
(bC,/dx)t = 0
(3b)
i =n where t h e partial molal voliime of component i, i i constant throi1r.hout the cell.
vi,
( w r ) %= (R,/An)J:m
x r ( a ~ , / a x ) cix t
(12)
so that summation yields the rth moment of the refractive index gradient curve, which is a measurable quantity. r = 1
The flows, J,, are now introduced from the continuity equation (aCJm
where An is the total change in refractive index across the freely diffusing boundary. In order to relate ( d n / d ~ t)o~ the solute flows, we first express this derivative in terms of the q solute concentration gradients. For this purpose it is assumed t h a t the dependence of refractive index, n , on solute concentrations, Ci, expressed per unit volume of solution, may be adequately represented by the first y 1 terms of a Taylor expansion n = nc
T o facilitate the derivation we define an rth moment of each concentration gradient curve by
= -(dJ,/bX)t
(14)
by first differentiating equation 12 with respect to time and then inverting the order of differentiation with respect to t and x.
These operations are permissible whenever the derivatives [b(bC,/bt),/b.~]~ and [ d ( b C ,/bx),/bt],are both c o n t i n u o ~ s . ~Substituting ~,~~ equation 14 into equation 13, and integrating once the righthand side by parts, yields
since, for free diffusion, ( b J , / b ~ is ) ~zero a t the limits x = a . Another integration by parts, remembering that J , is also zero a t s = + m , gives
*
Equation 16 is useful when r = 1 while equation 17 will be used when Y 3 2. Relation of the Moments to the Diffusion Coefficients.-We now substitute equation 3, describing the flow of each solute, into equation 17 and integrate the resulting equations in order to obtain expressions for the moments. T o avoid serious complications in handling the equations, we restrict our discussion to experiments in which the concentration dependence of each diffusion coefficient may be neglected,j so t h a t each D!j may be taken outside the corresponding integral sign. ( 2 6 ) I n taking Ri for each solute from under t h e integral sign we are assuming t h a t t h e concentration differences, A C i , are all so small t h a t each Ri can be considered constant (see ref. 3 ) . (27) W.Kaplan, “Advanced Calculus,’’ Addison-Wesley Publ. Co., Inc., Cambridge, >lass., 1963, p. 218. ( 2 8 ) R. Courant, “Differential and Integral Calculus,” \‘(>I. I 1 Rlackie and Son Ltd., London and Gl%sgow, 1430, p. 5 5 .
INTERACTING FLOWS : DIFFUSION COEFFICIENTS FROM A ~ O M E N T S
Oct. 20, 1853
4
=
T(T
-
1)
(R~/Rl)Dt,(~T-2)l
(18)
j = l
Equation 18 is seen to be a recursion formula relating the time derivative of any even moment ( r ) to a sum of the next lower ( r - 2 ) even moments, and the time derivative of any odd moment to a sum of the next lower odd moments; its use depends on determining values for (m0)jand ( m l ) j . An expression for (mo)jis obtained by integration of equation 12 and substitution of equation 10. (mol, = a,
(19)
Integration of equation 16 for r = 1 leads to the result t h a t the first moment of each concentration gradient curve is also independent of time
5237
We restrict the remainder of this discussion to two-solute systems ( q = 2) because it seems unlikely that, in the immediate future, systems with nine or more diffusion coefficients (q 3 3) will be investigated by using optical methods to study free diffusion. For this case we denote two naturally occurring combinations of diffusion coefficients by
+ (R?/RI)D~I = D?? + ( R I / R s ) D I ~
61' =
and 62
Dli
(25) (26)
so that, when equations 22 and 23 are summed over all components by means of equation 13, the following important relations are obtained.30 D 2 m = a161 f a262 (27)
