a007
Anal. Chem. 1882, 64, 3007-3012
Two-Dimensional Conductometric Detection in Ion Chromatography, Postsuppressor Conversion of Eluite Acids to a Salt Ingemar Berglund and Purnendu K. Dasgupta' Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061
A rbnuitaneour catWanion-exchangescheme k proposed where the effluent eluite acids from a suppressed ankn chromatography system proceed through the annular channel of a dual membrane converter device composed of an anion and a catlon-exchange membrane tube. The optimum regenerantsconsist of LIF doped with LiOH for the cation rlde and LIF doped with HF (or NH4HFz) for the anion side. Sensnivtty to very weak aclds is far better than detection in the suppressed mode and Is a160 improved markedly over a previouslydescribed system involvingsequential lonexchange to NaOH. A modelIspresentedfor the procem, and qualitative agreement to the experimentaldata Is established. The twodimendonal information permits the estimation of the pK. of unknown eiuites.
and later refined by Renn and S y n o ~ e c .In ~ terms of the specific strategy, it was clear, however, that improvements are necessary. The fundamental limitation of the technique is that for the crucial HX MX (M = Na, K, etc.) exchange step to proceed at an appreciable rate, HX should be dissociated to the maximum extent permissible by ita pK,. Unfortunately, the nature of the exchange system is such that as soon as some HX is exchanged to form NaX, the NaX-HX buffer system thus formed serves only to depress the further ionization of HX. As a result, the conversion efficiency decreases undesirably fast with increasing pK,. Should it be possible to more directly convert HX into a pH neutral salt (e.g., NaCl),this limitation would not exist (albeit admittedly the fundamental limitation posed by the pK, of HX cannot be overcome). The purpose of the present paper is to describe how HX may be converted to an alkali-metal halide salt and to elucidate the nature of the information available from deployment of such a technique in the context of detection in ion chromatography.
-
Suppressed conductometric anion chromatography has proven to be the analytical technique of choice for the determination of strong and moderately weak acid anions. However, weak and very weak acids are poorly ionized in the suppressed effluent. Consequently, the detection of eluite acids with pK, values of 1 5 is difficult and those with pK, values of 17 are virtually undetectable. Even in the PKa range 3-5, an important region that covers most carboxylic acids, the detector response is highly nonlinear with increasing concentration. As a result, quantitation errors become significant at higher levels for such analytes. In the previous paper in this series,l we therefore attempted the conversion of eluite HX to a base, NaOH. Earlier, in the context of ion exclusion chromatography, Tanaka and Fritz2 showed how a very weak acid (HX) can be converted into its potassium salt KX and thence to KOH. T w o sequential ion-exchange steps are necessary-in the first, a K+-formcation-exchange resin performs the crucial step of H+ K+ exchange and then the X- is exchanged for OH- by an OH--form anion-exchange resin. We had followed the same strategy, except in a membrane-based format. It was shown that the detection of the eluite HX and the converted NaOH by two independent conductometricdetectors provides valuable two-dimensional information. This can be processed to estimate the PKa value or the approximate concentration of an unknown eluite. In favorable cases, where the dispersion generated in the convertercanbe accountedfor, it waa also possible to generate ratiograms of the two detector outputs, as an indication of the purity of the eluite peak. The above study' represented a further step in generating two-dimensional information output in ion chromatography, extending the original scope and applicability of the broad general class of such techniques, pioneered by Wilson et al.3
It is obvious that if a two-step exchange HX MX MOH is limited by suppression of HX dissociation in the first step, it is pointless to consider similar sequential steps to form an alkali-metal halide from HX. However, one may consider a conversion step using a mixed-bed resin, containing, e.g., Na+-form cation-exchange and Cl--form anion-exchange resin. Cation and anion exchange will then occur virtually simultaneously and HX would be transformed to NaCl essentially in a single step without significant accumulation of NaX that can lead to the suppression of HX dissociation. Although conceptually this is straightforward, early on we experimentally discovered that the deployment of a mixedbed resin column in the desired ionic form results in little or no effluent signal from an input pulse containing a weak acid HX. Most likely, partition of the unionized acid to the interior of the resin (cf. ion exclusion chromatography9 broadens the peak too much for such a system to be practical. Nevertheless, it should be possible to implement the simultaneous ionexchange concept in a membrane-basedanalog of the mixedbed resin column. Such a system,in which an anion-exchange membrane tube is inserted within a cation-exchange membrane tube, has been previously describeda as the membrane counterpart of a mixed-bed deionization system. In the present application, the fluid stream of interest is to flow in between the two membranes, with appropriate electrolyte solutions flowing outside the cation-exchange membrane and imide the anion-exchange membrane to supply the replacement cation and anions, respectively. To our knowledge,
(1)Berglund, I.; Daagupta, P. K. Anal. Chem. 1991,63, 2175-2183. (2)Tanaka, K.;Fritz, J. S. Anal. Chem. 1987,59, 708-712. (3)Wileon, S.A.; Yeung, E. S.; Bobbit, D. R. Anal. Chem. 1984,56, 1457-1460.
