2628 Pi
r RO
v,
JOHN G. ALBRIGHT concentration (probability) of sites i of energy Ei an adsorbate group linear equivalent retention volume, ml./g. adsorbent surface volume, ml./g.
x,,x, 01
ee
Vol. 67 adsorbate concentrations in adsorbed, nonsorbed phases, g . / g and g./ml. adsorbent activity function adsorbent linear capacity
THEORIES FOR THE EXPERIMENTAL STUDY OF ISOTHERMAL FREE DIFFUSION IN TERKARY LIQUID SI'STEMS INT'OLVISG A CHEMICAL REXCTION1~2 BY JOHN G. ALBRIGHT Department of Chenzistry and the Institutejor Enzyme Research, Cniversity of Wisconsin, Jladison 6 , ?Tiisconsin Received June 3, 1963 The linear differential equations that describe an ideal case of isothermal diffusion of two mutually isomerizing components in a ternary system are solved for the case of free diffusion in one dimension. Expressions relating the solute concentrations t o the time, t , the position, 2, the four diffusion coefficients, the two rate constants, and the initial conditions are presented. These results are used to obtain expressions for the reduced heightarea ratio, D A , which show it to be a function of time but independent of Ap, the solute concentration difference between the two initial solutions. I t is shown that the results will also describe free diffusion of a ternary system with a general type of chemical reaction in the limit that Ap 40. Illustrative graphs of DA UF. l / t for a set of hypothetical systems are presented and used to indicate the manner in which the time dependence of fuA might be employed in the analysis of data from diffusion experiments on systems with chemical reactions.
Introduction In recent years theories and techniques have been developed for the measurement of the four diffusion coefficients that describe diffusion in isothermal liquid systems of three c0mponents.3-~0 These theories all suppose that t,he three components do not chemically react with one another. Thus these theories may only be applied to the investigation of systems involving (at most) chemical reactions that are either so fastll that the reacting species niay always be considered to be in chemical equilibrium during an experiment or so slow that for the duration of an experiment the reaction may be disregarded. The free diffusion of an isomerizing solute in a single solvent may be described by equations that appeared in recent art,icles12-14on the electrophoresis and sedimentation of an isomerizing solute, if in these equations the electrical (or gravitational) force is set equal to zero. These articles present the concentration-gradient distribution of the two isomer solutes as fuiictions of time and position for any set' of arbitrary values of the rate constants and the main diffusion coefficients; however, the cross-term diffusion coefficients, which are a measure of the interaction of the flows of the forms of t'he isomer, were assumed to be zer0.~5.~6Since these articles (1) This work has been extracted from material presented in the Ph.D. thesis of J. G. Albright, University of Wisconsin, hladison, Wis. ( 2 ) A portion of this work was presented a t the 141st National Meeting of t h e American Chemical Society, Tashington, D. C., March, 1962. (3) R. L. Baldmin, P. J. Dunlop, and L. J. Gosting, J . Am. Chem. Soc., 77, 5235 (1955). Gosting, ibid., '7'7, 5238 (1965). (4) P. J. Dunlop and L ( 5 ) H. Fujita and L. J. Gosting, ibid., '78, 1099 (1956). (6) H. Fujita, J . Phys. Chem., 63,242 (1969). (7) H. Fujita and L. J. Gosting, i b i d . , 64, 1256 (1960). (8) E. R. Gilliland, R. F. Baddour, and D. J. Goldstein, Can. J . Chem. Eng., 85, 10 (1957). (9) F. J. Kelly, Ph.D. Thesis, University of S e w England, Armidale, Kew South Wales, Australia, 1961. (10) J. K. Burchard and H. L. Toor, J . P h y s . Chem., 66, 2015 (1962). (11) Sn this case one or more of the three components consists of a group of clieniical species t h a t are i n equilibrium. (12) .J. R. Cann, J. G. Kirkwood, and It. rl.Brown, An:iz. Biochem. Biophys.. 78, 37 (1957). (18) T. A . Rak and W. 0. Iiautnan, Y r a n s . Puradag Soc., 56, 1109 (lW9). (14) 1'. C. Scholten, A r c h . Biochem. B i o p h y s . , 93, ,568 (l9cjl). (15) Tlmt, for some systems. such a n assuniption inay be seriouslyin error is auggcsted h y the large values of the cross-term diftusiun coefficients observed for tlie Bystenis discussed in ref. 9 and la.
were not primarily concerned wit'h the case of free diffusion, they do not, consider the effect of an isomerization reaction on qmntities that are experimentally measured by t'he currently used interferometric methods such as the G O Umethod ~ or the Ra,yleigh method. The purpose of this art'icle is to present, theories that apply to the experimental investigat,ion of the general case of free diffusion in a three-component system for which the cross-term diffusion coefficients may be nonzero and in which the components chemically react with one another. It is hoped t,liat t'hese theories will coiit'ribute to a better understanding of diffusion in more complicated systems such as those encountered in biology and chemical engineering.
