Sodium also depresses the calcium reading (Table I). This effect varies little at low concentrations of sodium but is significant when comparing samples differing widely in sodium concentration. In particular, one can use the same set of standards to compare distal tubule fluid samples (35-75 meq Na+) to proximal tubule fluid samples (140 meq Na+), but it is necessary to use separate standard curves in order to compare the calcium concentration of proximal or distal tubule fluid to that of fluid obtained from the bend of Henle’s thin loop (ca. 200-700 meq Na+) (10). For measuring distal and proximal samples we use standards that contain 105mMNaCl. Samples containing concentrations of sodium and urea similar to that of fluid in proximal and distal renal tubules were compared to aqueous samples for their calcium content (each with an actual calcium concentration of 4 rneq). The mean photometer reading for “proximal” fluid was 95.90 i 3.0 SD (standard deviation) and that for the “distal” fluid was 96.60 + 2.17 SD. Physiological concentrations of glucose (6mM), K (4-25mM), Mg (0.5-25mM), SO, (1.54.5 mM), PO4(4-15mM), and HCOI (10-25mM) had no effect on the calcium determination. Any effect of chloride was minimized by the presence of approximately 100 meq C1 in the diluent and the similarity in chloride concentration between sample and appropriate standard. A set of 12 solutions was prepared containing 0.3 to 3.5 meq calcium in Tyrode’s solution. Aliquots of 5 nl were handled exactly as micropuncture specimens. The standard curve was linear between 0 and 4 meq/l. of calcium. The results are presented in Table 11. The mean percentage recovery was 99.5 + 3 . 9 5 z standard error; the mean error was 0.02 meq/l. ; the relative error, 1.4%. DISCUSSION
The higher boiling point for calcium necessitated several modifications of the helium-glow photometer. A well regulated and finely adjustable power supply was incorporated to heat the wire. Since there was a significant time lag between the burn-off of other compounds in the samples and the vola-
tilization of calcium, a delay timer was incorporated before the phototube integrator to eliminate variation in background emission. Despite these modifications, sodium and urea still had an effect on the readings. The urea problem is handled by addition of urea to the diluent. The sodium influence can be minimized by addition of sodium to the standard solutions. These results confirm Vurek’s findings (5) that quantities of calcium as low as 0.3pM can be measured with a reasonable degree of accuracy using the modified helium-glow photometer. [The helium-glow photometer constructed by American Instrument Company (Bulletin 2416) and by Clifton Technical Physics could presumably easily accommodate the modifications reported here.] This sensitivity is nearly a hundred times greater than the next most sensitive chemical method (6) and permits the measurement of calcium in samples obtained from a punctured renal tubule after only 1 to 2 minutes collection time. Although the electron probe also measures these small quantities (7, S), the helium-glow photometer is less expensive ($6500 for the Aminco helium-glow photometer, as opposed to about $80,000 for the electron probe made by Applied Research Labs, Sunland, Calif.) and is much simpler to operate (7,8). The lower limit for detection of calcium by the atomic absorption spectrophotometer is higher than the detection limit of the helium-glow photometer by approximately two orders of magnitude (11). ACKNOWLEDGMENI
We are greatly indebted to Gerald Vurek for his invaluable advice and for the loan of filters. We also wish to thank Oliver Wever for his advice.
RECEIVED for review December 9, 1971. Accepted March 7, 1972. This work was supported by American Heart Association Grant-in-Aid No. 68714. R. L. Jamison is a recipient of Research Career Development Award No. 5 KO4 HE 42685 and is a Markle Scholar in Academic Medicine. (1 1) “Analytical Methods for Atomic Absorption Spectropho-
(10) R. L. Jamison, Amer. J . Physiol., 218,46 (1970).
tometry,” Perkin-Elmer, Norwalk, Conn.
Determination of Composition of Mixtures of Diastereoisomeric Forms of Weak Acids by pH Titration Neil Purdie, Mason B. Tomson, and Gwendolyn K. Cook Department of Chemistry, Oklahoma State Unioersity, Stillwater, Okla. 74074
IN THE ANALYSIS of potentiometric titration curves of polymeric polybasic acids (e.g., polyacrylic acid) to characterize the ionization equilibrium, the molarity of the acid is defined in terms of the molecular weight of the monomeric repeating unit. Since the calculated ionization constants vary with the degree of ionization, results are usually reported in terms of two other defined ionization constants, the intrinsic constant, pKi, and the apparent constant, pK,,,. Their meaning can be found in standard reference works on the ionization of polymeric acids ( I ) . Because of the definition of the molarity (1) C. Tanford, “Advances in Protein Chemistry,” Vol. 17, Academic Press, New York, N. Y . , 1962,Chap. 2.
