J . Phys. Chem. 1985,89, 1659-1664
1659
Calculation of Liquid Junction Potentials Harry Wms. Harper The Rockefeller University, New York. New York 10021 (Received: October 9, 1984)
Equations are given and procedures described for calculating liquid junction potentials over wide ranges of concentration and composition. Use is made of the Henderson continuous mixture assumption, avoiding treatment of the codiffusion problem. Ionic mobilities and activities are incorporated realistically, making use of standard data on binary electrolyte solutions. For ionic strengths up to about 0.1, an equation is derived in terms of ionic properties specified for the two end solutions; typical accuracy is 0.1 mV for simple junctions between two concentrations of the same electrolyte, and 1-2 mV for more complex junctions. Higher concentrations require specification of ionic properties at points intermediate between the end solutions; approximations are developed which prove accurate to about 0.5 mV in simple cases at ionic strengths as high as 5 , while in complex cases an accuracy of 1 to a few mV is found at ionic strengths as high as 4.5. Ionic strength ratios across the junctions examined range from 1:l to 94:1,typically being about 20:l. A rule is given for determining mobilities and activities of ions in mixtures, using standard data for binary solutions. An expression is obtained for the activity coefficients of single ion constituents in terms of mean activity coefficients, and an estimate of the associated error is derived. This is the error in dividing total cell potentials into a component due to electrode potentials and a component arising at the liquid junction; it appears to be of comparable magnitude to the errors, stated above, due to other approximations in the calculationsdescribed.
introduction The wtential which arises at the junction between two electrolyte solutions, A and B, containing ion constituents i = 1, 2, ..., n, is given (in mV) by
-EJ =
RT ?-I
B
Fz,
A
ti d In ai
where R is the gas constant, T the absolute temperature, F the Faraday, z the signed valence, a the activity, and the transference number t is the fractional conductance of the ith ion, given by
where u is the electrical mobility, M is molar concentration, and j ranges over all the ions, just as i in eq 1. The derivation usually given,' which treats the junction as a reversible system, has been criticized? as has the assumption of electrical neutrality often used in solving the equation. Relatively recent theoretica13q4and numericals analyses, which do not appeal to reversibility or charge neutrality, find that E, develops with a risetime on the order of s, after which the state of the system is consistent with the classical result; the characteristic time is that required for an ion to diffuse over a Debye length. These theoretical developments, however, do not yield convenient practical calculations. In fact, in spite of great interest in liquid junction potentials for over a century, the only computational procedure to see widespread use is that due to Henderson,' who formulated a modification of eq 1 which is readily integrated. Henderson simplified the problem in two major respects. First, the electrolyte solutions are treated as if ideal, having constant ionic mobilities (those found for infinite dilution) and unit activity coefficients, y , (so a, = yiMi becomes simply Mi). Second, it is assumed that the junction may be represented as a continuous series of mixtures of the two end solutions, A and B. The composition of the junction at any point may then be described in terms of the mixing fraction x , defined as the proportion of the total concentration contributed by end solution B. Thus, the concentration of the ith ion a t any point is given by (3) M i= Mf X ( M - Mf)
+
TABLE I electrolyte p 19 E , (obsd) eg 8 Henderson ref NaCl 0.0995 0.00499 -15.9 -15.9 -15.6 13 HC1 0.100 0.00345 53.9 53.8 55.5 14 CaC1, 0.290 0.00545 -39.6 -39.5 -35.0 15 ZnSOl
0.400
0.0186
-5.1
-5.2
-8.1
16
where superscripts designate end solutions. With these provisions eq 1 is easily interpreted but remains awkward to solve because of the logarithmic singularity encountered when an ion has zero concentration in one end solution. It is simple to change variables and integrate with respect to x ; the result is first order in x , and the analytical solution is straightforward. Henderson's equation is simple to apply but the counterfactual assumptions employed severely compromise its utility. Useful accuracy is obtained only in dilute solutions, where the assumption of ideal behavior is approximately satisfied, and in special cases, such as junctions between univalent salt solutions of moderate and nearly equal ionic strength, where the derivatives of the neglected nonideal components of the ion parameters are small. The approach followed below is to relax these assumptions as far as possible without introducing such mathematical complication as would constitute an obstacle to application. The points addressed are those relating to the nonideal behavior of the ions: the variations of u and y in solutions of differing concentration and composition. Henderson's use of the mixing fraction x is retained, avoiding quantitative treatment of the intractable codiffusion problem. The improvements achieved are of significant consequence;the resulting computational procedures are more accurate than Henderson's and have a much wider range of application. To proceed as indicated requires the determination of the activity coefficients for single ion constituents, whereas only the mean coefficients of neutral ion combinations are directly measurable; single ion values can be assigned only by adopting some rule for partitioning the mean coefficients. Theoretical considerations which evaluate the contribution of long-range electrostatic interactions to the chemical potentials of ions (Debye-Huckel theory and variants6) indicate that to a first approximation this partitioning is governed by the condition that the logarithm of the activity coefficient of each ion is proportional to the square of its charge.' For a binary electrolyte which dissociates- into (v+) cations of valence (z+) and (v-)anions of valence (z-), there being (v*) ions altogether, an expression for the cation activity coef-
(1) D. A. MacInnes, "Principlesof Electrochemistry", Dover, New York, 1961. .. .-.
