191
WEAKELECTROLYTE MOVING BOUNDARY SYSTEMS
Jan. 5 , 1954
[CONTRIBUTION FROM THE CHEMISTRY
DEPARTMENT, UNIVERSITY
OF WISCONSIN]
Weak Electrolyte Moving Boundary Systems Analogous to the Electrophoresis of a Single Protein' BY EDWARD B. DISMUKES~ AND ROBERT A. ALBERTY RECEIVED JULY 20, 1953 Moving boundary systems obtained in the electrophoresis of proteins are compared with various types of weak electrolyte moving boundary systems. The latter are suitable analogs for interpreting certain phenomena encountered with proteins. A Kohlrausch regulating function applicable to monovalent weak electrolytes is derived, and the number of boundaries permitted in a weak electrolyte moving boundary system is discussed. A theory which deals with boundary displacements, compositions of phases, boundary sharpening factors and the number of boundaries is derived for a weak electrolyte system analogous t o that of a protein above its isoelectric point in a buffer of the uncharged-acid type. The results of moving boundary experiments with sodium aspartate in sodium acetate buffers are compared with theoretical predictions.
Introduction The moving boundary theory for strong electrol y t e predicts ~~ some of the effects encountered in the electrophoresis of proteins. These include the existence of E - and &boundaries, the difference in velocity of ascending and descending protein boundaries, and the variation in apparent analysis with total protein c ~ n c e n t r a t i o n . ~ -However, ~ proteins are not strong electrolytes, and in solution they are in acid-base equilibrium with the buffer. Simple weak acids and bases are useful as models for proteins since their constituent mobilities are p H dependent.' Furthermore, when weak electrolyte constituents are caused to migrate by the passage of current, acid-base reactions occur which are similar to those in protein systems. An analog of a protein above its isoelectric point is the weak acid HS and its anion S-, since the mobility of the S constituent increases as the p H is raised. Similarly, a base C and its cationic form CH+ is an analog of a protein below its isoelectric point. Electrophoresis experiments may be carried out in buffers of the uncharged-acid and the unchargedbase types. These buffer types are represented symbolically by HR, AR and B, BHX, respectively, where A is the cation of a strong base and X is the anion of a strong acid. The two protein analogs and these two types of buffers lead to eight distinct moving boundary systems since there are ascending and descending cases for each combination. TYPEI, descending
TYPE111, ascending
HR,AR,C,CHR(a)::HR,AR,C, CHR(P)+HR,AR(y) TYPEIV, descending
B, B H X
(y):
:B,B H X (6) + B, BHX, C, C H X
(a)
TYPEIV, ascending
B,BHX,C,CHX(a)::B,BHX,C,CHX(/3)+B,BHX(y) The notation used in these moving boundary systems is that introduced by L o n g ~ w o r t h . ~ , ~ The Moving Boundary Equation and the Kohlrausch Regulating Function for Monovalent Weak Electrolytes.+vensson9 and Alberty'o have independently derived the moving boundary equation for weak electrolyte systems, which may be written
- ~ J B C J B / K B= v
ajaZja/~a
~ (BC J ~- CJS)
(I)
This equation applies to any constituent J in the phases a and p which are separated by a boundary produced by the passage of current. The quantity vu@ is the volume in milliliters swept out by the boundary during the passage of one coulomb of electricity. The specific conductances of the two phases are K~ and KB. The quantities ~ Z Jand EJ are the constituent mobility and constituent concentration (moles l i t e r 1 ) , respectively, for the J constituent, and they are given by expressions
E
ti
i-1
and
H R ,AR, HS,AS (a)+ HR,A R ( p ) : :HR,A R (y) TYPEI, ascending
HR,A R (y)+ HR,AR, HS,AS (8): :HR,AR, HS,AS (a) The subscript i in equations 2 and 3 denotes one of the n species in solution which comprise the J TYPE11, descending constituent; a given constituent is comprised by a B, BHX, HS, BHS (a)+ B, B H X (a): :B,B H X (y) species which cannot undergo dissociation and all TYPE11, ascending forms which axe in equilibrium with that species. B, B H X (y) + B,B H X ,HS, BHS (a): :B,BHX,HS,BHS The Kohlrausch" regulating function w for strong (a) electrolytes has been redefined by Dolea in terms TYPE111, descending of relative ion mobilities as HR,A R (7)::HR,A R (8) 4 HR, AR, C,CHR (a) n (4)
(1) Presented to t h e Graduate School of t h e University of Wisconsin in partial fulfillment of t h e requirements for t h e degree of Doctor of Philosophy. (2) Shell Fellow 1952-1953. (3) V. P. Dole, THIS JOURNAL, 67, 1119 (1945). (4) L. G.Longsworth, J . Phys. Colloid Chcm., 61, 171 (1947). (5) R.A. Alberty, J . Chcm. Educ., 26, 619 (1948). (6) J. R. Cann, THIS JOURNAL, 71,907 (1949). (7) J. C. Nichol, ibid., 72, 2367 (1950).
