1220
J. Phys. Chem. 1996, 100, 1220-1226
Binary Mixing Approximations and Relations between Specific Conductance, Molar Conductance, Equivalent Conductance, and Ionar Conductance for Mixtures† Donald G. Miller Chemistry and Materials Science Department, Lawrence LiVermore National Laboratory, UniVersity of California, P.O. Box 808, LiVermore, California 94551 ReceiVed: April 25, 1995; In Final Form: July 10, 1995X
Simple linear approximations to the specific conductance of a mixture can be written in terms of various solute fractions (molar, equivalent, or ionic strength) and the specific conductances of its constituent binary systems. These binary conductances are evaluated at some type of “constant” concentration which characterizes the mixture (constant total molarity, constant total equivalents, or constant total ionic strength). These expressions can be made exact for fitting experimental data by including a correction term. General forms have been derived for transforming these binary approximations for the specific conductance and their corresponding correction terms to the analogous binary approximations for “concentration conductances” such as molar, equivalent, or ionar (ionic strength) conductance. Conversely, simple binary approximations for any concentration conductance in terms of arbitrary fractions for an arbitrary binary evaluation strategy lead to binary approximations for the specific conductance. In both cases, simpler forms result when “natural” fractions or “natural” binary evaluation strategies are used. The specific conductance is the basic physical property. The NaCl-MgCl2-H2O system is used as an example.
I. Introduction Electrical conductance is one of the principal transport properties of electrolyte solutions, not only for its intrinsic interest but for technical and industrial applications such as batteries and plating. In electrolyte conductance measurements, the specific conductance (σ) is the quantity actually obtained. Since σ is nearly proportional to the concentration at lower concentrations, for more than a century it has been customary to divide it by some volume concentration measure to yield a “concentration conductance”. The one most used has been the equivalent concentration N, which yields the equivalent conductance ΛN. IUPAC recommends the use of molarity C, which yields the molar conductance ΛC. However, in a recent discussion of Young’s cross square rule for transport properties, Wu et al.1 effectively divided σ by the volume ionic strength S to obtain the ionar (ionic strength) conductance ΛS. Similarly, Pikal’s theoretical equations for the ionic Onsager coefficients2 can be recast in terms of ionar conductances. Thus, one motive for this research was to clarify the relationships among these types of tabulated data for mixtures. Because conductance can be measured with quite high precision, it is often used to test electrolyte theories. The most demanding of such tests are on mixtures. Typically, the deviation from a “linear mixture rule” is examined. Consider the example of a ternary system containing two electrolytes in water: Typically, a linear mixture rule for conductance has been based on comparing the experimental equiValent conductance ΛN of the ternary solution with a binary solution estimate in terms of the equiValent conductances ΛN1 in a binary solution of solute 1 and ΛN2 in a binary solution of solute 2. Both ΛNi are evaluated at the same ionic strength S as of the mixture, and each is multiplied by its corresponding equiValent fraction xi. The usual mixture rule thus consists of † This paper is dedicated to Prof. Harold Friedman, who has inspired and encouraged much of our work over the years. X Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-1220$12.00/0
a binary approximation part ΛN0 and the deviation δ from the experimental value ΛN:
ΛN0 ) x1ΛN1 + x2ΛN2
(1)
ΛN ) ΛN0 + δ
(2)
Thus, δ is the deviation from the approximate linear mixture rule, to be compared with electrolyte theories. There are three issues connected with the above mixture rule. 1. The evaluation of the solute conductances in their binary solutions at the same S as in the ternary (at constant S for short) is a binary eValuation strategy. The choice of S is based on the Debye-Hu¨ckel model, which is valid in dilute solutions. However, other binary evaluation strategies may be useful in very concentrated solutions. For example, using the binary evaluation strategy “at constant N” may give a smaller value of δ. Alternatively, evaluation “at constant C” can reduce the problem of extrapolation of ΛNi beyond the solubility limit of one or more of the solutes, even if the δ will typically be larger in this case. 2. The use of xi, a “natural fraction” for equivalent conductance, is a reasonable choice and also has some theoretical basis at infinite dilution. However, other choices are possible, such as ionic strength (ionar) fraction or solute fraction (moles of solute i divided by total moles of all solutes). 