(2) F. 0. Koenig; J . Phys. Chem., 44, 101 (1940). (3) H. J. Hickrnan, Chem. Eng. Sci., 25, 381 (1970). (4) J. L. Jackson, J . Phys. Chem., 78, 2060 (1974). (5) D. R. Hafemann, J . Phys. Chem., 69,4226(1965).
(6) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions", 2nd ed, Butterworths, London, 1959. (7) R. G. Bates, "Determination of pH: Theory and Practice", Wiley, New York, 1954, Chapter 3, eq 23.
0022-3654/85/2089-1659$01.50/00 1985 American Chemical Society
1660 The Journal of Physical Chemistry, Vol. 89, No. 9, 1985
ficient (y+), in terms of the mean activity coefficient obtained as follows: If
In (y+)
0:
In (y-)
( z + ) ~ and
b&),is
a (z-)~
by eq 4 is the only readily accessible and generally applicable estimation whose physical basis can be clearly justified.
An Equation for Moderate Concentrations We require to know u and y as continuous functions of x for each ion. Fortunately there is a simple empirical relationship, valid for dilute to moderate concentrations, which well serves this need: for ions in aqueous solutions, the equivalent conductance, A, and the logarithm of the activity coefficient both decrease in linear proportion to the cube root of the ionic strength
then
Since (yf)(U*)
Harper
= (y”‘”+’(J+)‘“’
by definition, substitution and rearrangement yield
O+) = (yf)(z+~~~[(u+)+(‘)l/[(v+)(Z+)’+(~)(Z-)21)
Observation agrees with theory that the dependence goes like 11/2 as the limit of infinite dilution is approached; however, in the range 0.001 < I < 0.1 (roughly) the Z1/3 relation is found to hold.’O (The physical basis is a matter of dispute.”) Knowing Xf and A!, and also IAand IB (by eq 5 ) , use of the mixing fraction gives
By electroneutrality
I(v+)(z+)l
=
l(p)(z-)l
or
l(z+)/(z4l = I(z-)/(v+)l which may be used to eliminate The final result is (y+) =
Y’S
from the exponent of (yi).
(yf)((z+)2/l(r+)(z-)I)
(4)
A like expression obtains for the anion activity coefficient. It has been suggested that single ion activity coefficients might be calculated by taking into account variations in ionic radii, in the context of Debye-Huckel theory.8 However, the determination of ionic radii in aqueous solutions, which is inseparable from the problem of ionic hydration, is a controversial undertaking. Different methods give very different results, and their relevance to ionic activities is not clearly ~nderstood.~ Morever, the potential distribution assumed by Debye and Huckel is of questionable applicability for asymmetrical electrolytes, and in any case at higher concentrations, while improved treatments are both technically elaborate and problematical.6J0 The relation indicated (8) J. Kielland, J . Am. Chem. Soc., 59, 1675 (1937). Related ideas are discussed by H. S. Frank, J . Phys. Chem., 67,1554 (1963); R. G.Bates, B. R. Staples, and R. A. Robinson, Aml. Chem., 42,867 (1970); R. Tamamushi, BUN. Chem. SOC.Jpn. 47, 1921 (1974). (9) J. E. Desnoyers and C. Jolicoer in J. O M . Bockris and B. E. Conway, Eds., “Modern Aspects of Electrochemistry, No. 5”, Plenum Press: New York, 1969, pp 1-89; J. F. Hinton and E. S. Amis, Chem. Rev., 71,627 (1971); B. E. Conway, ref 1 1 below, Chapter 5.
A similar expression obtains for In yi. Since the mobility of an ion is directly proportional to its equivalent conductance, and mobilities are involved in eq 1 only as ratios, one may simply substitute A’s for u’s. Making use of the expressions just given for X i and yi, and those previously given for M iand ti (eq 3 and 2), enables one to write eq 1 in the form of eq 7 . After the variables are changed, eq 7 becomes eq 8. (The reason for the subscript S will be given later.) Like the Henderson equation, eq 8 is couched entirely in terms of properties specified for the two end solutions. The equation proves to be fourth order in x and could be integrated analytically. A simple expression would not be anticipated, however, and we shall see below that in order to explore the possibilities of the present approach for higher concentrations, eq 8 must assume different forms. Given the ready availability of computing facilities, numerical integration was (10) H. S. Frank and P. T. Thompson in W. J. Hamer, Ed., “The Structure of Electrolytic Solutions”, Wiley, New York, 1959, pp 113-134; H.C. Anderson in J. O’M. Bockris and B. E. Conway, Eds., “Modern Aspects of Electrochemistry, No. l l ” , Plenum Press: New York, 1975, pp 1-31; ref 1 1 below. (1 1) R. M. F’ytkowicz, Ed., “Activity Coefficients in Electrolyte Solutions”, CRC Press, Boca Raton, FL, 1979, Chapters 2, 5, and 8.