(4)Renn, C. N.;Synovec, R. E. Anal. Chem. 1989,61, 1387-1393. (5)Smell, H.Zon Chromatography; Plenum Press, New York, 1990; pp 127-132. (6) Dasgupta, P. K.; Bligh, R. Q.; Mercurio, M. A. Anal. Chem. 1985, 57,484-489.
-
PRINCIPLES
0 1982 American Chemlcal Society
- -
3008
ANALYTICAL CHEMISTRY, VOL. 64, NO. 23, DECEMBER 1, 1992
PT
except for the well-known example of deionization, simultaneous catiodanion exchange has never been explored for analytical applications, whether in a membrane-based format or otherwise. We can consider such a membrane-based ion-exchange system as a multiplate device' and assume that at each plate a fraction f of the ions H+ and X- is exchanged. If C is the concentration of HX with a dissociation constant of Ka, the fraction of C that is ionized, al, is simply
t I
a1= Ka/(K,+ [H'I)
(1) where [H+l is given by, unless C and K, are both very low
+
F
P
Flguro 1. Detalls of the converter device: N, Naflon tube,A, anionexchange membrane tube; J, Jackettube; I, Inletloutlet connectkn to jacket; M, male fitting; F; ferrule; P, PEEK Insert; U, 10-32 threaded female unianwlth drilled and threadedhde T; PT, threaded PEEKtubhg; ST, syllnse needletuMnginm%n, Tefbnhrbe;PVC, poly(vinylchkrlde)
+
[H'] = (-Ka (K: 4KaC)"2)/2 (2) The equivalent concentration exchanged a t the first plate ( C J ) is therefore
tublng.
Ce,1 = Calf (3) and the remaining concentration exiting the first plate (Co,l) is given by Co,l = C - Caj (4)
It is readily apparent that the equivalent concentration exchanged a t the ith plate is given by C , i = Co,i-laJ
(5)
where
c:,,= c ~ ~--ad ~(I (6) Combination of eqs 4 and 5 leads to the final outlet concentration after n plates: C,,= C ( 1 - alnn
(7)
Note that each plate muat have some absolute maximum capacity, due to membrane transport limitations, as to how much concentration it can exchange under any given experimental conditions. Cescan never exceed this value. The total exchanged concentration Ce is therefore given by
C, = C - C,,= C(1- ( 1 - a d " )
(8)
the overall conversion efficiency E being defined by
E = C$C = 1 - (1- a d "
(9)
Interpretation of experimental data on the basis of the above is complicated by the fact that a chromatographic eluite band represents an ensemble of fluid elements containing a continuous range of concentrations of HX. As a f i t approximation, in interpreting experimental data we use a concentration of C* for C where C* is the average concentration in the eluite band, obtained from
c* = 2cbj v b j / (vT,sup + vT,co,,)
(10) where C* and V a are the concentration and volumes of the are the triangulated injected sample and V T , and ~ ~ ~ band volumes at the suppressed and converted detectors, respectively. Experimental overall conversionefficienciescan be computed from the observed peak area, known mobilities of the replacement ions, and detector calibration.
EXPERIMENTAL SECTION Converter Device. The dual-membraneion-exchange device was fabricated by inserting a radiation-grafted poly(ethy1vinylacetate)-based anion-exchange membrane tube* (CFS-1 refii fiber, Dionex Corp., Sunnyvale, CA) inside a Ndion tube (type 811X, Perma-Pure Products, Toms River, NJ). Referring to (7) Dasgupta, P.K.Anal. Chem. 1984,56, 96-103. (8)Dasgupta, P.K. InZon Chromatography;Tarter,J. G.,Eds.;MarcelDekker, New York, 1987; pp 220-224.
