Ho-Leung Fung and Lawrence Cheng School of Pharmacy SUNY at Buffalo Buffalo, New York 14214
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Linear Plots in the Determination of Microscopic Dissociation Constants
Many compounds of biological interest, e.g., catecholamines, amino acids, etc., possess more than one dissociahle proton. In order to calculate the concentration of each possible ionization state at a particular pH, it is necessary to obtain accurate estimates of the microscopic ionization constants (micro-constants) involved. The method developed by Edsall et al.,l is often used in such determinations. For most compounds, the Edsall method is found to give accurate results. However, as will he shown later, for compounds whose micro-constants are in certain unfavorable ratios, application of the Edsall method gives rise to large uncertainties. In these cases, it is instructive to see whether i t is possible to obtain improved values through manipulation of the equations and plotting method without resorting to computer techniques. In this report, we describe linearized plots of Edsall's equation which, in principle, will allow for the unambiguous determination of micro-constants. The application of this plotting technique to experimental data of L-cysteine ethyl ester and simulated values of dopamine is shown. The merits and limitations of these new plotting procedures are then discussed. Materials presented in this report may be incorporated into a biochemistry course while discussing the physical properties of amino acids, or into a physical chemistly course in which solution equilihria and pK, measurements are discussed. The experiment described here also gives the students an appreciation of the application and limitations of data treatment using graphical techniques.
are related to the micro-constants as shown in eqns. (2-4)
and
Edsall et al.,' have developed equations to calculate values of the micro-constants from spectral data when the macro-constants are also unknown. In their derivation, it is assumed that species (+-) and (0-) both have the same molar absorptivity, different from the ahsorptivities of (+0) and (00). The absorptivity of each of the latter two species is assumed to he equal, and ordinarily assumed to be zero. They then define an experimentally determinable quantity a, the fraction of all the chromophoric groups which are ionized (viz., species ( + -) and (0-)) a t a narticular DH. In the case of an aminouhenol, the chrokophoric is the phenol. Thus
Background
The dissociation of a dihasic acid, e.g., an aminophenoL2 may be described by equilibria shown below.
+
HN
-0(+ -)
Usingeqn. (11, it can be shown that a =
*.*
k, [Htl
[HT+ (k, + k,) [ P I + k,k,,,
Another function, M is defined such that
or pM = pH - log - = 1-0 Q
where each molecular species may be readily abbreviated by the charges residing on the amino nitrogen and phenolic oxygen atoms, respectively. Thus, (+ -) represents the zwitterion, (00) represents the completely neutral molecule, etc. Each of the four dissociation steps is defined by the appropriate micro-constant, i.e.
The two macroscopic dissociation constants (macroconstants), KI and Kz, can be ohtained by titration and 'Edsall, J. T., Martin, R. B., and Hallingwotth, B. R., Ploc. Not. Acnd. Sci., 44,505 (1958). 2Martin,R.B., J. Phys. Chern., 75,2657 (1971). 108
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+ k,k*,i
-
-
- log k,[H7 + %,, [H+1 + k,
It can be seen from eqn. (7) that when [H+] is very high, a kl/[H+] 0. In such a limiting case, eqn. (9) can he simplified to p M = pkl because kl/[H+] >> k ~ k and ~ , [H+] ~ >> kz. On the other hand, when [H+] is very low, a will approach to unity. In this limiting case, then, eqn. (9) becomes p M = pkz.1 because now kzkz.1 >> k~ [H+] and k2 >> [H+l. Thus, the values of pkl and pk2.1 can he ohtained by extrapolation in a p M versus u plot. Once these two constants are known, it is possible to solve for k2: for example, when a = 0.5, p M = pH (i.e., M = [H+]); and from eqn. (a), it can he shown that
Table 1. Ratios k l l k and ~ k z . ~ / k lof Various Compounds
T ~ m i n e ~ tsstep h~l Cyrbinylglyeineo Cy.teine ethyl Tyrosine* Dopab
cyrteine0 Isoprnpylcpinephrine" p-Tyramine" Norepinaph"ea Epinephrinea Dopamineb
'From footnote 1. " Fmm faotnotez.
Values of KI, Kz and h1.z are now easily calculated from eqns. (2-4). The shape of the p M versus a plot is completely determined by the ratios h z , l / k ~and kllhz. For a series of amino-phenols and amino-~atechols,~ the ratio hz,l/hl was found to be relatively constant at about 0.5. The ratio hl/hz, however, varied tremendously, ranging from 0.008 for tvrosine ethvl ester to 15 for dooamine (Table 1). Fieure i shows a f;lmily of curves of (I;M - pk;) versus a f& different arbitrary ratios of kl/hz, using a value of 0.5 for kz,l/hl. It is apparent that when kl/hz > 1, ph, can he determined accurately from the graph, but extrapolation to p k z ~is extremely uncertain. In these cases, uncertainties in the values of other micro-constants and macro-constants will follow, since they are calculated from both h l and kz.1. Edsalll estimated that this problem will arise when 5 < kilkz < 0.2 for compounds whose h z . l / h ~ratio is 0.1. For compounds with lower hz.l/hl ratios, such as are found in the cysteine derivatives (Table l ) , this unfavorable range for kl/kz will become smaller. This is apparent on comparing Figure 1 with Figure 2 in which values of 0.5 and 0.02 for kz.l/kl are used, respectively, to construct the family of curves of (pM - pk,) versus a. When hz,l/hl is small (Fig. 2), extrapolation at the steep end of the curves becomes increasingly less accurate because the curves are more precipitous. Thus, on examination of Table 1, it is evident that for a large number of compounds the Edsall plotting method will not be accurate.