Theory Basic Equations.-In this development different>ial equat,ions are solved t.hat describe diffusion in one dimension of an isothermal system composed of an isomerizing solute in a single solvent. The init'ial and boundary conditions for which these equations are solved are t'hose of free diffusioii1' with the added assumption that t'he two forms of t'he isomerizing solute are a t equilibrium within the initial solutions.l* The differential equations are linear and hence describe systems which are ideal in the sense that t.here is no volume change caused by mixing or by the isomerization rea,ction, and in t'hat the rate constants and diffusion Coefficients are independent of concentration. These differential equations when written to express the conservation of mass for diffusion relative to a volume-fixed frame of reference are 2 , 1 9 , 2 0 (16) R . P. Wendt, J . Phys. Chem., 66, 1279 (1962). (17) S e e , for example, L. J. Gosting, "Advances i n Protein Cheniistry," Vol. XI. Academic Press, Inc., New York, N.P., 1956. (18) This is a practical requirement because for the case of noneyuililrrium, special experimental techniques must be developed to determine tho concentrations of the initial solutions a t the s t a r t of a n experiment. (19) F o r this special case the volume-fixed f r a m e of reference is c(lnivalcnt,
to the cell-fixed frame of reference of a n actual experiment. (20) These equations may be obtained, for example, from e y . 1 on 1). U4 of S. R. de Groot, "Tlierirlodylialnics of IrrerersiblP I'rocessrs," Intorsuicnce Publishers, Inc., New York, S . Y., 1951. I n chis eclualion PliVk = J* where J1, is llie flow relatire to a cell-fixed flame of re~ert!rioc. For the t w o solutes in the one-dimensional system being considered Iicrc, i i ia perinissiblo l,o write Jlr = - L & ( a p i / a x ) - DkZ(am/dx) wltero IC = 1,2
ISOTHERNAL FREEDIFFUSIOX
Dec., 1963
2629
4, = bP,/bX
(P
=
(9)
1,2)
The partial derivatives with respect to x of the initial conditions described by eq. 4 and 5 represent the initial conditions for eq. 8 and may be written Here Du, 0 1 2 , D21, and Dz2 are the four diffusion coefficients necessary to describe diffusion of three coniponents; kl and l c ~are the rate constants in the isomerization reaction as expressed by the equation
=
(&)i=O
[(PB)B- (Pp)A]a (P = 1,2)
Here 6 is the Dirac delta function defined to be zero for all x except for z = 0 a t which point it becomes infinity such that
SIm6dx
L2
where -41and A2 denote the two isomer forms of the solute, p1 and p2 are the concentrations in mass per unit volume of hland A,, t represents time, and x is the distance along the axis of diffusion. The first two terms on the right of each of eq. 1 and 2 may be recognized as the terms which would describe the change in the concentration of the solutes with time if there were no chemical reaction.6 The remaining terms on the right of each equation indicate the rate of change of the solute concentration directly caused by the chemical reaction.21 The initial conditions of free diffusion for the system under consideration may be defined by the equationsz2 (PdA
(Pl)t=o =
(P2)A
(Pl)t=O =
(P1)R
(PZ)t=O
i
1
- fm
+a
o
(4)
(5)
(Pz)t=o = (P2)B
(Pl)A~cl=
(P2)Ah
(6)
=
(P2)Bh
(7)
(P1)Bkl
=
1
By defining (PdB f (P2)B - (Pl)A - (P2)A (12)
AP
f , = k,/(lCl
+ k2)
(p = 1,2)
(13)
and applying eq. 6 and 7, eq. 10 may be miitten (&)t=O
=
(1 - L,)Aps
(p
=
l,2)
(14)
The Fourier transform, u ( r ) , of a function, j(x), may be defined by the equation24
By denoting u1 and u2 to be Fourier transforms with respect to x (treating t as a parameter) of 91 and 42, respectively, eq. 8 may be transformedz4 to first-order differential equations ; whence
_ bu, - at
where the initial boundary is a t z = 0. The boundary conditions are implicit in the assumption that z extends from --co to + m ; a,t the limits that z = f a , the concentrations \Till not change in a finite length of time and will be equal to the initial concentrations specified in eq. 4 and 5 for all finite values of t. Because equilibrium is assumed between the two forms of the isomerizing solute in the initial solutions
(10)
2
E
D,,?u,
+ (-l)p+qks~q
(p = 1,2)
q=1
(16)
To obtain eq. 16, the quantities +1 and $2 for t > 0 were assumed to be continuous functions of 5 from --co to +-co which are zero a t x = f m . With this assumption, the condition of free diffusion that the diffusion system (diffusion cell) is infinitely long is included in the derivation. The Fourier transforms (ul),=o and ( U Z ) ~ = O of the initial conditioiis ($I), = O and (4z)t=o, respectively, become, with the use of Euler's identity
Integration.--The partial derivatives with respect to .2: of eq. 1 and 2 may be writtenz3
where $1 and A P X (RIA12
(p