Provided that relative ion mobilities are constant, the function w has the same value in successive (8) L. G. Longsworth, ibid., 67, I109 (1945). Svensson, I. Acta Chcm. Scand., 2, 841 (1948). (9) € (10) R. A. Alberty, TEISJ O U R N ~ L , 79, 2361 (1950). (11) F. Kohlrausch, Ann. Physik, 69, 209 (1897).
VOl. 76
EDWARD B. DISMUKES AND ROBERT A. ALBERTY
192
phases that are separated by moving boundaries. It is possible to derive a similar function which is applicable to systems containing monovalent weak acids and bases. If the mobility of the uncharged form is zero, then it may be seen from equations 2 and 3 that GjEj = ujcj, where j designates the singly charged form of the constituent. It is now possible to rewrite the moving boundary equation 1 in the following way ujaCjp/Ka - l L j 6 C j @ / K P = .daP ( F J a - cJ@) (5) 'l'he relative mobility r, of any ion j is defined as ? ~ j l ' uwhere , urnis the mobility of some particular ion 111, and similarly the relative conductance c is defined as K , ' u ~ .With the use of these definitions and the assumption that relative ion mobilities are constant, equation 5 may be rewritten as cj" /ua
- cjP/aB
= W B (EJJa/rj - EJJB/rj)
(6)
The sumination of equations of the form 6 for all constituents is
(7)
The electroneutrality condition states that Ccja
Cc,B = 0.
=
j
Therefore, if
# 0
j
x(EJa/rj) = C(cJ@/rj) J J
The function C(EJ/rj) is designated as
(8)
w and will be
T
referred to as "the Kohlrausch regulating function for monovalent weak electrolytes. For successive phases which are separated by moving boundaries, the function G has the same value in each phase. For the case of a stationary boundary (that is, tiaB = 0), equation 6 shows that all ions are diluted by the same factor across the boundary. In the expression for the Kohlrausch regulating function, concentrations and relative mobilities always appear as ratios, and it is not necessary to follow the sign convention described in connection with equation I .lo In the following calculations, the convention that all concentrations, mobilities and boundary displacements are positive quantities will be used. The only modification required in using this convention is that the term ( E J ~- CJ@) in equations 1 and 5 must be changed to (Ef - E J ~ if) the boundary moves in the direction opposite to that of the constituent J. The Number of Boundaries in a Weak Electrolyte Moving Boundary System.-For strong electrolytes Dole3 has proved that a maximum of ?a - 1 boundaries can form in a system containing n different ionic species. One of these boundaries must be stationary if relative ion mobilities are constant, leaving a maximum of n - 2 moving boundaries. By reasoning similar to that used by Dole, the maximum number of boundaries permitted in a weak electrolyte system may be derived.I2 The total number of constituents contained in a weak electrolyte system is designated as N . Of these the concentrations of N - 2 constituents in each phase are independently variable. It is con(12) 0 J. Plescia and R. S Alberty, presented at t h e Chicago Meeti n g of t h e American Chemical Society, September, 1960.
venient to regard the constituent concentrations of hydrogen and hydroxyl, which are present in any aqueous solution, as the only concentrations which are not independently variable. Another way of expressing this number of degrees of freedom is in terms of the number S of species, including water molecules, which are present. There are less than S degrees of freedom in determining the composition of a phase since the definition of a concentration scale imposes one restriction on the number of independently variable concentration values, the electroneutrality condition imposes another, and li additional restrictions are imposed if there are 2; independent expressions for chemical equilibria among the S species. The number of degrees of freedom N - 2 may also be expressed as S - E 2. I n the case of the Q! solution of the Type I system, made up by adding HR, HS and AOH to water. there are five constituents (R.S. A. H and OH) and eight species (R-, HR, S-,'HS, k+,H+, OH- and H.0). If B is tie'number of boundaries which form, there are B 1 phases, of which B - 1 appear after the current has been turned on. The number of unknowns which must be evaluated in order to describe the system completely is ( B - l) (,V - 2 ) B , where ( B - 1) ( N - 2 ) is the number of unknown concentrations required to give the compositions of the new phases and B is the number of boundary displacements. For each boundary, IV - 2 independent moving boundary equations can be written, and there are therefore a total of B(-IT3 ) relationships that are available. I n order that the values of the unknowns may be unique and determinate, the number of unknowns must be equal to the number of available relationships. Therefore
+
+
(B
-
l)(N
- +
2) B = B(N B=lV-2
- 2)
I t should be stressed that LV- 2 is the niaxinium number of boundaries permitted, but in certain experiments all of them may not exist or they may not be detected. For any particular system, a complete description can be obtained by solving (-V- 2)2relationships for the unknown quantities in terms of the compositions of the two original phases. I n view of the fact that in each of the weak electrolyte moving boundary systems formulated above there are five constituents present, three boundaries are permitted. The absence of one of the boundaries implies that each initial system has some special property that prevents the formation of the second moving boundary. I n the analogous protein systems which are encountered experimentally, i t is probable that there are actually three boundaries in many cases, but the concentration gradients across one of the moving boundaries are too small to be detected by the usual experimental methods. Weak Electrolyte Systems Analogous to Those Containing a Protein Above Its Isoelectric Point in a n Uncharged-acid Type Buffer A theory will be derived for the moving boundary systems of Type I which form when current is passed through initial boundaries between the s o h tions HR, AR, HS, AS (a)and HR, AR (7).The