Calculation of Liquid Junction Potentials chosen instead. Simpson’s rule is adequate to the case, but it may be noted that the integrals become highly skewed when the concentrations of the end solutions are very different. Accordingly, a Fortran program was written which first inspects the integrals and then segments them, allocating an appropriate number of intervals in each segment to achieve a specified accuracy (0.1 mV) for the overall evaluation when Simpson’s rule is applied to each segment.12
Simple Junctions, to Moderate Concentrations The simplest liquid junction is that between two solutions of the same binary electrolyte, differing only in concentration, a case of great interest in the present context. Accurate mobility and activity data are available for many binary electrolytes. Also, it is obvious that such a junction may in fact be described at any moment as a series of mixtures of the two end solutions. This means that application of the mixing parameter x to interpret eq 1 is justified without reservation. Since our treatment is in other respects a close approximation, we expect nearly exact results. The observations with which we are to compare calculations are potentials recorded from electrochemical cells with liquid junctions, measured between reversible electrodes placed in the end solutions. Equation 8 is so written that the sign of EJ will be positive when end solution B is positive with respect to solution A. Observing the same convention for the electrode potentials, the cell potential is given in obvious notation by
The results in four cases are presented in Table I, which also includes the values calculated by using Henderson’s equation for comparison. Thoughout this paper, liquid junction potentials are in mV at 25 O C . Sources for mobility and activity data are grouped under this paper’s final reference.33 Sources for mobility and activity data are grouped under the final reference.33 Observed potentials are taken from the published sources indicated.
Equations for Higher Concentrations When concentrations approach Z L 1, the generally applicable linear Z1I3 dependence of x’s and In y’s is lost, as ions come under the influence of short-range forces and exhibit marked individual differences in behavior. This poses difficulties for extending the success of eq I to higher concentrations; in fact, we need to generate a family of similar equations, having a new member for each junction to be evaluated. An approximate response was formulated as follows. Since the functions relating x’s and In y’s to Z usually exhibit only one major inflection, provision is made in eq 7 to hinge these functions, that is, to break each function in two parts a t a selectable point. For example, each function which originally has the form of eq 6 is hinged at an ionic strength and value of Xi intermediate between ZA,Xf and P,Xy. If the common intermediate value of Xi is designated by superscript h (for hinge), and the associated ionic strength by superscript X (to show it relates to equivalent conductance), we can write
The parameters specifically relevant to Xi which enter the cal(12) Fortran IV source listings of programs developed for this application are available for nominal charge from Duck Engineering Dtriign, 500 E. 63rd St., 19-B, New York, N Y 10021. There is no copyright. (13) A. S. Brown and D. A. MacInnes, J . Am. Chem. Soc., 57, 1356 (1935). (14) T. Shedlovsky and D. A. MacInnes, J . Am. Chem. SOC.,58, 1970 (1936). (15) T. Shedlovsky and D. A. MacInnes, J . Am. Chem. Soc., 59, 503 (1937). (16) R. E. Lang and C. V. King, J. Am. Chem. SOC.,76, 4716 (1954).
The Journal of Physical Chemistry, Vol. 89, No. 9, 1985 1661
TABLE I1 electrolyte HCI
HCI MgClz
ZnSOI
IA 0.0585 0.0585 0.277 0.333
E , (obsd)
eg 8h
ref
-75.6 -99.6” 39.5 4.9
-76.5 -99.0 38.9 5.0
17 17, 18 19 16
2.820 5.00 5.127 4.116
”This potential is taken from splined data from two sources; the resolution is k0.5 mV.
culation are therefore Xf, A?, A! and Z.! These values are taken from graphs of X i against Z1l3, which must be available for each ion over the range of ionic strengths for which EJ’s are to be calculated. For a particular junction and ion, the portion of the graph between ZA I’ and PI’ is examined, and a hinge point is selected in the region of the major inflection, such that the two straight lines (chosen by eye) whose coordinates are ZA1”,Xf Z;1’3,A;, and Z?”’,X! - ZB” ,A!, best approximate the graphed function. These coordinates supply the values needed for the calculation. When the numerical integration requires that an ordinate of eq 8 be evaluated for a given value of x , the program must determine whether the corresponding value of I lies in the interval ZA - Z; or Z; - P,and select part 1 or part 2 accordingly. A similar change is made in the expression used to calculate y’s. However, since at high concentrations In y tends to vary directly with Z rather than Z1/3, with the major inflection in the relation between Z and In y occurring between these two regions, the affected portion of eq 8 (the term between brackets subscripted by S) becomes
In yp)(P - ZA) ...[ (In y: -(IB - Iy)
part 2 when ZA
< ZB or
. .[
part 1
part 2
I..