-
02 0
40
KO
Regeneront Conceitrotion m N
80
20 A
1 Segeoeron:
60
Concentration
80
5
mv
Flguro 2. (a)Rackgroundconductance and (b) responcw,(areacounts) to 1.5 mMborate as aamplefordmerent sake replace" ebcWyte8 as a functbn of electrotyte concentratlon In the converter dew.
Figure 1, the Ndion tube N was swelled in hot ethanol and the anex tube A was inserted within it in the swelled condition. Polyether ether ketone (PEEK) tubing (1.6-mm o.d., 1.0-mm i.d., Upchurch Scientific, Oak Harbor, WA) segments P were inserted at either end of N to secure leak-free connections. The assembly was connected to a 10-32 threaded PEEK union U (Dionex) with a male fitting M and ferrule F. The union was converted into a tee by drilling a hole into the central partition and providing it with 1-72 threads T to which appropriately threaded PEEK tubing PT could be directly ~ ~ n n ~ t Ae d . ~ polypropylenejacket tube J was connected to the fitting U by hotmelt adhesive and provided within inlet/outlet aperture I. The end of the tubing A protruding through U is connected by a Teflon tubing 'IT acting as a sleeve, a syringe needle tubing segment ST acting as an insert and a male nut and ferrule. The electrolyte containing the replacement anion is pumped through 'I",and the effluent from the Suppressed detector is brought into the converter through PT. The electrolyte containing the replacement cation flows through I. The active length of the converter was 43 cm. Chromatography. The chromatographicpump was a Beckman llOA, followed by a 4.6- X 250-mm column packed with unfunctionalized poly(styrenediviny1bemne) particles (Hamilton Co., Reno, NV) functioning as a pulee dampener. A flow rate of 1 mL/min was used throughout. An electropneumatidy driven dual-stack slider valve (Dionex Corp., Sunnyvale, CA) equipped with a 25-pL loop was used for sample injection. Separations were carried out on a Dionex IonPac AS6A-6fi 100X 4.6" column using an NaOH eluent. The suppressor used was a f i i e n t - f i i e d helical tubular Ndion device (400pm i.d. tube, Ndion 020, Perma-Pure) externally resin-packed, and a c& 20 mM dodecylbenzenesulfonic acid solution (Bio-Soft S-100, Stepan Chemical Co., NortWield, IL)was used as regenerant. After the suppressor, one Model 213 conductivity detector (Wescan Instruments, Santa Clara, CA) served as the suppressed signal detector followed by the converter and then by a second (9) Morris, K.;Dasg~pta,P. K. LC-GC 1992,10, 149.
ANALYTICAL CHEMISTRY, VOL. 64, NO. 23, DECEMBER 1, 1992 800R'
"
'
'
'
'
"
'
'
'
'
'
'
"
"
'
'
6 0;
304
3009
'
22.5
U
Chloride
Bromide Propionate Sulfide 1 ox
7oa
Borate 1o x
m
m
Bromide
PI VI
e I 525 1 B
a !I
! a
350
PI
2
175
Y
m
PI
PI
a Figure 3. Response behavior with different regenerants in the converter device. 10 mM LiF constituted the base case (A). The other regenerant electrotyte solutions contained in addition on the anion and cation sue, respectively (B) 0 mM HF, 1 mM LIOH; (C) 1 mM HF, 1 mM LIOH; (D) 2 mM HF, 2 mM LiOH; (E) 0 mM HF, 10 mM LIOH; and (F) 10 mM HF, 10 mM LiOH. (a) Response to different anions (1 mN) with regenerants A and D. Note that the response is 1OX magnified for propionate and borate. (b) Backgroundconductance observed with the different regenerants. (c, d) Response to 1 mM bromide and borate, respectively, with the different regenerants.
identical detector for measuring the converted signal. In the converter, except as stated, a 10 mM LiF + 1mM NHdF-HFand a 10 mM LiF + 2 mM LiOH solution were respectively used as the anion- and cation-exchange electrolytes. Both liquids flowed in the same direction, countercurrent to the principal flowstream of interest, at flow rates of 1-2 mL/min. Sodium hydroxide eluent solutions were prepared from 50% stock solution (Fisher Scientific). To prevent COn intrusion, a soda-lime trap was installed. Analyte solutions were made from the corresponding alkali-metal salts or occasionally from the corresponding acids. All reagents were of reagent grade. Deionized water (specificresistance 2 17 M k m ) was used throughout for the preparation of eluent and sample solutions.