3. The equivalent conductance is not the only possible one. Other choices include the molar conductance and ionar conductance, as well as the specific conductance. These issues become important in practical applications, since it is often necessary to estimate the conductance from only binary data. In this case, the smallest value of δ is the best, and it could come from using a binary evaluation strategy other than evaluation at the S of the ternary, using a fraction other than xi, or using a type of conductance other than equivalent. Thus, it is worthwhile to generalize the linear mixture rule above to include all these possible choices and their relationships. © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 4, 1996 1221
We suggest, following Weinga¨rtner,3 that the specific conductance is the fundamental quantity in considering binary solution mixing approximations. Consequently, there are two significant questions. (1) What do the various binary solution approximations for the specific conductance and their correction terms imply about the binary approximations for the different “concentration conductances” and their corresponding correction terms. (2) Conversely, what does a particular binary approximation for a concentration conductance and its correction term imply for the binary approximation to the specific conductance and its correction term? To include these cases, we follow Wu et al.1 by estimating an arbitrary property of a ternary electrolyte mixture in terms of its binary constituents with a linear approximation plus a correction term
P ) P0 + δ ) a1P1 + a2P2 + δ
(3)
or in general for multicomponent mixtures
P ) P0 + δ ) ∑aiPi + δ
(4)
where P is the solution property, P0 is the binary solution approximation to P, Pi is the corresponding property of binary solution of constituent i evaluated by some binary evaluation strategy, and ai are arbitrarily chosen composition fractions such as the solute fractions zi, equivalent fractions xi, or ional (volumetric ionic strength) fractions yi. Equations 3 and 4 are essentially the generalization of Young’s rule4 to properties other than volume or heat of mixing, in particular to transport properties. Typical binary evaluation strategies are to evaluate Pi at either the same total molar concentration C, the same total equivalent concentration N, or the same total ionar concentration (ionic strength) S as the mixture, i.e., at constant C, N, or S. Although not used here, osmolarity or the corresponding mass concentrations could also be considered. The δ represents the deviation from experiment of the binary approximation using one of these strategies and a choice of fractions. The intuitive choice for ai with the specific conductance is the natural fractions corresponding to the binary evaluation strategy: i.e., zi with constant C, xi with constant N, and yi with constant S. However, in principle, it could come from any one of the nine possible combinations of (C,N,S) with (zi,xi,yi). For the concentration conductances, the example illustrated by eqs 1 and 2 suggests that the natural fractions are those associated with the type of concentration conductance, e.g., equivalent fractions xi with equivalent conductance ΛNi , without regard to the binary evaluation strategy. However, again in principle, any of the other fractions could be used with a given concentration conductance in the search for a smaller δ. In this paper, we explore all these issues and provide some general theorems. The results are given for ternary mixtures but are easily extended to any number of solutes. The final expressions for the binary approximations depend on both the choice of binary evaluation strategy and the choice of concentration fractions. They take simple forms only when natural fractions or natural binary evaluation strategies are used. A consequence of taking the specific conductance as fundamental is the clarification of the relationships among the correction terms of the various concentration conductances. An interesting result of our analysis is that the use of the natural fractions based on the binary evaluation strategy for the specific conductance implies that the fractions for a given concentration conductance correspond to that type of concentration conductance regardless of the binary evaluation strategy,
just as in the example of eqs 1 and 2. Conversely, the use of the fractions corresponding to the type of concentration conductance for any given binary evaluation strategy implies that the fractions for the specific conductance correspond precisely to that binary evaluation strategy. II. Notation and Definitions Because we use sets of arbitrary concentration types, arbitrary composition fractions, and arbitrary concentration conductances, the notation is necessarily complex. We shall also present a table of factors to convert one concentration type to another. Let a binary electrolyte i ionize as