(In y? - In y:)(ZB - IA I]...
( E - ZA) (In y,” - In yp)(ZB- ZA)
..e[
( P i 3-
zy)3[ZA +
X ( P
I..
- ZA)I2/)
when ZA > ZB. In this case, graphs plotting In yf against both Z1/3 and Z are needed. After a pair of intersecting straight lines are selected, as before, except that the lower ionic strength segment is from the Z1l3plot, and the higher ionic strength segment from the Z plot, appropriate single ion activity coefficients may be obtained from the coordinates with the aid of eq 4. Once again the integrating program must select the appropriate expression and also must accommodate the fact that eq 8 in this form is discontinuous at Z = 8. Since the program already segments each integral in order to increase computational efficiency for skewed cases, it is straightforward to require that one segment boundary fall at the value of x such that the corresponding Z = 8, which fully answers to the problem.12 The equation, or really family of equations, represented by these modifications of eq 8 will be referred to as eq 8h.
Simple Junctions, to Higher Concentrations Several comparisons of calculated and observed potentials are given in Table 11. Henderson’s equation is inappropriate at these concentrations; discrepancies in excess of 25 mV appear. Activity coefficients range from 0.045 to 3.1 in these examples, and equivalent conductances from 5.8 to 334. In no case were the (17) H. S. Harned and E. C. Dreby, J . Am. Chem. Soc., 61,3113 (1939). (18) S. Lengyel, J. Giber, and J. TamBs, Acto Chim. Hung., 32, 429 (1962). (19) S. Phang and R. H. Stokes, J . Solution Chem., 9, 497 (1980).
1662 The Journal of Physical Chemistry, Vol. 89. No. 9, 1985 TABLE 111 ion binary constituent component
M
I
A+
A-
1.5 1.5 22.0 1.5 15.8 0.5 0.75 2.25 15.6 0.5 2.0 6.4 1.5 1.5 0.5 1.5 0.75 2.25 0.5 2.0
y+
y-
0.819 0.442 0.419 0.057 40.4 38.6 26.5 20.5
0.819 0.805
0.038 0.057
approximations made by graph and eye adjusted in any way to obtain these results; such differences as careful individuals might introduce in carrying out those procedures should not greatly exceed the differences between calculated and observed values seen here.
Complex Junctions: General Considerations Junctions between two different electrolytes, or mixtures of electrolytes, cannot generally be described as a series of mixtures of the end solutions, and use of the mixing fraction x is an approximation with associated error. Granting this, the problem of realistically specifying the mobilities and activities of the ions is approachable. Since data are generally available only for neutral binary solutions, it is appropriate to resolve every solution encountered into such components. What is then required is a rule which relates measured mobilities and activities to the ions of these binary components. Two general principles are found useful to this end. The first is the ionic strength principle. We have seen that at low to moderate concentrations, X and y vary in an orderly way with respect to I. This principle is also applicable at high concentrations, to apportion specific ionic interactions. Second, Bronsted's principle of specific interaction will be applied.20 Bronsted reasoned that since specific interactions depend on short-range forces, which operate between ions at close approach, and since in general ions of like charge do not approach closely, it is sufficient to consider only interactions between ions of unlike charge. This means that we are indeed justified in considering only neutral binary components, independently. Also, for this purpose the concentration of a given ion constituent, in each binary component in which it participates, is taken to be that of its counterion. (In this case, of course, we are speaking of equivalent concentrations.) The rule which emerges from application of these principles may be stated as follows: the value ofh or In y of an ion constituent in solution is a linear combination of those which obtain for all binary components in which it participates, at the ionic strength of the total solution, weighted according to the ionic strength of each component. An example serves to illustrate how this rule is used. Suppose we consider a mixture which is 1.5 M in NaCl and 0.5 M in MgS04. Table I11 shows how the solution is described in terms of binary components; the values of X and y are from measurements on these components at the total ionic strength of 3.5. The needed values of X and In y are then calculated as follows: 1.5(22.0) + lS(15.8) for N a X= = 18.9 1.5 + 1.5 1.5(ln 0.819) + 1.5(ln 0.442) lny = = -0.508 1.5 + 1.5 2.25(15.6) + 2(6.4) A = = 11.3 for Mg 2.25 + 2 2.25(1n 0.419) + 2(ln 0.057) Iny= = -1.81 2.25 + 2 and so forth. The following considerations participated in the formulation of the rule just described. In the case of activity coefficients, for (20) J. N. Bronsted, J . Am. Chem. SOC.,44, 877 (1922).