of a polymeric acid, pKt, or pK,,,, may be taken as equal to some average value for all ionizations within the macromolecule, and a comparison can then be made with the values for the analogous monomolecular derivative. For example, in this laboratory the ionization of ethylenemaleic acid copolymer (EMA) of molecular weight 20-30,000 was compared with the ionization of 2,3-dimethylsuccinic acid (DMSA). The variation in the ionization constant with degree of dissociation of a macromolecular polyacid is usually attributed to a monotonically increasing electrostatic work function, which must be overcome before protons can be progressively removed from the negatively charged macroion ( I ) . An almost equivalent variation in pK vias calculated from the titration ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1525
Table I. Thermodynamic Ionization Constants for Substituted Succinic Acids This work 1.21 x 10-4 (6) 1.12 x 10-4 (7) Ki 1.36 + 0.02 x 10-4 1.10 x 10-6(6) 6.4 X lO-7(7) 1.13 f 0.02 x 10-6 KP Kl 1.88 f 0.03 X 1.69 X (6) 1.70 x 10-4 (7) 5.3 x 10-6(6) 1.20 x 10-6 (7) K2 4.27 =t0.02 x 10-6 Ki 9.21 =k 0.03 x 10-4 9.0 X lO-4(6) 9.25 x 10-4 (8) 3.5 x 10-5 (6) 4.30 x 10-5 (8) KP 4.27 i 0.02 x 10-5 5.2 x 10-4(~ 6.9 x 10-4(9) Ki 5.89 i 0.01 x 10-4 1.2 X lO-5(6) 1.26 x 10-5 (9) K2 1.29 i. 0.01 x 10-5
Acid dl-DMSA meso-DMSA dl-Tartaric meso-Tartaric
KOH, ml 0 0.50
1 .oo 1.50 2.00 2.50 3.00 3.50
Emf, exper. 0.2492 0.2264 0.2070 0.1888 0.1694 0.1470 0.1235 0.1015 W E )
nuT =
1.023 x 10-*M;
Table 11. Example Calculation for 2,3-Dimethylsuccinic Acid Mixturee (45.8 % meso, curve ( e ) Figure 1) Emf (calculated), meso 0 20 40 60 80 0.2476 0,2466 0.2494 0,2503 0.2486 0.2222 0.2240 0.2256 0.2271 0.2285 0.2020 0.2040 0.2059 0.2078 0.2096 0.1832 0.1856 0.1878 0.1900 0.1921 0.1617 0.1652 0.1683 0.1711 0.1738 0.1333 0.1399 0.1453 0.1497 0.1536 0.1058 0.1140 0.1215 0,1279 0.1332 0,0850 0,0922 0.0998 0.1073 0.1142 0.0728 0.0403 0.0100 0.0175 0.0425 b = 0.1033M; vol.acid = 25.00ml; Nernst slope = 0.0589; E4.01 = 0.1852volt.
curve of a commercially available sample of DMSA, which turns out to be a mixture of the dl and meso diastereoisomers. Although the first and second ionizations are interfering for both isomers, this does not explain the observed variation and one is led to the conclusion that the successive ionization constants are weighted averages of the values for both forms. The accepted mechanism of an increasing electrostatic work function is not applicable in this simple case. The mechanism of polymerization of EMA is free radical addition (2) and it is conceivable that diastereoisomeric forms of the monomeric repeating unit could be produced. Their presence might also account, at least in part, for the variation of pK with degree of dissociation. The objective of this work was to develop an analytical method by which the composition of a mixture of closely related acids could be determined, with the further but more difficult objective in mind of applying it to a copolymer of the type of EMA. There is no restriction to its present application to weak acids other than its limitation to a two-component mixture. It could have a distinct advantage over methods which require separation of the isomers in that the analysis is done in situ and the possible interconversion, except by the action of strong base, is eliminated.