CricAria f ricCizic + riaAizia
(5)
ri ) ric + ria
(6)
riczic + riazia ) 0
(7)
for which
where ric and ria are the cation and anion stoichiometric coefficients of ionization, zic and zia are the cation and anion valences with due regard to sign, and ri is the total number of ions of i. To make the derivations both general and more compact, we denote arbitrary volume concentrations of solutes by K, K′, K′′, and L, depending on context. These all refer to any of (C, N, S) for the mixture. Similarly, Ki, etc., refer to the various possible concentrations of electrolyte i in its binary solution, (Ci, Ni, Si). To avoid factors of 1000 in the derivations, these concentrations are in mol cm-3, equiv cm-3, ional cm-3, and in kayal cm-3 for C, N, S, and K, respectively. The term kayal is introduced to refer to an arbitrary solute quantity such as mol, equiv, or ional. The volumetric concentrations C, N, S, and K are called molar, equivalent, ionar, and kayar.5 The customary units, which are per dm3, can be obtained by substituting K/1000 or Ki/1000 for K and Ki everywhere. The various fractions are denoted by
zi ) Ci /C ) aiC
(8)
xi ) Ni /N ) aiN
(9)
yi ) Si /S ) aiS
(10)
Ki /K ) aiK
(11)
The specific conductance of the mixture with binary evaluation strategy K is denoted by σK, and the specific conductance of binary i evaluated at that same K as the mixture (i.e., at constant K) is denoted by σiK. The various concentration conductances are denoted in general by Λ or Λi with a superscript K′ and specifically with superscript C, N, or S. ΛK′ or ΛK′ i is called a molar, equivalent, ionar, or kayar conductance. The expressions relating any Λ to σ are in general
ΛK′ )
σ K′
ΛiK′ )
σi K′i
(12)
Consequently, the various concentration conductances are all directly related to each other through σ or σi. The binary evaluation strategy for concentration conductances is denoted in general with subscript K and specifically with K′ for the equivalent subscript C, N, or S. For example, ΛiK
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Miller
conductance of solute 2 evaluated at the same ionic strength as the ternary solution is denoted by ΛN2S and its corresponding specific conductance by σ2S. Concentration conversions are necessary to relate the various K′ , these Λ to each other. For the concentration conductance ΛiK conversions are also required to define that concentration K′i* which corresponds to the value of K held constant in the binary evaluation strategy. An example conversion from molar to equivalent is
N1 ) r1cz1cC1
(13)
An example conversion to C1* for evaluating σ1 at constant N is
C1* ) N/r1cz1c
(14)
A third example is the conversion to N1* from the total S of the mixture. This is necessary to get ΛN1S, the equivalent conductance of solute 1 at constant S, from σ1S. It is
2 S N*1 ) z1c - z1a
Li* ) PiLKK
(16)
where L and Li are also arbitrary concentrations. It follows that
PiLK ) 1/PiKL;
III. Relations of Concentration Conductances to Binary Approximations for σ Consider the binary approximation to σ (eq 18) with σi evaluated at K, using fractions based on K′′, and including the correction term. What are the forms of the binary approximations to the concentration conductances? In terms of the K′ conductance for the ternary, we have
ΛKK′ )
σ σK a1K′′σ1K a2K′′σ2K δK′′K ) ) + + K′ K′ K′ K′ K′
K′ ) ΛiK
σi K′i*
(21)
evaluated at K, we need to transform the i terms, making use of Table 1, as follows. Thus,
( )
K′ aiK′′σiK PiK′KK aiK′′ΛiK PiK′KK aiK′′σiK aiK′′σiK K′i* ) ) ) ) K′ K′ K′i* K′ K′i* K′
()
K′ PiK′KK aiK′′ΛiK
K′
PiKLPiLM ) PiKM
(20)
To obtain the corresponding binary K′ conductances
(15)
More generally, a conversion factor PiLK which converts Ki into Li or K into Li* is defined by
Li ) PiLKKi;
strength fractions and constant ionic strength do not always lead to the smallest correction term. In particular, evaluation at constant equivalents sometimes gives better results for certain concentrations or certain properties.8
(17)
K′ aiK′′ΛiK Ki K′iK aiK′′aiK′ K′ ) ) ΛiK (22) Ki K′Ki aiK
Therefore, our final result from eq 18 is Table 1 contains the various conversion factors derived from the standard definitions7 of Ci, Ni, and Si. TABLE 1: Transforms PiLK Which Convert K into L K L
Ni
Ci
Ci
1
1/(riczic)
Ni Si
riczic (-riziczia)/2 ) [riczic(zic - zia)]/2
1 (zic - zia)/2
Si 2/[riczic(zic - zia)] ) -2/(riziczia) 2/(zic - zia) 1
ΛKK′ )
a1K′′a1K′ K′ a2K′′a2K′ K′ δK′′K Λ1K + Λ2K + a1K a2K K′
The coefficients of Λi are a composite combination of the fractions. For example, if (a) evaluations of the binaries are at constant ionic strength, i.e., K ) S, (b) equivalent fractions xi are used, i.e., K′′ ) N, and (c) molar conductances are used, i.e., K′ ) C, then
ΛCS )
Analogous to eq 3, we write the binary solution approximation to σ of a ternary solution as