=
1,2) (tlc, >> 1) (43)
The condition, lli, >> 1, is obtained by noting from an examination of eq. 20-22, 41, and 42 that Q1 and Qz may be expected to be of the same magnitudes as D and D2,respectively (if not of lesser magnitudes). Hence this condition should ensure that the last termin eq. 43 will be small compared to the first terms,32 and thus that the higher order terms not included in the expansion will be negligible. The Reduced Height-Area Ratio.--A quantity which is basic t,o free-diffusion experiments performed with the use of interferometric optical systems such as the Gouy optical Eiystem is the reduced height-area ratio, Dzi. This quantity is defined by the equation (30) See integral No. 432 in "Mathematical Tables from the Handbook of Chemistry and Physits," 11th Edition, C. D. Hodgman, Ed., Chemical Rubber Publishing Go., 'Cleveland, Ohio, 1969, p. 283. (31) For the intewation of e q . 40, it is useful to note t h a t e x p ( - D t r Z ) = --D-'r-ebexp(--Dtrz)/at = D-%-462exp( - D W ) / b t 2 . With the use of these expressions and the inversion of the order of integration and differentiation, the integrals i n eq. 40 may be reduced to the form shown in ref. 30. (32) This is more clearly seen by inspect,ion of eg. 49.
+ RzA2Je(a-b)t]dr(48)
Equation 44 with eq. 46 and 48 yields an exact expression of a)A4for the free diffusion of an isomerizing solute. An approximate expression for a)* may be obtained, by use of the following expression for the maximum index of refraction derived by substituting eq. 43 into eq. 47. X
(49) The direct use of the above expressions for the analysis of a system from measured values of the reduced height-area ratio would require that the system under investigation have a perfect starting boundary which could be described exactly by eq. 4 and 5. (33) This m a y be shown by eq. 20 a n d 21 of ref. 34. Equation 21 has the same form as the continuity equation for binary diffusion, and the soliitions for all values of i subject t o the initial a n d boundary conditions of ficc diffusion should yield a value of 9~which is independent of time. Thls has been shown explicitly for ternary systems in ref. 5. (34) L. Onsager, Ann. N . Y.Acad. Scz., 46,241 (1945).
JOHX G. ALBRIGHT
2632
Vol. 67
le 1 However, in actual diffusion experiments the starting noAo nlAl n2-42 (53) boundary is always slightly diffuse. Consequently, the k2 time, t', which is measured from the start of an experiment is always less than the time, t, used in this where no, nl, and n2 denote the number of molecules of development. The difference in time, t - t', is recomponents Ao, .A1, and A?, respectively, which are ferred to as the starting-time correction and is denoted involved in the chemical reaction. By convention the subscript 0 will be used to denote the solvent component. by At. The values of the apparent reduced heightHere the rate constants mill be defined to describe area ratio obtained by using t' instead of t in eq. 44 are slightly erroneous. These values, denoted by D ~ ' , systems wherein the concentrations are expressed in are related to D A by the e q ~ a t i o n ~ 5 - ~ ~ moles per cc. Thus for the chemical reaction in an isothermal system without diffusion DA' = a)A (1 f 4t/t') (50)
+
This expression should be used in conjunction with eq. 44-49 in the analysis of experimental data. In the study of systems which are not complicated by where co, cl, and c2 are the concentrations in moles per a chemical reaction and for which the quantity D A cc. of components Ao, AI,and A2, respectively. may be assumed t o be independent of time, At is By now expressing the concentration of solute 1 in simply obtained from the slope of D'I us. 1, t'. Howpi = Xlcl) and the mass per cc. of component A1 ( k , ever, for the case of diffusion of an isomerizing solute, concentration of solute 2 as nlMl/n2Mztimes the mass it may38not be possible to separate the time dependence per cc. of component A2 (Le., pz = nlM1c2/ne), the of a*'resulting from the chemical reaction from the following differential equations may be ~ b t a i i i e d ~ ~ b ~ ~ time dependence of D A ' resulting from the startingfrom eq. 54 time correction. In this case the value of At must be measured in some other manner39 or estimated from values of At for experiments on systems involving solutes with similar rates of diffusion but without a chemical reaction. When eq. 44 with eq. 46 and 49 is inserted into eq. 50, [kl (E)"'- lc2 (p = l,2) (55) 110 dl1 nlMl an expression relating DA' and At to the parameters describing the diffusion of an isomerizing system is obtained. Of particular interest are the values of 3 ~ ' where ilfo, AIl,and M z are the molecular weights of the solvent and the two solute components. and da)*