10.4 x 10-4 (10) 4.5 x 10-5 (IO)
Actual % 45.8 0.2488 0.2260 0.2065 0.1885 0.1691 0.1466 0.1234 0.1020 0.0028
100 0.2511 0.2299 0.2113 0.1941 0.1763 0.1570 0.1379 0.1201 0.0649
mp = 209-11 "C, which are in good agreement with the literature values of 129 and 209 "C, respectively (3). Titrations. Emf measurements were made using a Beckman research pH meter Model 1019 on a cell with liquid junction of the type Agl AgC1, 0.2N HCl/glass1solution under study1 Satd KCljSCE The solution was purged of carbon dioxide by a constant stream of nitrogen gas and maintained at a temperature of 25 i 0.01 "C. The emf scale was calibrated with Beckman standard buffer solutions of pH 4.01 and 9.18, respectively, before and after the titration. The total acid concentration was always around 10-2M. Experiments were conducted in the absence of inert electrolyte but there is no reason why the analysis could not be done in solutions of constant ionic strength. RESULTS Individual Ionization Constants. The ionization constants for the four pure isomeric forms were calculated by a computer analysis of the titration curves as described for adipic acid (4), by reiteration around the equations
EXPERIMENTAL Reagents. The acids used in this work were 2,3-dimethylsuccinic acid (Aldrich Chemical Co., Inc.) and dl- and mesotartaric acids (Aldrich Chemical Co., Inc.). All were reagent grade and were used without further purification, although rneso-tartaric acid was stored in a vacuum desiccator over silica gel for a prolonged period prior to its use. Separation of DMSA (mixture mp 1.180 "C) into its isomeric forms was accomplished by fractional crystallization from deionized water. The effectiveness of the separation was checked by determining the melting points, dl mp = 126-8 "C and meso
(2) S. Machi, T. Sakai, M. Gotoda, and T. Kagiya, J . Polym. Sci., A4,821 (1966). 1526
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
and
(3) R. P. Linstead and M. Whalley, J . Chem. Soc., 1954,3722. (4) D. Litchinsky, N. Purdie, M. B. Tomson, and W. D. White, ANAL.CHEM., 41,1726(1969).
or
I
I
L 2a-L [a(Hi
1 + /3{ Hi )I = CK2
I
I
40
20
I
1
I
I
I
80
60
(4)
where a and b are analytical acid and base concentrations, p is the ionic strength, [HS] and {Hi] are the hydrogen ion concentration and activity, respectively, y , is the activity coefficient of the ith ion calculated from the Davies equation (3, KOand y~ are both unity, and L = b [Hi] - [OH-]. To obtain Kl and Kz from Equation 4, two such equations are constructed using { H i ] values from two randomly chosen experimental points and solved simultaneously. The results are given in Table I where a comparison is made with other values taken from the literature (6-10). Agreement is good in all cases. Analysis of Mixtures. Mixtures of varying composition were prepared by volume from stock solutions of the pure acids and titrated with standard KOH. In the mathematical analysis for the composition of any one of these mixtures, potentiometric curves were calculated for a number of solutions of arbitrary composition which usually differed by intervals of 10% from pure dl to pure meso. The rationale for the analysis is that the best fit of a calculated curve to the experimental curve would be obtained for the correctly chosen analytical mixture of dl to meso acid. The criterion used for the best fit is that the sum of the absolute value of emf (experimental) minus emf (calculated) = Z(AE) for a number of points in the buffer region would be a minimum in a plot of Z(AE) cs. composition. This more quantitative criterion is preferred to the straightforward superposition of titration curves. The calculation of emf is essentially the reverse of the reiterative method employed to calculate the individual thermodynamic ionization constants with one important modification. Equations 1 through 4 must first be revised to include contributions from both acids. The terms involved in the revision are the respective acid concentrations and the ionization constants. These are distinguished by primes as are the coefficients /3 and u in which the terms appear. The revised forms of the equations are as follows :
+
UT
b
= u
+ a’
=
+
U[H~A] .’[(HzA)’]
(5)
+ [Hi] = [HA-] + [(HA’)-] + 2[A*-] + 2[(A’)2-l
+ [OH-]
(6)
and
Equation 8 is set up for a number of points in the buffer region by choosing b to correspond with actual experimental conditions and by substituting in the values of the appropriate ionization constants from Table I. Activity coefficients are calculated in the reiterative procedure by which convergence to ( 5 ) C. W. Davies, “Ion Association,” Butterworths, London, 1962. (6) J. F. J. Dippy, S. R. C. Hughes, and A. Rozanski, J . Cliem. SOC., 1959,2492. (7) R. Gane and C. K. Ingold, ibid.,1931,2158. (8) R. G. Baies,J. Res. Nut. Bur. Staid., 47,343 (1951). (9) H. Bode and K . Peterson, Ber., 71,871 (1938). (10) I. Jone‘ and F. G. Soper, J . Cl7em. SOC.,1934,1836.