σK ) a1K′′σ1K + a2K′′σ2K + δK′′K )
σK0
+ δK′′K
(18)
where K represents an arbitrary binary evaluation strategy (i.e., at constant K) and K′′ an arbitrary choice of composition fraction. Of course, the experimental value of σ depends only on concentration, not on K; the subscript K only indicates the type of binary evaluation concentration. The value of δK′′K necessarily depends on both the binary evaluation strategy and the choice of composition fractions. For example, δNS applies to the binary approximation when using equivalent fractions and evaluating σi at constant S. The general multicomponent case is
σK ) ∑aiK′′σiK + δK′′K
(19)
i
For the three usual concentration choices, there are nine possible binary approximations using K and K′′. Which combination leads to the smallest value of δKK′′ depends on the particular experimental system. We emphasize that ionic
(23)
x2z2 C δNS x1z1 C Λ + Λ + y1 1S y2 2S C
(24)
There are 27 possibilities, in view of the 3 ways to define concentration conductances for each of the 9 ways to assign fractions and binary evaluation strategies for σ. However, only the nine involving σ are independent, because δK′′K does not depend on K′. Consequently, the concentration conductance correction terms δK′′K/K′ are all related. With a given choice of fractions K′′ and binary evaluation strategy K for σ, δK′′K is determined. Consequently, the deviations for the K′ conductances, δK′′K/K′, always decrease (or are all equal) in the K′ order molar, equivalent, and ionar because C e N e S. Therefore, the smallest deviations observed for the ionar conductance have no essential significance. The significant deviation is δK′′K and should be determined from σ. There are two situations where the Λi have simple coefficients. These are easily found by seeing how fractions can cancel in eq 23. (a) Let the natural fractions aiK′′ for σ correspond to the constant evaluation concentration; i.e., K′′ ) K. Then from eqs 18 and 23, we get
σK ) a1Kσ1K + a2Kσ2K + δKK
(25)
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J. Phys. Chem., Vol. 100, No. 4, 1996 1223
and K′ K′ ΛKK′ ) a1K′Λ1K + a2K′Λ2K +
δKK K′
(26)
Consequently, in this case, the simple natural fractions for the K′ conductance always correspond to that type of concentration conductance rather than to those fractions of the binary evaluation strategy which were natural for σK. This is the principal result mentioned in the Introduction. There are nine such cases with simple coefficients. For example, if the σi are evaluated at constant S and the corresponding natural ionar fractions yi are used, then
σS ) y1σ1S + y2σ2S + δSS
(28)
ΛNS ) x1ΛN1S + x2ΛN2S +
δSS N
(29)
ΛSS ) y1ΛS1S + y2ΛS2S +
δSS S
(30)
)
z1ΛC1S
+
z2ΛC2S
Our analysis makes explicit the relationships among the correction terms. Consequently, any one can be obtained directly from any of the others. We discuss below estimates of σ with different binary evaluation strategies K and different fractions K′′ for the system NaCl-MgCl2-H2O at 25 °C. Comparisons show that generally the δK′′K are smallest if K′′ ) K, i.e., if the natural fractions are used. Therefore, the δKK can then be compared to see which binary evaluation strategy gives the smallest δ at a given composition. For example, consider the comparison of δ at a fixed composition with the binary evaluation strategies of constant N and constant S, using their corresponding natural fractions. The appropriate expressions are
σN ) x1σ1N + x2σ2N + δNN
(31)
σS ) y1σ1S + y2σ2S + δSS
(32)
However, we can also compare the correction terms from ΛNN and ΛNS just as well, because our theorem above shows that the expressions
δNN N N + x2Λ2N + ΛNN ) x1Λ1N N
(33)
δSS N
(34)
ΛNS ) x1ΛN1S + x2ΛN2S +
correspond to eqs 31 and 32, respectively. The comparison from eqs 33 and 34 differs from that of eqs 31 and 32 only by the common factor N, proVided the xi are used with both evaluation strategies. The same holds true for molar and ionar conductances if the zi and yi are used, respectively. Analogously, molar conductances with zi yield δNN/C and δSS/C, and ionar conductances with yi yield δNN/S and δSS/S. (b) Let the σi be evaluated at constant K for our arbitrary choice of aiK′′ and let the type of concentration conductance also be K; i.e., K′ ) K. Then eq 23 yields K K + a2K′′Λ2K + ΛKK ) a1K′′Λ1K
δK′′K K
σC ) y1σ1C + y2σ2C + δSC
(35)
(36)
and C C + y2Λ2C + ΛCC ) y1Λ1C
(27)
δSS + C
ΛCS
K Consequently, in this case, the coefficients of ΛiK are explicitly the same as those used for σK, however “unnatural” these may seem. There are nine such cases, of which three overlap with case (a). For example, if ionic strength fractions yi are chosen, if the σi are evaluated at constant C, and if molar conductance is used, then
δSC C
(37)