Harper present purposes specific interactions may be ignored below Z N 0.1; that is to say, at constant ionic strength y+(Na) is about the same whether in NaCl, NaN03, NaCNS, Na2S04,.... At higher concentrations, large specific effects may be evident. Numerous studies have been made of the variation of the mean activity of a test electrolyte, whose concentration is varied, in an electrolyte mixture of constant total ionic strength. Under these conditions any change in y f of the test electrolyte~isdue to specific interactions. It is found that in most cases In y f is directly proportional to the concentration (hence ionic strength) of the test electrolyte.21 In other words, the logarithm of the mean activity coefficient of the test electrolyte is proportional to its ionic strength fraction. Our usage generalizes this result to the activity coefficients of single ion constituents in mixtures. With regard to mobilities the situation is different in two respects. First, ionic mobilities are specific properties which couple through the long-range electrostatic force, in a complex way, to produce significant effects even at low concentrations.22 Second, there is no generalization in the literature, similar to that for activity coefficients, which may readily be adapted to our purpose. Accordingly, such a simplification was sought for on a purely empirical basis; it is helpful that interactions between oppositely charged ions are again most important. The hypotheses entertained were that the mobility of an ion constituent is a linear combination of those which obtain for all binary components in which it participates, at the ionic strength of the total solution, weighted by its molarity and the zeroth, first, or second power of its charge, that is, according to its molar fraction, conductance fraction, or ionic strength fraction. Data on the total conductance of various ion mixtures were exa m i ~ ~ ethese d ; ~ details ~ are omitted for brevity. It was found that the ionic strength fraction gave best results, accounting for total solution conductance to better than 1% except in cases where pronounced ion association would be expected.
Complex Junctions, to Moderate Concentrations Equation 8 may be applied to complex junctions by determining values of X and y for each ion constituent, in each end solution, according to the directions given above. This mode of use is appropriate when ionic strengths are roughly equal in the end solutions and will be signified by the suffix a: eq 8a. If the ionic strength of one end solution is several times that of the other, it is better to let this side dominate. Suppose ZA >> p . Then values of XA and y A are determined as before, but XB's and yB's are assigned as if the counterions were those of end solution A, at the total ionic strength of solution B. This is mode b: eq 8b. In some cases binary species may be encountered for which data are lacking and must be supplied by reasonable analogy; this is mode c: eq 8c. A liquid junction which has been extensively examined is that between 0.1 M solutions of KCl and HCl. The best experimental measurement is probably that of Roberts and F e n ~ i c k who ,~~ found EJ = -28.0 mV. By eq 8a, EJ = -28.2 mV. MacInnes and LongsworthZ5carried out a graphical integration of eq 1, using measured values of transference numbers and activities in the relevant mixtures, and assuming that the neutral electrolytes of each end solution diffuse independently. Their result is the same as ours. Potential measurements on junctions of more interesting composition are difficult to find in the literature, perhaps because obstacles to the theoretical treatment of even simple cases did not encourage such explorations. An exception is the study by Chloupek, Dane;, and Dane'Sova,26using the system [calomel, 0.1 (21) H. S.Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions", 2nd ed,Reinhold, New York, 1950, Chapter 14. (22) L. Onsager and R. M . Fuoss, J . Phys. Chem., 36, 2689 (1932). (23) A. K. Smith and R. A. Gortner, J. Phys. chem., 37,79 (1933); P. Van Rysselberghe and L. Nutting, J. Am. Chem. SOC.,56, 1435 (1934); P. Van Rysselberghe and L. Nutting, J. Am. Chem. SOC.,59, 333 (1937); P. Van Rysselberghe,S.W. Grinnell, and J. M. Carlson, J. Am. Chem. SOC., 59, 336 (1937).
(24) E. J. Roberts and F. Fcnwick, J . Am. Chem. SOC.,49,2787 (1927). (25) D. A. MacInnes and L. G. Longsworth, Cold Spring Harbor Symp. Quant. Biol., 4, 18 (1936).
The Journal of Physical Chemistry, Vol. 89, No. 9, 1985 1663
Calculation of Liquid Junction Potentials TAR1.E IV
-
bridne NaCl
K2S04 Na2S04
MgC12
I 0.01 0.1 0.03 0.3 0.3 0.03
0.3
ZnS04 a
0.02 0.2
E .,. , Iobsd)
ea 8
-57.7 -29.7 -51.1 -29.7 -28.7" -48.6" -16.4 -65.1 -37.4
-60.2 -28.2 -51.4 -28.9 -28.3 -49.9 -17.8 -66.4 -40.2
I
mode b a b, c a, c a, c b a b, c a, c
Henderson -60.4 -26.9 -48.9 -22.8 -22.3 -48.4 -16.1 -60.0 -27.4
where the ion is absent. Bearing these considerations in mind can greatly simplify the treatment of a junction. The examples in Table V involving SO-: ions present a special problem because of the presence of the partially dissociated species H2S04in the acid junctions; for instance, in the junction between N a 2 S 0 4and HC1, we must expect to find the constituents Na+, In brief (for a full account is lengthy), H+, C1-, HS04-, and SO-.: these junctions are interpreted as follows. Selected values of the mixing parameter x are used to calculate total H+ and S042concentrations, naively. At each point the equilibrium
From graphed data. The tabled values are obvious misprints.