RESULTS AND DISCUSSION Converter Device. The residence volume and the dispersion induced by the device for a 25-pL injected band was measured by the usual procedurelo to be 265 and 140 pL, respectively. This dispersion is larger than the previous NaOH sequential exchanger device.' This undesirable performance largely results from the linear configuration. Unfortunately, the device could not be coiled into a helical configuration (attempts to thermoset any formed shape resulted in excessive swelling of the anion-exchange tube). Neither could it be woven into the serpentine-I1 pattern;ll the deformation of the anion-exchange tube greatly increased the resistance to flow through it. Conceivably, a lower dispersion converter, performing simultaneous anion/cationexchange can be made with sheet membranes based on the design by Stillian.12 However, initial efforts to achieve this were not successful with sheet membranes available to us. (10) Dasgupta, P.K. Anal. Chem. 1984,56,103-105. (11) Curtis, M. A.; Shahwan, G. J. LC-GC 1988,6,158-164. (12) Stillian, J. R. LC Mag. 1985,3,802-812.
Choice of Cation-and Anion-ExchangerElectrolytes. The choice of the regenerant electrolytes containing the replacement ions is important in governing both the extent of ion exchange attained and the background conductance, since some counterion penetration through the membranes is inevitable. To attain a greater extent of exchange, we reasoned that the replacement ions should not have too large an affinity for the membrane ion-exchange sites. As such, we chose to experiment with Li+ and Na+ as the replacement cations and F and C1- as the replacement anions. Resulta are shown in Figure 2 using NaF, LiF, LiC1, or NaCl as the regenerant electrolyte in 25-75 mM concentration. The same solution was used as the ion replacement regenerant for either the cation- or the anion-exchange membrane. The lowest background conductances were observed with LiF (Figure Za), presumably because Li+ and F- have lower equivalent conductances than Na+ and C1-, respectively. Otherwise, the background conductance increases monotonically with electrolyte concentration in all cases. Using borate, a representative very poorly ionized eluite acid as the test solute, the analyte response was measured as a function of the regenerant concentration for each of the four regenerants (Figure 2b). In each case, the response initially increases with increasing regenerant concentration. For the fluoride-basedregenerants (where the solution is alkaline) we believe that the primary contributor to regenerant penetration is the alkali-metal hydroxide. In this case, ionization of the eluite weak acid is assisted by regenerant penetration; analyte response essentially reaches a plateau value a t higher regenerant concentrations. With the alkali-metal chloride regeneranta on the other hand, response decreases sharply a t higher regenerant concentrations where the penetrated regenerant ions compete increasingly for the available exchange sites. If the signal/
3010
ANALYTICAL CHEMISTRY, VOL. 64, NO. 23, DECEMBER 1, 1992
Table I. Linearity of Response for Different Analytes. suppressed system
pK,
anal*
linear r2
LiF conversion system
NaOH conversion system
,
% std error ?& std error % std error uncertainty of estimation, uncertainty of estimation, uncertainty of estimation, of slopeb mMc linear9 ofslopeb mMc linear9 of slopeb mMC ~
chloride nitrate sulfamate chloroacetate fumarate azide acetate propionate arsenite silicate sulfate oxalate selenite succinate
Sd
S 0.99 2.86 3.02 4.72 4.76 4.87 9.16 9.77 S,1.96 1.27,4.27 2.64,8.27 4.21,5.64
0.9962 0.9991 0.9989 0.9968 0.9849 0.9373 0.9653 0.9441 0.9951 0.9937 0.9898 0.9824
2.7 1.7 1.9 3.3 7.1 11.6 8.5 10.9 insensitivef insensitive 4.0 4.6 5.9 7.7
0.07
0.03 0.04 0.06 0.13 0.31 0.22 0.29 0.08 0.09 0.11 0.15
0.9999 0.9996
0.4 1.4 nae na na
0.02 0.06
0.9982 0.9987 0.9860
2.5 2.5 6.9 na 1.8 3.0 na 7.5
0.05 0.10 0.25
0.9993 0.9973 0.9888
0.996 0.995 0.9979 0.9984 0.9958 0.9924 0.9996 0.9982 0.9833 0.9835 0.9977 0.9959 0.9931 0.9927
0.07 0.10 0.08
0.9 1.2 2.6 2.3 3.7 3.9 0.85 1.9 7.5 9.2 2.8 3.7 4.8 5.0
0.02 0.02 0.05 0.04 0.07 0.10 0.02 0.05 0.14 0.18 0.05 0.07 0.09 0.09
a Over 0-3 mM concentration range, n = 5, linear fit of the area response in units of volt seconds vs the injected concentration is considered. The percent uncertainty of the linear slope is the uncertainty of the slope divided by the mean value of the slope times 100. Standard error resulting from the use of a linear regression equation for quantitation; see, e.g., ref 14 or any other standard text on statistics. Strong acid, fully ionized. e No experimental data are available. f No reliable measurement is possible.