This case is uncommon, but it does have a simple form. Thus, there are 15 distinguishable cases within (a) and (b), plus 3 overlaps. The remaining nine have composite coefficients. IV. Relations between σ and Binary Approximations to Concentration Conductances Let us examine the converse of section III, the determination of the binary approximation to σ from concentration conductances. Consider the binary approximation to an arbitrary concentration conductance ΛK′ K in terms of arbitrary fractions aiK′′. We denote the correction term for a concentration conductance by ∆K′′K because it is not necessarily related to the δK′′K of the specific conductance (eq 18). Then, we can write K′ K′ + a2K′′Λ2K + ΛKK′ ) a1K′′Λ1K
∆K′′K K′
(38)
To obtain an expression for the corresponding binary approximation to σ, we use transformation arguments similar to those in section III. Thus, multiply eq 38 by K′, get σiK from its definition in terms of Ki′* (in the denominator), replace this with PiK′KK, multiply by Ki/Ki, use PiK′KKi to get Ki′ (also in the denominator), and use the definitions of fractions. The result is
σK )
a1K′′a1K a2K′′a2K σ + σ + ∆K′′K a1K′ 1K a2K′ 2K
(39)
Analogously to section III, the coefficients of the σi are composite. Comparison of eq 39 with eq 18 shows that the ∆K′′K will only be related to δK′′K when the coefficients are not composite. Again there are two situations with simple coefficients for σi, i.e., where fractions can cancel in eq 39. (a) Let the concentration conductance K′ be evaluated at constant K, and let the natural K′ fractions be used; i.e., K′′ ) K′. Then the natural fractions for the coefficients of σi are obtained; these fractions correspond to evaluation at constant K. Thus, if K′ K′ + a2K′Λ2K + ΛKK′ ) a1K′Λ1K
∆K′K K′
(40)
then
σK ) a1Kσ1K + a2Kσ2K + ∆K′K
(41)
This case, the intuitively obvious one, is the converse of (a) of the previous section. Consequently, ∆K′K ) δKK. This result
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Miller
is the converse of the principal result mentioned in the Introduction. (b) If the natural conductance form for constant K is used, i.e., K′ ) K, then the original fractions used for ΛKK are the coefficients of σiK, even if these are not natural for σK. Thus, if K K ΛKK ) a1K′′Λ1K + a2K′′Λ2K +
∆K′′K K
(42)
then
σK ) a1K′′σ1K + a2K′′σ2K + ∆K′′K
(43)
This case is the converse of (b) of the previous section. Consequently, ∆K′′K ) δK′′K. There are nine cases in both (a) and (b), but three overlap. This leaves 15 distinguishable cases; the other 9 are composite. For the 15 distinguishable simple cases, we have the relationships between ∆ and δ. For the composite cases, there are no such relationships. L V. Transform of One Simple Type of ΛK′ K to Another ΛK, Both at Constant K
Now consider the conversion of one concentration conductance K′ with arbitrary fractions to another concentration conductance L. This only makes sense if both cases are evaluated at the same constant K. Suppose we have the simple form for ΛK′ K K′ K′ + a2K′′Λ2K + ΛKK′ ) a1K′′Λ1K
∆K′′K K′
(44)
The transform of this to another conductance type is found by an argument analogous to that of the previous sections. The result is
ΛKL )
a1K′′a1L L a2K′′a2L L ∆K′′K Λ + Λ + a1K′ 1K a2K′ 2K L
(45)
There are simple forms for ΛLK only if the natural fractions for the original conductance ΛK′ K are used; i.e., K′′ ) K′. In this situation, the natural fractions for ΛLK are obtained. Thus, if K′ K′ + a2K′Λ2K + ΛKK′ ) a1K′Λ1K
∆K′K K′
(46)
then L L + a2LΛ2K + ΛKL ) a1LΛ1K
∆K′K L
(47)
This case is obviously related to the (a) cases of both previous sections. Comparison with those results shows clearly that ∆K′K ) δKK. The other possible simple case with cancellation in eq 45 is K′ ) L. However, this yields the original conductance, so nothing is gained. VI. Discussion The theorems of previous sections allow us to see clearly the connections of all the concentration conductances to the more fundamental specific conductance. Thus, all the 27 variants of Λ involve only the 9 possible δK′′K. At the same time, we found all the relationships among the correction terms. Consequently,
a search for the smallest δK′′K (i.e., the best linear approximation) is best done with specific conductance. It is interesting to examine some numerical calculations of all the δK′′K for a particular system. A convenient one is NaClMgCl2-H2O at 25 °C, for which conductance data have been obtained by Bianchi et al.9 at 25 compositions as part of an international collaboration to obtain all the transport properties of this system.10 These data consist of five sets of equivalent conductances ΛN. Each set was measured at the same total concentration C for five different solute fractions, z1 ) 0, 0.25, 0.5, 0.75, and 1 (mole ratios 0, 1:3, 1:1, 3:1, and ∞). The five total concentrations C for the five sets are 0.5, 1.0, 2.0, 3.0, and about 3.7 mol dm-3. There are two ways to use these sets of data at constant C to compare binary evaluation strategies. The first is to use the