Figure 1. Plot of Z(AQ (mv) us. % composition for DMSA (a) pure meso-
commercial sample, 78 ( e ) 45.8z meso(c)
I
-
I
20
I
I
zmesoI
40
I
60
(6) 83.5% meso-
( d ) 64 % meso( f ) pure dl-
I
I
I
80
% mesa-
.^
Figure 2. Plot of Z(AQ (mv) us. acid (a) pure meso( c ) 7 0 z meso( e ) 3 0 z meso-
yo composition for tartaric (b) 80% meso(d) 50% meso( f ) pure dl-
{Hi is achieved by a half interval search method. The calculation is repeated for the same points for all ratios of ala’. Data typical of the analyses form the content of Table 11. The results are shown graphically as Z(AE) os. composition for this and the remaining DMSA mixtures and tartaric acid mixtures in Figures 1 and 2, respectively. Included for comparison are the results for solutions of the pure acids alone calculated by the same method. In the tartaric acid case, all solutions were prepared from stock solutions of the pure isomers and the estimated composition is within + 1% of the actual composition in all cases. To the 50% mixture a known weight of pure dl isomer was added to change the composition to 65 % dl; the new minimum in the Z(AE) os. composition plot again agreed to better than 1 %. For DMSA a further test was possible in that the commercially available sample could be analyzed (Figure 1 , curve c . ) . This mixture is 78 f 1 meso, a result which is contrary to recent conclusions about the relative stabilities of the two ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
e
1527
E t volt)
-. 1800 -.I 700
% meso10
20
Figure 3. Plot of emf (volt) 6s.
composition for tartaric acid
The solid curve is the calculated calibration curve at 75 neutralization. The open circles are the experimental emf values for the mixtures listed in Figure 2, also at 75 neutralization.
isomers (11, 12), although it is consistent with earlier reports (3, 13). However, the details of the conditions of the preparation are not available to us and the result could be misleading. In the event that a high speed computer is not available, results of comparable accuracy (+2 %) can be obtained from the consideration of only one point in the buffer region. Without the computer facility, it is imperative to work at constant total acid and a fixed standard base concentration, and the pure acids must have been separated so that their respective titration curves are available. In this way a calibration curve can be constructed from experimental data taken at a predetermined degree of neutralization and the composition of the unknown mixture obtained by interpolation. For greatest accuracy, the point of interest should correspond with the greatest difference in emf between the two pure forms. This is illustrated for tartaric acid at threefourths neutralization in Figure 3. (In our case the calibration curve was obtained by calculation and as such the comparison of the results with the more protracted analysis is more objective.) DISCUSSION
The procedure is applicable to a mixture of any two weak acids, and presumably weak bases. Extension to a mixture of more than two constituents has not been attempted. All (11) P. J. Sniegoski,J. Org. C/zem.,36,2200(1971). (12) L. Eberson, Acta Chem. Scand., 13,203 (1959). (13) D. H. R. Barton and R. C. Cookson, Quart. Reo., 10, 44 (1956).
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
that is required, which is different, for work in constant ionic strength media, is that the individual ionization constants be determined under the chosen conditions if they are not already known. In the present case, almost all of the previously published ionization constants would have sufficed for the accuracy obtained in the analysis. In contrast, if the second method is used, it is not even required that the ionization constants be known, but only the molecular weight. The examples chosen to illustrate the method are representative of the most restrictive condition in that K I , ~ ~=/ K ~ , ~ 0.73 and 1.57, and K2,dl/K2,meso = 0.26 and 3.31 for DMSA and tartaric acid, respectively. For ratios larger than these, the individual ionization constants would not have to be known with as much precision to obtain an equivalent accuracy. In the limit, the error in the analysis is determined by the magnitude of emfdl minus emfmesofor equimolar concentrations of the pure acids and the relative accuracy capability of the potentiometer used. For tartaric acid the maximum difference (ca. 30 mV) occurs at three-fourths neutralization. With a capability of measuring the emf to 0.1 mV, the theoretical limit to our analysis was on the order of 1 part in 300, although in practice this was not achieved, Results from the analysis of the isomeric composition of EMA are at best inconclusive. Two principal difficulties should be self-evident. Primarily the present analysis requires a prior knowledge of the individual ionization constants of the monomeric acids within the macromolecular framework. There is not a unique solution to this problem using the procedure described although it is conceptually possible to solve the problem when neither ionization constants nor composition is known. In reality, however, this is a much more difficult problem. Furthermore, if differences in ionization constants and an electrostatic work function are both operating to produce the variation in pK with degree of ionization, there is no apparent way to partition the effect between the two causes. We could only consider the condition of zero contribution from the electrostatic work term. Nevertheless, from the work on DMSA, the indication is that the complicating feature of differences in ionization parameters of diastereoisomeric forms is real and should be recognized as an important contribution to nonconstant pK values in polyacid ionizations. We intend to continue the study of DMSA and tartaric acid ionizations as a function of temperature to obtain A H and AS" of ionization in an attempt to better understand the reasons for the ionization differences. RECEIVED for review January 1 1 , 1972. Accepted March 17, 1972. The work described is abstracted from the doctoral dissertation of MBT, Oklahoma State University, May 1972. Copies of the computer programs and directions for their use are attached as appendices and will be available from University Microfilms. We wish to express our appreciation for the financial assistance given to this project and to MBT, respectively, by the Research Foundation and the Graduate College, Oklahoma State University.