background ratio rather than the absolute value of the signal is considered, LiF yields the best results virtually at any concentration. Further experiments were then limited to tailoring the LiF regenerant for obtaining maximum conversion of very weak acids. It was reasoned that the exchange of the eluite H+for Li+ at the cation-exchangemembrane will be facilitated by making the cation replacement electrolyte alkaline such that the H+-OH-neutralization reaction will be an additional driving force. Similarly, it was thought that the anion regenerant electrolyte could be made acidic to favor the anion-exchange process. In each case, both replacement electrolytescontained 10 mM LiF (base case, A). The other regenerant electrolyte solutions contained in addition on the anion and cation side, respectively. (B) 0 mM HF, 1mM LiOH; (C) 1mM HF, 1 mM LiOH; (D) 2 mM HF, 2 mM LiOH; (E)0 mM HF, 10mM LiOH; and (F) 10 mM HF, 10 mM LiOH. The results are shown in Figure 3. Figure 3a shows the data for a number of eluite acids with different pKa values. Regenerants A-D give approximately the same results for all test solutes except borate. In the case of borate, a very weak acid anion, the formulation of the acidic/basic anion/cation regenerants obviously makes a dramatic difference over the unmodified salt. While the addition of acid and base to the anion and cation replacement electrolytes is beneficial for improving the conversion of weak acids, there are limits to which this can be done. If added acid or base concentrations are too large, forbidden ion penetration through the membrane13 occurs to such a degree that the background conductance increases undesirably (casesE and F, Figure 3b). Partsc and d of Figure 3, respectively, show the response to a strong acid anion, bromide, and a very weak acid anion, borate, for the six different replacement electrolyte situations (A-F). High backgrounds, such as with solution F, completely mask any low-level signals resulting from the conversion of the eluite. With solution E, the background conductance is due to the significant presence of LiOH in the effluent. Consequently, when a weak acid such as boric acid elutes and is superimposed on this LiOH concentration, a large negative signal,resulting from the replacement of hydroxide by borate, (13) Dasgupta, P. K.; Bligh, R. Q.;Lee, J.; D'Agostino, V. Anal. Chem. 1985,57,253-257.
(14)Weatherburn, C. E. A First Course in Mathematical Statistics; English Language Book Society Edition, Cambridge University Press: Cambridge, U.K., 1961; pp 69-74.
Chloride, Nitrate
0.90
o s u lfate dichloroacetate malonate
0.70 Osuccinate propionate0
0.50
0.30
0.10
1
L silicate n i t e
I 1 ' 1 1 [ ( ' I1 ' 1 1
0.00
~
0.20
"
~
0.40
"
~
~
0.60
~
"
~
0.80
~
"
~
1.00
~
"
"
1.20
Observed Conversion Efficiency Flgure 4. Observedvs predictedconversion efficiency. The predictions are based on a best fit to eq 9, as amended by eq 10.
100.0
.-
75.0
0 v)
a
s
50.0
Y C
a
g a e
25.0
0 SulfaIe
Chloiitle
hiiinle
O d n i e Maionate Sutciiiale
Pio~iioiiale Atelnie
r\rsPiiiie
Flgure 5. Comparisons of the conversion efficiency observed for the NaOH vs LiF conversion systems for various analytes.