measured mixture data at the experimental concentrations and interpolate the binary data at the Ci* appropriate to each specific solute fraction. This Ci* also depends on the binary evaluation strategy and, in turn, depends on the different values of C, N, and S of a given solute fraction. However, the comparisons are for the same value of C for a given set of solute fractions. The second way is to compare the conductances of the set of solute fractions at the same C, the same N, and the same S. This requires interpolation of the mixture data for each solute fraction, as well as of the binary data at its appropriate Ci*. Note that for this system N ) 1 equiv dm-3 corresponds to a lower total concentration than C ) 1 mol dm-3 at any solute fraction z1 < 1, and S ) 1 ional dm-3 corresponds to a still lower total concentration than N ) 1 for the same solute fraction. This second approach was used by Wirth and Bangert to evaluate binary mixing approximations for volumes.8 In this second way, interpolations for the mixtures are relatively uncertain because the ternary data for the NaClMgCl2-H2O system9 are at relatively large separations for a given solute fraction. In contrast, there are binary data at much closer separations, making interpolations safer. Consequently, we have chosen the first way. We use the C ) 1 and C ) 2 mol dm-3 sets as examples because the concentrations are high enough to show the deviations from the binary approximation but not so high as to have the binary Ci* for NaCl above its solubility limit. Our specific conductance comparisons are made for all nine possible cases of the three different kinds of fractions (K′′ ) zi, xi, yi) with the three binary evaluation strategies (K ) C, N, S). Note that as z1 varies from 0 to 1, the equivalent concentrations N of the mixtures at constant C vary from 2C to C, and the ionic strengths S vary from 3C to C. The values of σiK were obtained from ΛNi of the binaries in absolute conductance units. These in turn were obtained from graphical interpolations of unpublished smoothings of ΛN for NaCl (1988) updated from an earlier smoothing12 and for MgCl2 (1989) updated from an earlier smoothing.13 Both smoothings were adjusted graphically to provide consistency with the binary data of Bianchi et al.9 for NaCl and MgCl2 so as to be compatible with their ternary data. The binary values σiK were then obtained from N 103σiK ) N* iKΛiK
(48)
where N*iK is the value of Ni in the binary corresponding to the constant concentration K. It is given by
N*iK ) PiNKK
(49)
where N*1K is (C, N, S) and N* 2K is (2C, N, 2S/3) when K ) C,
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J. Phys. Chem., Vol. 100, No. 4, 1996 1225
TABLE 2: Specific Conductance Deviations for NaCl-MgCl2-H2O at 25 °C (C ) 1 mol dm-3)a 103σexp 103σ1‚C 103σ2‚C
C
116.10 110.37 103.50 95.325 85.71
116.10 116.10 116.10 116.10 116.10
1.0 1.0 1.0 1.0 1.0
103σexp 103σ1‚N 103σ2‚N
C
116.10 110.37 103.50 95.325 85.71
85.71 85.71 85.71 85.71 85.71
103σC°C δCC δCC/(z1z2)
0.0 116.10 0.0 0.25 108.50 1.87 0.5 100.90 2.60 0.75 93.31 2.02 1.0 85.71 0.0 z1
103σC°N
δCN
149.36 116.10 1.0 0.0 116.10 0.0 135.05 106.63 1.0 0.25 113.74 -3.36 119.72 96.00 1.0 0.5 107.86 -4.36 103.29 84.19 1.0 0.75 98.52 -3.19 85.71 71.02 1.0 1.0 85.71 0.0
103σexp 103σ1‚S 103σ2‚S 116.10 110.37 103.50 95.325 85.71
z1
C
z1
196.65 116.10 1.0 0.0 175.03 103.23 1.0 0.25 149.38 88.27 1.0 0.5 119.72 71.08 1.0 0.75 85.71 51.34 1.0 1.0
103σC°S
δCS
116.10 0.0 121.18 -10.81 118.82 -15.32 107.56 -12.24 85.71 0.0
9.96 10.38 10.76
N
103σN°C
x1
2.0 1.75 1.5 1.25 1.0
δNC
S
y1
-11.34 -11.12 -10.59
3.0 2.5 2.0 1.5 1.0
0.0 0.1 0.25 0.5 1.0
0.0 116.10 0.0 0.1429 111.76 -1.39 0.3333 105.97 -2.47 0.6 97.87 -2.54 1.0 85.71 0.0
δCN/(z1z2)
N
-17.95 -17.44 -17.01
2.0 1.75 1.5 1.25 1.0
δCS/(z1z2)
N
x1
-57.65 -61.30 -65.25
2.0 1.75 1.5 1.25 1.0
0.0 0.1429 0.3333 0.6 1.0
x1
103σN°N
δNN
0.0 116.10 0.0 0.1429 110.69 -0.32 0.3333 103.91 -0.41 0.6 95.65 -0.32 1.0 85.71 0.0 103σN°S
δNS
116.10 0.0 113.49 -3.12 108.64 -5.14 100.26 -4.93 85.71 0.0
103σS°C
δNC/(x1x2)
δNN/(x1x2)
S
-2.61 -1.83 -1.35
3.0 2.5 2.0 1.5 1.0
δNS/(x1x2)
S
-25.46 -23.13 -20.58
3.0 2.5 2.0 1.5 1.0
y1
δSC
116.10 0.0 113.06 -2.69 108.50 -5.00 100.90 -5.58 85.71 0.0 103σS°N
δSN
0.0 116.10 0.0 0.1 109.47 0.90 0.25 101.93 1.57 0.5 93.74 1.58 1.0 85.71 0.0 y1
103σS°S
δSS
0.0 116.10 0.0 0.1 110.41 -0.04 0.25 103.55 -0.05 0.5 95.40 -0.075 1.0 85.71 0.0
δSC/(y1y2) -29.90 -26.68 -22.32
δSN/(y1y2) 9.98 8.37 6.34 δSS/(y1y2) -0.444 -0.253 -0.300
a δK′′K ) σexp - σK°′′K, where σexp is the experimental value and σK°′′K are the binary estimates for the various binary evaluation strategies K using fractions aiK′′ . σ‚iK are the specific conductances of the binary solutions at the K corresponding to the total concentration C at its appropriate solute fraction. Units of C, N, and S are per dm-3 and of σ and δ are absolute conductance units (S cm-1).