CErn= ai[D1161+ (R2/Ri)D216z] +
a2[(Ri/R2)D1261f D L ~ ~ (28) z]
. . . . . . . . . . . Here the symbols
since J, = 0 a t x = It 03 ; hence the first moment coincides with the position of the initial boundary. For convenience the origin of x is now chosen to be the position of the initially sharp boundary, thus making the constant arising from integration of equation 20 equal to zero (m1)J
0
(21)
If now equation 18 is integrated for each successive odd value of r (after first substituting equation 21 into it for the case r = 3) it will be seen t h a t every odd moment of each concentration gradient curve is zero. Thus we have the same result that is well known for systems without interacting flows: when all the diffusion coefficients (Dz3)and differential refractive increments ( R z )are independent of concentration, then every concentration gradient curve is an even function of x, and therefore the refractive index gradient curve is symmetrical about the position of the sharp initial boundary. Expressions for the even moments are obtained by the same procedure. Thus, after substituting equation 19 for the case r = 2, equation 18 is integrated for each successive even value of r . Q
( ~ 2 ) '
C (Rt/R,)Dh,aj
= 2t
(22)
3 = 1
4
a
( m d s = 12t2
(Rt/&)DI$,kak
3 = l
(23)
k = l
anrn m2/2t a:,,, m4/12t*
(29) (30)
denote experimentally measurable quantities which we choose to call, respectively, the reduced second moment, the reduced fourth moment, etc., of the refractive index gradient curve. Equations for the Diffusion Coefficients in Terms of the Moments.-When q = 2, measurement of Ezmand for two diffusion experiments suffices to determine the four diffusion coefficients, provided R1, Rz, al and az-are known and provided that the same values of Ci (equation 6) are used in both experiments. Denoting the two experiments by subscripts I and 11, we first solve equation 27 for el and 02.
Because these linear combinations of diffusion coefficients can be determined from the reduced second moments, without use of the less accurate reduced fourth moments, their values are more accurate than those which can be obtained for Dll, 0 1 2 , D21 and Dzz. To obtain expressions for the individual diffusion coefficients, equation 28 is written in the form 5% = 011D11(61- 62) f
6162
+ a2D22(62 - 61)
(33)
After writing this equation for experiments I and 11, solutions for Dn and 0 2~. 2 Ri, (24) are readily obtained-and sim... _ _ D i,zz, . D . . , , D~ , r + 2 a i ,+ * i? = 1 i3 = 1 i 1+2 Ri C?) (I)(-) (7) plified by means of equations ( d = l 31 and 32; ( R I / R ~ ) Dand ~z where the running indices, i, j , k, etc., have been (Rz/Rl)D21 are then obtained from equations 25 replaced, for simplicity, by il, 22, is, e t c z 9 Substi- and 26. tution of these expressions into equation 13 yields D,,= the desired equations for moments of the refractive [ ( a z ) I - (a2)Ir]B102 - [(a2)I(D;rn)Ir - (a2)II(a;m)rl (34) index gradient curve in terms of the diffusion coef[(Dxrnh - ( 2 a m ) 1 1 1 ficients. (30) If there is no interaction of flows ( D Z J= 0 for i # j ) it is seen or, in general (r)!W (mr)i,= __
(i)!
q
4
1213.
(29) This same approach has been used [R.L. Baldwin, J. Phys. Chcm., 68, 1081 (1954)) to find expressions for the moments of the refractive index gradient curves observed in velocity ultracentrifuge experiments. T h e case considered was t h a t in which Ji = - D i . (dCi/Dx)r CisioQx,where each diffusion coefficient, D,,and each sedimentation coefficient, si, is constant.
+
t h a t equations 27 and 28 reduce t o
Dzm
=
E;,,,
3
WDII aiD:l
+ azD22
+ CU~DI,
in agreement with earlier derivations (ref. 8). ref. 22.)