TABLE V bridge NaCl
Na2S04
MgCl2 ZnS04
is then assumed, where I 1 .o 3.0 3.0 4.5 2.8
El
(obsd)
-5.2 2.9 -22.8 7.4 -21.0
eg 8h
K2
=
a(H+) a(S042-) - y(H+) Y ( S O ~ ~[H+] - ) [SOq2-]
2.4 -21.9
10.5 -22.3
M KCl (bridging solution) 0.1 M HCl, calomel] and a wide variety of bridging solutions, each of course giving two liquid junctions. Table IV compares the measured sum of the two junction potentials for several bridging solutions with the values obtained by using eq 8; the results from Henderson's equation are also given. The authors appear to have prepared their calomel electrodes under oxygen-free conditions, but the -27.1-mV potential they find for a 0.1 M KCl bridge solution indicates that dissolved air must have reached the acid electrode; accordingly, a 1.2-mV correction has been applied to their data.27 E: - E t is taken to be -0.85 mV, based on the mean activity coefficients for 0.1 M KCl and HCl, and eq 4. Complex Junctions, to Higher Concentrations At higher concentrations, eq 8h is used. For each junction, and in general for each ion, graphs are prepared which plot X against Ill3 and In y against Z1/3and I. The graphs range from I = IA to I = IBand are based on values of X and In y calculated at convenient intervals in I, using the rule previously given for binary components weighted by ionic strength. (After appropriate intervals in I are selected, each corresponding value of x is given - IA). The concentrations of the ions are then by x = ( I - IA)/(a given by eq 3; the sum of these for each x constitute a mixture to which the rule for determining A's and y's applies.) These graphs (called meshes) are then approximated by hinged pairs of straight lines, just as described for simple junctions at high concentrations. Sample results are shown in Table V, compared with data from the same source as Table IV. In practice it is not usually necessary to prepare meshes for every junction and ion. Junctions between binary electrolytes with one ion in common, such as those above where NaCl is the bridge solution, are less demanding. Here the cations are all present as chlorides, and graphs of their properties as chlorides over a wide range of ionic strengths, like those used for simple junctions at high concentrations, may be employed. Moreover, in this case where only univalent ions are present, chloride may be treated as NaCl throughout; no meshes are needed. In general, an ion's mobility will change significantly only when the counterion changes to one of different mobility and at least one of the ions is multivalent; that is to say, the mobilities of ion constituents interact more strongly when the ionic charges are greater. Activity coefficients are significantly altered by changing counterions only at higher ionic strengths; in the NaCl examples above, in every place the ionic strength is great enough for specific counterion effects to be important, the sodium ion predominates. Also, the values of A and y assigned to an ion are less critical when the concentration of the ion is relatively low, as in an end solution (26) J. B. Chloupek, VL. Z. Dane;, and B. A. DaneSova, Collect. Czech. Chem. Commun., 5, 469, 521 (1933). (27) M. Randall and L. E. Young, J . Am. Chem. Soc., 50, 989 (1928).
-
a(HS04-)
-5.5
y(HS04-)
iHSO4-1
or QyQc. We take K2 = 0.Ol2*and calculate Qy for each x from activity coefficient data on appropriate binary solutions at the corresponding ionic strength. Knowing total [H+] and [SOq2-], we can write Qc
=
([H+1 - [HSOCI)([S042-l - [HSOd-]) [HS04-1
A quadratic in [HS04-] is obtained, where the desired root is
which we easily evaluate knowing [H+], [S042-], and Qc = O.Ol/Qy. Thus we can graph the concentrations of H+, S042-, and HS04- as functions of x throughout the junction. The HSO; concentration, which is of course zero in both end solutions, reaches a maximum at some value of x. It is natural to divide the junction in two parts near the maximum, approximate the ion concentration functions with straight lines which intersect at the common boundary, and obtain the value of the total integral as the sum of these two parts integrated separately. (Note that this complication was simply ignored at the lower concentrations of Table IV. Production of HSO; affects E, principally through reduction of highly mobile H+; such reduction is much less in these cases.)