TABLE 3: Specific Conductance Deviations for NaCl-MgCl2-H2O at 25 °C (C ) 2 mol dm-3)a 103σexp
103σ1‚c
103σ2‚C
C
z1
103σC°C
δCC
160.48 149.38 157.84 2.0 0.25 155.72 4.76 160.35 149.38 157.84 2.0 0.5 153.61 6.74 156.80 149.38 157.84 2.0 0.75 151.50 5.30 103σexp 103σ1‚N 103σ2‚N
C
z1
103σC°N
δCN
160.48 214.58 152.39 2.0 0.25 167.94 -7.46 160.35 196.65 143.91 2.0 0.5 170.28 -9.93 156.80 175.02 131.98 2.0 0.75 164.26 -7.46 103σexp 103σ1‚S
103σ2‚S
C
z1
103σC°S
δCS
δCC/(z1z2) 25.36 26.96 28.29 δCN/(z1z2) -39.77 -39.72 -39.79 δCS/(z1z2)
N
x1
103σN°C
δNC
3.5 0.1429 156.63 3.85 3.0 0.3333 155.02 5.33 2.5 0.6 152.76 4.04 N
103σN°N
x1
δNN
3.5 0.1429 161.27 -0.79 3.0 0.3333 161.49 -1.14 2.5 0.6 157.80 -1.00 N
x1
103σN°S
δNS
160.48 247.20 149.90 2.0 0.25 174.22 -13.74 -73.31 3.5 0.1429 163.80 -3.32 160.35 228.92 136.37 2.0 0.5 182.64 -22.30 -89.18 3.0 0.3333 167.22 -6.87 156.80 196.65 116.10 2.0 0.75 176.51 -19.71 -105.1 2.5 0.6 164.43 -7.63
δNC/(x1x2) 31.43 23.98 16.82 δNN/(x1x2) -6.49 -5.13 -4.18 δNS/(x1x2) -27.11 -30.91 -31.79
S
y1
103σS°C
δSC
5.0 0.1 156.99 3.49 4.0 0.25 155.72 4.62 3.0 0.5 153.61 3.19 S
y1
103σS°N
δSN
5.0 0.1 158.61 1.87 4.0 0.25 157.10 3.26 3.0 0.5 153.50 3.30 S
y1
103σS°S
δSS
5.0 0.1 159.63 0.85 4.0 0.25 159.51 0.84 3.0 0.5 156.38 0.425
δSC/(y1y2) 38.73 24.67 12.76 δSN/(y1y2) 20.79 17.36 13.2 δSS/(y1y2) 9.44 4.49 1.70
aδ K′′K ) σexp - σK° ′′K, where σexp is the experimental value and σK° ′′K are the binary estimates for the various binary evaluation strategies K using fractions aiK′′ . σ‚iK are the specific conductances of the binary solutions at the K corresponding to the total concentration C at its appropriate solute fraction. The table is simplified compared to Table 2 by omitting the data for z1 ) 0 and 1 since the deviations are all zero. The units are as described in Table 2.
N, S, respectively. Note that we haVe now conVerted concentrations from the per cm3 used in the deriVations to the customary per dm3 units; these customary units will be used in the tables and comparisons. According to Friedman,1,14 δSS can be written in the general form
δSS ) y1y2S2[b0 + b1(y1 - y2)S + b2(y1 - y2)2S2 + ...] (50) where the bi are functions of S. This can be generalized for arbitrary K to
δKK ) a1Ka2KK2[b0 + b1(a1K - a2K)K + b2(a1K - a2K)2K2 + ...] (51) where the bi are now functions of K. These functions may not necessarily be polynomials in K. We will give values of δKK/ (a1Ka2K) in the tables for our example. We note that Timmermann15 has extensively analyzed the equivalent conductance of mixtures at constant K ) N using eq 51 with aiK ) xi. He examined δNN and δNN/(x1x2) for various
earlier mixture data, and based his work on the semiempirical result of Redlich and Kister,16 which has exactly the same form as eq 51. Table 2 contains the comparisons at C ) 1 mol dm-3 for all nine possible δK′′K in terms of the three aiK′′ and the three binary evaluation strategies K. We see that the natural fractions (K′′ ) K) overall give the smallest deviations, δKK, for a given binary evaluation strategy K. Among the δKK, we see that δSS is smaller than δNN, and both are substantially smaller than δCC. All δK′′K are negative except δCC and δSN at C ) 1 mol dm-3. We have also calculated δK′′K/(a1K′′a2K′′). Note that the values of δCK/ (z1z2) are all fairly constant with a given K. Table 3 contains the analogous comparisons for C ) 2 mol dm-3. Here among the δKC, the δCC is not the smallest, but δSC is. However, among the others, δKK is the smallest of the δK′′K. Note that all the δSK are positive now, as is δNC. Note also that δSS is now positive, in contrast to C ) 1 mol dm-3. Among the δKK, δSS is slightly smaller than δNN, and both are significantly smaller than δCC. Part of the reason that δSS is smaller than δNN is that it has changed sign between C ) 1 and C ) 2 mol dm-3. Once again, the values of δCK/(z1z2) are fairly constant.
1226 J. Phys. Chem., Vol. 100, No. 4, 1996
Miller evaluation strategy K, then the fractions for ΛK′ K are the natural ones corresponding to the type K′ of the concentration conductance, and conversely. (2) If the fractions for σK are arbitrary with respect to the binary evaluation strategy K and the type of concentration conductance is also K, then the fractions for K are the same as for σK, and conversely. In both (1) and ΛiK (2), the relationships among the correction terms are explicit. In view of the fundamental importance of σ in testing binary approximations, we recommend that all conductance data be presented as σ as well as Λ.