is observed (Figure3d). In the case of solution F, simultaneous penetration of HF does not allowthe pH of the internal stream to drop as much as with solution E and, consequently, significant ionization of boric acid does not occur. On the
~
'
"
~
'
ANALYTICAL CHEMISTRY, VOL. 64, NO. 23, DECEMBER 1, 1992
basis of these results, the regenerant electrolyte represented by D was used in all further experimentsexceptthat NH$.HF was substituted for HF. The latter substitution avoided the need to handle HF solutions directly. No difference in performance from such a substitution was observed. Exchange Efficiency and Linearity of Reeponw. For relatively strong acids that are fully ionized, 80-100% of the eluite acid was converted to LiF. For such anal*, conversion efficiencywas independent of the concentxation. For example, the observedconverted to suppressed signal (peak area) ratio was 0.2672 f 0.0016 and 0.2266 f 0.0018 for chloride and sulfamate, respectively, over a 0-3 mM concentration range. Statistical data pertinent to the linearity of response behavior are shown in Table I for a variety for analytes, in the suppressed mode, after conversion to NaOH using the sequential membrane converter' and after conversion to LiF using the present system. The most important parameter, the standard error of estimation that resulta if the linear regression equation is used for quantitation, is uniformly lower in the LiF conversion system for weak acids. The improvementa for monoprotic carboxylic acids over the Suppressed system are remarkable; of course, very weak acid anions such as arsenite and silicate cannot be determined at all by the suppressed system. The observed overall conversion efficiency for a number of analyte acids was fit to the model represented by eq 10 with C being replaced by C* in computing [H+l (eq 2). The number of plates, n,was assumed to be 430 (plate height = 1 mm). As a crude approximation, the average of the two PKa values was used for diprotic acids. The results are shown in Figure 4. Given the very approximate nature of this model (e.g., no account is taken of the fact that sulfate will be much more efficiently exchanged than dichloroacetate by the anionexchange membrane due to the much higher affinity of the former ion for the ion-exchange sites), it would appear that the model can semiquantitatively predict the observed behavior. The resulta shown in Figure 4 are based on a best fit value off = 4.58 X (i.e. of 0.46% conversion at each plate) obtained by a nonlinear least squares minimization routine based on a Marquardt-Levenberg algorithm.ls For the plot shown, the linear correlation coefficient is 0.9663 and the slope is 1.04 f 0.08. Figure 5 compares the overall conversion efficiencies for the NaOH and LiF conversion systems. For strong acids, the NaOH system is equal to or better than the LiF system but for the weak acids, LiF conversion is better. For the very weak acid anion arsenite, a 6-fold improvement is observed. The reason for the significantly better conversion of acetate in the LiF vs the NaOH conversion system relative to the other carboxylic acids may lie in the very weak affinity of this ion for anion-exchange sites. With respect to a performance comparison with the NaOH conversion system, although the limits of detection were not explicitly determined, the improvement of S/N at low concentrations appears to be linearly related to the improvement in conversion efficiency. However, because the equivalent conductance of LiF is significantly lower than that of NaOH, the overall gain is sacrificed some. For very weak acids, an improvement by a factor of 2-3 should be expected. Estimation of the pK. Value of a n Unknown Eluite. Although the conversion efficiency is a function of the pK, of the eluite acid and the latter can be estimated from the former, it is not possible to determine the conversion efficiency for an eluite band in a sample under test because the concentration is unknown as well. However, the ratio of the suppressed/converted signal shows a clear dependence on
SO11
i (u
2.0 --
0
hy
1.0
--
cn
ar L a a 0.0
$
7
0.0
2.0
4.0
8.0
6.0
10.0
pKa Fburo 6. Dependence of the suppressed to converted signal area ratk upon analyte p&. The sdld line is a best flt to eq 11.
p"-1.00
1.50
I 1\ 1.00
1.50
2.d0
2.50
Suppressed
2.50
2.00
Time, min Figure 7. Overtapped chromatogram of chkr#e and glyoxylate. The converted peak was corrected for dispersion, and the ratlo was then Computed.
PKa, decreasingwith increasing pK, (i.e., the detection in the converted mode becomes increasingly attractive for weaker acids). This suppreseed/converted signal ratio, R , is shown in Figure 6 as a function of PKa with the best fit line shown for an equation of the type
+
R = R*ATK/(ATK B ) (11) .where R* is the ratio observed for strong acids and A and B are constants. Note that eq 11 is merely a general representation of speciesdistribution functions ('a-fractions") used in ionic equilibria. For example, if A = 10 and B = [H+], the multiplier of R* in eq 11 is the fraction of a weak acid HX (15) MINSQ. Nonlinear Curve Fitting and Model Development,Ver. 3.1, Micromath Scientific Software, Salt Lake City, UT, ISSO.
3012
ANALYTICAL CHEMISTRY, VOL. 84, NO. 23,MCEMBER 1, 1902
that exists in the X-form at a given H+concentration. In generating Figure 6, we arbitrarily assumed the pK, of strong acids to be zero. For diprotic acids, denoted in Figure 6 with asterisks, with the exception of selenite, the two pK, values are