( 2% 1 ( 28a 1
(Also see equation 33,
PETERJ. DUKLOP A N D LOCISJ . GOSTISG
523.3s
)II - (Dh )I(Th) I I Ill> = (T?,,, (Ti,,)ll - (Th,)l
Yol. T i
(13)
The above equations may be used to calculate diffusion coefficients from measurements made with any diffusion apparatus which yields the reduced second and fourth moments of the refractive index gradient curve in free diffusion. In a companion Experiments in which the concentration of one paper measurements of 22m and 2:mwith the Gouy solute is the same on both sides of the boundary diffusiometer are presented for two three-compoare of special interest because the value calculated nent systems with interacting flows. for each D,,is then influenced least by experimental Acknowledgments.-\Ye would like to thank uncertainty in the reduced moments. Further, such experiments can be used t o evaluate R1 and R?. Dr. J. G. Kirkwood for valuable suggestions in According to equations 31 and 33, when ( a ? ) = ~ 0 connection with evaluating diffusion coefficients from the moments of the gradient curves, Dr. C. F. so that ( a 1 ) 1= 1 , and ( c q ) = ~ ~0 so that ( a g ) ~ =~1 Curtiss for his helpful discussion of the flow equael = (n,,,iI (38) tions, and Dr. J. LV.LYilliams for support and enand couragement of this work. This research was sup82 = (T?,n)lI (39) ported in part by the National Institutes of Health For such experiments equations 34-3'7 reduce, after and by the Research Committee of the Graduate School from funds supplied by the n'isconsin substitution of equations 31 and 32, to Xlumni Research Foundation. D?? =
~IADISOS, \\'ISCO\
SI\
[COSTRIRUTIOS FROM THE DEPARTMEXT O F CHEMISTRY, VXIVERSITY O F \\*ISCONSIN]
Interacting Flows in Liquid Diffusion : Expressions for the Solute Concentration Curves in Free Diffusion, and their Use in Interpreting Gouy Diffusiometer Data for Aqueous Three-component Systems BY PETER J. DUNLOP AND LOUISJ. GOSTING RECEIVED ~ Z A R C H 30, 1956 By using a n extension of Fick's first law t o represent interacting flows in three-component systems, series expansions are developed for the concentration of each solute as a function of time, position and four diffusion coefficients. These expressions converge rapidly when either of the two cross-term diffusion coefficients is small. From these series expansions, equations are derived relating the diffusion coefficients to the reduced height-area ratio, D.4, and to the fringe deviation graphs which provide a measure of deviations of the refractive index gradient curve from Gaussian shape. To test these relations, data are presented from several experiments with the Gouy diffusiometer in which lithium chloride and potassium chloride diffused simultaneousll-, An electrolyte system was chosen with the hope that it would exhibit greater interaction of flows than a non-electrolyte system through electrostatic coupling of the flows of ions of different mobility; assuming t h a t the flow equations are valid, the series expansions are applicable t o either case. I n all experiments the mean concentratioii of a given solute was the same, b u t the ratio of their concentration increments across the boundary was varied from experiment t o experiment. Values were chosen for the four diffusion coefficients which best fit the reduced moments (consult companion paper), reduced height-area ratios, and fringe deviation graphs of the several experiments. The results provide evidence for the validity of the flow equations. Data are also reported for the simultaneous diffusion of lithium and sodium chlorides in aqueous solution. The four diffusion coefficients for this sl'stern are calculated from reduced moments of the refractive index gradient curves: neither cross-term diffusion coeficient is small enough t o permit use of the equations for the fringe deviation graphs.
Throughout the past one hundred years, diffusion measurements in liquid systems have been interpreted by means of Fick's first law.' This law may be regarded as a phenomenological equation expressing proportionality between the flow of a component and the first power of its concentration gradient, the proportionality coefficient being called the diffusion coefficient. Many careful studies of free, restricted and steady-state diffusion have established the validity of this law to describe the process of diffusion in liquid two-component ( I ) A . Fick, Po,?,?.A W ~ I9,4, , .i8 (185.;).
systems.' Furthermore, it has also been found adequate to describe the flow of each solute in certain three-component On the other hand, throughout the last several ( 2 ) I t should be noted t h a t t h e diffusion coefficient may vary with concentration without invalidating Fick's first law. However, if t h e partial molal \-olume of t h e solute varies markedly with concentration, an additional term in t h e bulk flow must he added t o the flotv equation (see Onsager, ref. 1 5 ) . T h e effects of siich variations on t h e shape of the diffusing boundary can be diminished by reducing t h e concentration differences between t h e two solutions which are used tu form t h e initial boundary. (19.73). ( 3 ) D. F. Akeley and L. 1. Gosting, THISJ O U R X A I . , 75, i 4 ) P. I. Dunloll. ibid., 7 7 , 2994 (19.5.5)