WEAKELECTROLYTE MOVING BOUNDARYSYSTEMS
Jan. 6, 1954
compositions of the initial phases may not be arbitrarily chosen if only two boundaries are formed by the passage of current. The special properties that lead to the formation of one less than the maximum number of boundaries will be discussed. I n order to simplify the problem of describing these systems, the following assumptions are made: (1) the relative mobilities of the ions are constant throughout the system, (2) the mobilities of uncharged weak acid molecules are zero, and (3) the relative values of ionization constants for the weak acids expressed in terms of concentrations are constant throughout the system. An important corollary of assumption (1) is that ionic conductances are additive.8 The additivity of ionic conductances is expressed by K = (F/lOOO) Cuici, where F is the
193
The four unknown concentrations in the 0-phase of the ascending system may be calculated from the dilution ratio across the stationary a@-boundary (equation 14), the equality of Kohlrausch regulating functions (equation 15), conservation of weak acid (equation IS), and the mass action relationship (equation 17) CR@
+ rA
I
C S ~
CR@
+
Ts
(14)
CR8/CS8 = CRa'/cSa C H R ~+ C S ~
=5 7
CHS@
rR
+
rA
+
CHR~
(16)
CHS@ = CHRY
KHRCHRBICRB = K~~SCHSB/CSB
(17)
The solution of equations 14, 15, 16 and 17 leads to
i
faraday. The conductance due to hydrogen and hydroxyl ions will be ignored. A corollary of assumption 2 and the stoichiometry of reactions of the type RHS = HR S - which occur in the moving boundaries is that the total concentration of weak acid a t any level in the cell is constant. I n the descending system, the migration of Sions tends to leave a region adjacent to the initial boundary position in which there are no S - ions. However, in the adjustment of equilibrium, Rions react with HS molecules to form additional Sions in this region. If, as is the case for systems discussed in this paper, the continuous migration of S - ions and the adjustment of equilibrium results in the complete disappearance of the S constituent from this region, designated as the @-phase, the moving boundary equation for the S constituent simply reduces to
+
+
(19)
CRY
CRB
=
- rS) - CHs8 rA(rR rs(rA + rR) CSa I R ( I A
+
cs8 =
rS(rA
+ +
rA(yR
-
rS(?'A
Sa
f'R(rA
w
rs(rA
+
78)
- rS)
+
+ +
(20)
rR) rR)
rS)
(21)
TR)
Sharpening Factors at the Moving Boundaries.-A moving boundary in a weak electrolyte system will have sharp gradients in concentration if the constituent disappearing across the boundary has a greater velocity in the phase behind the boundary aZsa = va&a (9) In the ascending system, the migration of S - ions than in the phase ahead of the boundary. Differleads to the formation of the @-phase,which con- ences in field strength and pH across a boundary tains HS molecules as a result of the reaction of S - produce the conductivity effect13and the p H e f f e ~ t , ~ ions with H R molecules. The moving boundary which influence the sharpness of the boundary. illthough these effects may be discussed separately, equation for S a t the @ y-boundary is it is the net effect which determines the change in a s p = VBYKB (10) velocity of a constituent across the boundary. If the velocity of the constituent is greater behind In each system there will be a stationary boundary, the @y-boundaryin the descending system and the boundary than ahead of it, the boundary will the a@-boundaryin the ascending system, since (1) eventually reach a steady state in which the tendthere will be gradients in concentration near the ency to diffuse is balanced by the tendency of the initial boundary positions if the choice of concen- material in the region behind the boundary to overtrations for the two initial phases is arbitrary and take that in the region ahead of the boundary. For (2) the positions of these gradients will not change the descending a@-boundary,the condition may be during the passage of current if relative ion mobili- expressed as ties are constant. aZsBEB > asaEa (22) Compositions of Intermediate Phases.-The con- where Gs@ is the mobility that the S constituent centrations of H R and AR in the @-phaseof the would have a t the p H of the @-phase descending system may be calculated from the equality of Kohlrausch regulating functions of the a- and @-phases(equation l l ) , the electroneutrality condition, and the equation for conservation of Inequality 22 may be combined with equations 12 weak acid a t any level of the cell (equation 12). and 13 to obtain the following condition for the movement of the a@-boundaryin a steady state G.8 + C R ~ C H R ~= 41" + C R ~ C H R ~+ CSQ C H B ~
+
rA
rR
rA
+
+
rR
rs
+
C H R ~= C H R ~
(11) CHSQ
(12)
Thus, the concentration of AR in the &phase is
For any two acids, the only variable which determines whether the boundary moves in a steady (13) L. G.Longsworth and D.A. MacInnes, THISJOURNAL, 81,705 (1940).