Discussion The methods described here are much in the spirit of earlier attempts by Harr~ed;~Taylor,30and MacInnes and Longsw0rth,2~ but are more systematic and much wider in range of application. Equation 8 is easy to use in the context of its integrating program.12 The procedures described above for calculations at higher concentrations are somewhat clumsy, especially for complex junctions. It is interesting nonetheless to see results ,of useful accuracy emerge, even in difficult cases, when it is considered that the codiffusion problem is not explicitly treated. The form of the approximations chosen was influenced by the desire to make use of the computer environment developed to support eq 8, in conjunction with standard sources of data, to explore the high concentration region. If a computer data base of mobilities and activity coefficients for all relevant binary electrolyte solutions were compiled, the principles demonstrated above could be implemented in a more accurate and much more convenient way. The question of junction geometry has not been dealt with here. There is little systematic experimental work to draw upon, and theoretically this is the codiffusion problem we chose to avoid. A common laboratory practice is to form junctions in a tube, at a plane perpendicular to the long axis. It is easy to picture the role of the mixing fraction x in such a case. However, it is (28) M. Kerker, J . Am. Chem. SOC.,79, 3664 (1957); T. F. Young, L. F. Maranville, and H. M. Smith in W. J. Hamer, Ed., "The Structure of Electrolytic Solutions", Wiley, New York, 1959, pp 35-63; H. Chen and D. E. Irish, J . Phys. Chem., 75, 2672 (1971). (29) H. S.Harned, J . Phys. Chem., 30, 433 (1926). (30) P. B. Taylor, J . Phys. Chem., 31, 1478 (1927).
1664
J . Phys. Chem. 1985, 89, 1664-1670 tioning, which is essentially the error arising from specific interaction effects on the activity coefficients of the single ions. It seems reasonable to accept as relevant the extent of the variation in activity coefficients observed when a given ion is paired with a variety of c o ~ n t e r i o n s . Such ~ ~ a study was carried out. Frequency histograms of the logarithm of the activity coefficient of the given ion (by eq 4), paired with various univalent and divalent counterions at constant ionic strength, did indeed show a clear central tendency. Characterizing these distributions by their probable error, and expressing the results in mV, we obtained the values shown in Table VI. (The number of counterions used in each case is shown in parentheses.) It appears that the error due to interactions not taken into account in eq 4 is of comparable magnitude to that evidenced in calculations using eq 8 or q 8h, and related procedures.
TABLE VI
given ion
I = 0.1
I = 1.0
I = 3.0
K+
0.5 (14)
Na’
0.3 (19) 0.3 (14) 1.3 (12)
2.6 (12) 2.3 (19)
6.9 (10) 6.9 (17)
1.5 (14) 2.1 (12)
3.1 (14) 3.9 ( I 1)
c1SO,’-
unnecessary to give x any geometrical interpretation. In the equations developed here, x is merely a parameter of convenience; it should be obvious that the working parameter is I . The geometrical implications of this choice (that is, its relation to the codiffusion problem) remain to be worked out. The assignment of activity coefficients to single ions has been a controversial problem.31 It was thought at one time that with adequate theoretical understanding EJ’smight provide information leading to such determinations. However, HarnedZ9(tentatively) and Taylor30 (conclusively) showed that the total potential of electrochemical cells containing liquid junctions is given by an expression using only mean activity coefficients. Splitting the total potential into a component due to the difference in electrode potentials and a component arising at the liquid junction is equivalent to assigning activity coefficients to the individual ion constituents, and the other way around, but there is no experimental procedure, or complete theoretical analysis, which will accomplish either. We have made use of independently measured mean activity coefficients in the results described, and the comparisons between calculation and observation must be regarded as valid for the total cell potentials, without passing judgment on eq 4. (Had we tabled cell potentials rather than junction potentials, the absolute errors would be the same.) Clearly it is of interest to know the magnitude of the error associated with this parti-
(32) B. E. Conay, ref 11 above, Chapter 5, p 136. (33) Sources of physical data not cited above. General compilations: “International Critical Tables”, McGraw-Hill, New York, 1933, Vol. 3 and 6; “Landolt-Bornstein”, Springer-Verlag, Berlin, 1960, 11. Band, 7. Teil; ‘Handbook of Chemistry and Physics”, CRC Press, Cleveland, 1974. Con-
centrative properties of aqueous solutions. Compilations, activity coefficients: ref 6 above; K.S. Pitzer and G. Mayorga, J. Phys. Chem., 77,2300 (1973); J . Solution Chem., 3, 539 (1974). Compilation, transference numbers; E. Kaimakov and N. C. Varshavskaya, R u m Chem. Rev., 35,89 (1966). Other sources, equivalent conductance and transference numbers: T. Shedlovsky and A. S. Brown, J . Am. Chem. Soc., 56, 1066 (1934) (A, CaCI,, MgCI,); L. G. Longsworth, J . Am. Chem. Soc., 57, 1185 (1935) (t’, CaCI,, Na2S04);B. B. Owen and S. F. Sweeton, J . Am. Chem. SOC.,63,281 1 (1941) (A, HCI); L. G. Longsworth, J . Am. Chem. SOC.,54,2741 (1932) (f+, HCI, KCI, NaCI); J. F. Chambers, J. M. Stokes, and R. H. Stokes, J. Phys. Chem., 60,985 (1956) (A, KCI, NaCI); D. A. MacInnes and M. Dole, J . Am. Chem. Soc., 53, 1357 (1931) (t+, KCI); G. S. Hartley and G. W. Donaldson, Trans. Faraday Soc.,33,457 (1937) (t+, KZSO,); M. Postler, Collect. Czech. Chem. Commun., 35, 2244 (1970) (A, MgCI2); K. Lee and R. L. Kay, Aust. J . Chem., 33, 1895 (1980) (t+, MgCI,); D. M. Egan and J. R. Partington, J . Chem. Soc., 191 (1945) (t+, ZnCl,); B. B. Owen and D. W. Gurry, J . Am. Chem. Soc., 60,3074 (1938) (A, ZnSO,).