Figure 1. Deviations δKK in S cm-1 vs z1, where del on the drawing refers to δ. Circles are for δCC, squares are for δNN, and diamonds are for δSS. The open symbols are for the C ) 1 sets, and filled symbols are for the C ) 2 mol dm-3 sets. Note that δCC are all positive and δNN are all negative, whereas δSS are negative at C ) 1 but positive at C ) 2 mol dm-3 with an inflection point at about z1 ) 0.8.
Figure 1 is a plot of δKK vs z1 and shows their relationships more clearly. The open symbols are for C ) 1 and the filled ones for C ) 2 mol dm-3. The δCC are both positive and are nearly but not quite symmetric. The δNN are both negative, are a little less symmetric than δCC, and are skewed toward larger z1. However, the change in sign of δSS between C ) 1 and C ) 2 mol dm-3 leads to a significant skewing toward smaller z1 at C ) 2 mol dm-3. This skewing is more pronounced if plotted against x1 or y1. At some concentrations between C ) 1 and C ) 2 mol dm-3 (not shown), δSS will have “S-shaped” curves, being positive at lower z1 and negative at higher z1. Although they are not tabulated here, comparisons of δKK show that δSS is negative and the smallest at C ) 0.5 mol dm-3. At C of 3 and of about 3.7 mol dm-3, the Ci* appropriate to the equivalent and ional concentrations for z1 of 0.25 and 0.5 are beyond the solubility limit of NaCl; consequently, no binary value of σ1K is available. However, at z1 ) 0.75, it is possible to compare δNN and δSS. At C ) 3 mol dm-3, they have about the same absolute value (1.8), but δNN is smaller than δSS in absolute value at C ) 3.79 mol dm-3. That constant equivalents sometimes yield somewhat smaller δ at higher concentrations has been observed for other properties8 and for other systems. In summary, the results of sections III and IV are quite general and involve composite coefficients. However, the results for the simple cases are useful. These are as follows: (1) If the fractions for σK are the natural ones corresponding to the binary
Acknowledgment. I thank Dr. Hermann Weinga¨rtner for suggesting the pursuit of the ideas contained in this paper. I also thank Prof. John G. Albright and Dr. Joseph A. Rard for helpful comments on the manuscript. This work was begun in 1988 at the Institut fu¨r Physikalische Chemie, RWTH, Aachen, Germany, thanks to the hospitality of Profs. Hans Scho¨nert and Manfred Zeidler. The remainder of this work was performed under the auspices of the Office of Basic Energy Sciences (Geosciences) of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48. References and Notes (1) Wu, Y. C.; Koch, W. F.; Zhong, E. C.; Friedman, H. L. J. Phys. Chem. 1988, 92, 1692-1695. (2) Pikal, M. J. J. Phys. Chem. 1971, 75, 3124-3134. (3) Weinga¨rtner, H. Private communication, Oct 1988. (4) Young, T. F.; Smith, M. B. J. Phys. Chem. 1954, 58, 716-724. (5) In an analogous study of density approximations,6 we had to distinguish between volume and mass concentrations, so we used some other names and symbols there. (6) Miller, D. G. J. Solution Chem., in press. (7) Miller, D. G. J. Phys. Chem. 1981, 85, 1137-1146. (8) Wirth, H. E.; Bangert, F. K. J. Phys. Chem. 1972, 76, 3491-3494. These authors have compared excess volumes of mixing for NaCl-MgCl2H2O in terms of mass concentrations equal to 4.0 mol kg-1, 4.0 equiv kg-1, and 4.0 ional kg-1, using the second way of comparing binary evaluation strategies. They found that constant equivalents had the lowest deviations, followed closely by constant ionic strength, with significantly larger deviations for constant molality. (9) Bianchi, H.; Corti, H. R.; Ferna´ndez-Prini, R. J. Solution Chem. 1989, 18, 485-491. (10) This international collaboration was begun at the suggestion of Harold Friedman at the 1984 Gordon Conference on Water and Aqueous Solutions. Most of the work has been published. See ref 11 for references to the various results. (11) Miller, D. G.; Albright, J. G.; Mathew, R.; Lee, C. M.; Rard, J. A.; Eppstein, L. B. J. Phys. Chem. 1993, 97, 3885-3899. (12) Miller, D. G. J. Phys. Chem. 1966, 70, 2639-2659. (13) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Albright, J. G. J. Phys. Chem. 1984, 88, 5739-5748. (14) Friedman, H. L. J. Chem. Phys. 1960, 32, 1351-1362. (15) Timmermann, E. O. Ber. Bunsenges. Phys. Chem. 1979, 83, 257263, 263-270. (16) Redlich, O.; Kister, A. T. Ind. Eng. Chem. 1948, 40, 341-345, 345-348.
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