EDWARD B. DISMUKES AND ROBERT A. ALBERTY
194
state is the pH of the a-phase. The boundary should move in a steady state under the following conditions (1) KHS> KHRand r8 2 r R (2) KES> KHRand r~ < r~
( 3 ) K H S< KHRand
>
+ )= ( rK
IA
zzz)
Descending
> rR
IS
+
- 2 )C H ~ C H R Y
1
7s
cSQcRY
I t is of interest to consider the factors that determine whether the conductivity and pH effects are favorable for a sharp moving boundary. A boundary r q moving toward the r-phase will be sharpened by the conductivity effect if E?/Er > 1 where E is the field strength. If ion mobilities are equal on both sides of the boundary, which is the case if the ionic strengths are approximately equal, then the ratio of field strengths may be written in terms of relative conductances. Ev/Ef =
(26)
U ~ / U T )
Since the mobility of the anion constituent of a weak acid increases with the pH, the constituent mobility will be greater in the q-phase than in the (-phase if C H ~ / C H> ~ 1, and the {q-boundary will be sharpened by the pH effect if the constituent disappears across the boundary. For the descending ap-boundary, equations 12 and 13, which give the composition of the P-phase, may be used t o obtain the ratios of field strengths and hydrogen ion concentrations across the boundary. E@ _ -EQ
__
lr.4
+
+ + + r s ) c s ~+ -r s
r K h a
(r.4
(r.4
+ rs) +
(IR
- ~S)CHSQ
(27)
~HLQ
YR)
- rs)1
IA(~R
CSQ ? R ( ~ A
+
TK)
(28)
For the By-boundary in the ascending limb, the corresponding relations are EB Er
=
1+-- KHSC E ~ KHRCSQ I KHSC K ~ ~ A ( T K - r s ) CHRY KHRcsa ~S(YA Y R ) CRY
+
_ CH y CHB
z:
1+---
::{
1-
+
- or 0
+
+-
-
+ or 0
TR(IA
rY(rA
cRQ ( r A
sl; C R Q (?A CSQ ( T A
r4(rR
rs(r.4
rR)
;
rR)
I
+ rd + + rs)
- Y S ) CHRr
+
rR)
+ rs) 9 +
re (29)
(30)
TR) CRa
In Table I the directions of the conductivity and pH effects under various conditions are listed. As a general rule, the mobility of P protein is less than
Ascending Conductivity pH
+ -
+ or 0 - or 0 -
+
-
+ + -
that of the buffer ion of the same sign. Avoiding the extreme case in which KHS rs. From the table, it is seen that the directions of the conductivity and pH effects are opposed in each moving boundary system. Furthermore, if one of the effects is favorable in either the descending or ascending system, it is unfavorable in the other systern. These conclusions were reached by Longsworth4from a consideration of the theory for strong electrolytes by assuming the concentrations of undissociated species to be constant throughout the system. Longsworth showed that in the case of ovalbumin in an acetate buffer there is a higher degree of enantiography between the ascending and descending patterns when the protein is above its isoelectric point than when the protein is below its isoelectric point. Conditions under which Only One Moving Boundary IS Obtained.-Consider any two solutions ( and q which contain the same constituents and which are separated by the lo-boundary. The flow 5 of any constituent J in one of the solutions as current is passed through the boundary is ~ J E J / 1 0 0 0 ~moles coulomb-l, and so the difference in flow in the {- and q-phases is expressed by writing equation 1 as ziJf~J~/10o~ Kiij'IC?~?/1000K? ~
(al)
If each constituent J has only one ionic form j and if relative ion mobilities are constant, then equation (31) may be rewritten as A5 = (c$/d
+
pH
ductivity
K H S> K ~ Rrs. < rit KHSQ KRH,rs 2 r~ K H S>>KHR,rs > YK KEa