(31) E. A. Guggenheim, J . Phys. Chem., 33,842 (1929); H. S. Frank, ref 8 above.
1,P-Rearrangement in ,&Substituted Ethyl Radicals. A Molecular Orbital Study Tova Hoz, Milon Sprecher,* and Harold Bas&* Department of Chemistry, Bar- Ilan University, Ramat-Gan, Israel (Received: October 15, 1984) Ab initio self-consistent field (SCF) calculations in a Gaussian orbital basis set have been carried out on both the equilibrium (open) and bridging (ring) structures of various 8-substituted ethyl radicals (C2H4X,X = C1, SH, and PHz) in order to study substituent 1,2-migration. For X = C1 the C, symmetrically bridged (SB)structure is found at all geometric points to be above the CzH4+ C1 dissociation energy limit in a multiconfiguration(MC) SCF framework, although single-configuration unrestricted SCF calculations show very shallow minima (for ‘Al and ’B, states) at long C2H4-Cl distances. In the MC calculations on the steeply repulsive ZAlstate charge transfer (C2H4 C1) is found to be an important stabilizing feature, suggesting a method for stabilizing the ring structure. Bridging structures (both symmetric and slightly unsymmetric) involving relative twisting of the methylene groups are found to be more stable than the SB structures. For both X = SH and PH2 the unrestricted SCF results for the ring structures are similar to that for CzH4CI.
-
The 1,Zmigration of the chlorine atom in certain 0-chloroalkyl monoradicals to the free-radical terminus was first reported in 1951 and has been extensively studied and reviewed s i n ~ e . ~ - ~ In simple cases (Figure 1, R1= H or CH,; R2= R3 = CH3), the activation energy appears to be 5 5 kcal/mol, and the accumulated experimental data have been taken as evidence for an ‘s2
(1) Urry, W. H.; Eiszner, R. J. Am. Chem. Soc. 1951, 73, 2977. 1952, 74, 5822. (2) Nesmeyanov, A. N.; Freidlina, R. Kh.; Fintov, V. I. Izu. Akad. Nauk SSR, Otd. Khim. Nauk 1951, 505. Nesmeyanov, A. N.; Freidlina, R. Kh.; Zakharkin, L. I. Dokl. Akad. Nauk SSR 1951,81, 199. (3) Freidlina, R. Kh. Adu. Free Radical Chem. 1965, 1 , 21 1. (4) Wilt, J. W. In “Free Radicals”, Kochi, J. K., Ed.; Wiley: New York, 1973; Vol. 1, Chapter 8. Skell, P. S.; Shea, K. J. Ibid. Vol. 2, Chapter 26.
(5) Beckwith, A. L. J.;Ingold, K. U.In “Rearrangements in G :ound and Excited States”, de Mayo, P., Ed., Academic Press: New York, 1980; Vol. 1, Chapter 4.
0022-3654/85/2089-1664$01.50/0
intramolecular migration rather than an elimination-readdition mechanism.6 In analogous rearrangements of P-bromoalkyl radicals the findings are less clear-cut. Some data indicate elimination-readdition, consistent with the lower bond energy of C-Br as compared with C-C1, while in other cases an intramolecular shift appears to be a more reasonable explanation. Among the latter is the free-radical chlorination of optically active 2bromobutane which yielded, inter alia, optically active 1bromo-2-chlorobutane of retained configuration at C-2.’ The formation of this chiral product could, of course, not have proceeded via bromine atom readdition to free achiral 1-butene, and it implies that the intermediate rearranged radical (Figure 1, 2, (6) Reference 5, p 249, and references cited therein. (7) Skell, P. S.; Pavlis, R. R.; Lewis, D. C.; Shea, K. J. J . Am. Chem. Soc. 1973, 95, 6735. Skell, P. S.; Traynham, J. G. Acc. Chem. Res. 1984, 17, 160.
0 1985 American Chemical Society