0(
Figure 1. Modified pM versus u plot for different arbitrary ratios of kl/kz, which are reoresenled bv numbers attached to the curves. These curves areobtained by assuming k2,,/kl = 0.5.
Linear Plots
As was shown earlier, when h ~ / k zis low, hz.1 is easily extrapolated to give a value of high precision. This extrapolated value can now be treated as a constant in a rearranged equation (eqn. (11)) obtained from eqn. (8)
Figure 2. Modified pM versus n plot for different arbitrary ratios of kl/k2. which are represented by numbers attached to the curves. These curves areobtained by assuming k + ~ / h ,= 0.02.
A plot of M versus ( M - hZ.l)/[H+] will theoretically yield a straight line and from the ordinate intercept and slope, h l and hz can he determined, respectively. k, is also given by - (abscissa intercept X slope). Similarly, when hl/hz is high, the extrapolated value of hl from a p M versus a plot is a good estimate. An equation similar to eqn. (11) can now be written, taking hl as a constant. Thus
order of magnitude (i.e., ( M - h2.1) in eqn. (11) and (kl - M ) in eqn. (12)). The equations could he of such extreme sensitivity to the precision of M, h l and hz.1 that meaningful linear plots cannot be obtained. In the report, the use of eqn. (11) in the determination of micro-constants of L-cysteine ethyl ester is tested and the values thus obtained are compared with those from the classical Edsall's method. The application and limitation of eqn. (12) to the determination of micro-constants of dopamine is also illustrated through the use of simulated plots. Experimental
(kl - M ) versus M will be linear. Again, a plot of [H+] The slope of the line is hz, and kz.1 is given by - (ordinate intercept/slope) or simply the abscissa intercept. The usefulness of eqns. (11) and (12) as a practical means to determine micro-constants accurately is questionable, because the plot of each equation involves a quantity which is the difference of two numbers of similar
L-cysteine ethyl ester hydrochloride obtained from Mann Research Laboratories, Inc., was used without purification. The spectral measurements were performed at 237 nm with a Beekman DB-G spectrophotometer, using 0.1 M plycine and phasphate buffers (adjusted to an ionic strength of 0.16 M with KCI) which contained 8.75 r M of the amino acid ester. The pH values of these solutions were measured with an accuracy of +0.005 pH unit at ambient temperature (23 + 2°C). Volume51, Number 2, February 1974
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107
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Figure 5, p M versus a plat for dopamine constructed from literature values Of microcon5tant5.
cd
Figure 3, p M versus a plot for L-cysteine ethyl ester using eqn. (91. Figure 4. Plot of eqn. (11) tor L-cysteine ethyl ester using various estimates of pK2,,: pk.,, = 9.20. rioht bar:. ok, . . = 9.16 (solid dots): pk,,, = 9.12, left bar.
-
Table 2. Ionization Constants of L-Cysteine Ethyl Ester Obtained from Different Plots EdssllisPlof This Work
~ k l P ~ X ~k1.z pk2.1 PKI PK~
7.2 -7.6 6.41 - 6.94 8.37 - 6.50 9.16 6.34 - 6.85 9.18
1.42 6.64 6.34 9.1% 6.51 9.19
OFmm fmtnote 1 in text; r ,= 0.16:23.. Values used toconstruct linear p1or5.
Discussion
Figure 3 shows the p M versus a plot for L-cysteine ethyl ester constructed from absorbance data according to Edsall's meth0d.l It is evident that visual extrapolation at one end of the curve ( a = 1) can be accurately carried out. An average value of 9.162 could be easily assigned for pkz,1. On the other hand, visual extrapolation of the curve to a = 0 involved a rather high degree of uncertainty; any value within the range of 7.2 to 7.6 could, with quite equal justification, be assigned to pkl. All other constants will, in turn, he affected (Table 2). The extrapolation cannot he much improved by acquiring more data points near a = 0, since these points would be derived from absorbance readings not significantly different from zero. Figure 4 shows a plot of eqn. (11) using the same data and the same estimated value of 6.887 X 10-la for k2.1 (points represented by solid circles). Evidently, a linear plot does not result when all data points are considered. A reasonably good straight line can be drawn, however, through points which reside in the range 0.05 < a < 0.2. I t can be argued that points with a > 0.2 give M values too close to that of kz,l and, consequently, may be discarded because a relatively large error results on computing the difference (M - k2.1). On the other hand, points with a < 0.05 were derived from solutions whose absorbance readings were less than 0.05 and were, therefore, highly sensitive to the precision of absorbance measurements. For example, if the deviant point X in Figure 4 were allowed to have an absorbance reading of 0.048 instead of the observed reading of 0.044, the point would fall right on the straight line. Using the straight line thus drawn, it is possible to cal108
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Figure 6. Plat of eqn. (121 for dopamine using various estimates at p k l . CUWBA:p K , = 8.88; curve 8: pk, = 8.90: curve C : p k , = 9.92. culate estimates for the various constants. Their values, shown in Table 2 (3rd column), compare very favorably with those in the literature, which are presumably computer fitted. The values of pkl as determined from both intercepts are identical. The sensitivity of the plot to the value chosen for k2.1 can be tested by deliberateljl using "wrong" estimates of kz.1 to construct the plot. In Figure 4, the bars on each side of the points represent values of ( M - kz.d/[H+] ohtained by using a higher estimate (9.20; right bar) and a lower estimate (9.12; left bar) of k2.1. It is apparent that in this case, the plot as represented by eqn. (11) is not overly sensitive to slight errors in k2.1. These "wrona" estirnaies also dl, not lead to significantchanges in the magnitudes of the calculated constants [Table 2). The drmee of usefulness of linear d o t s in the determination of micro-constants of a compo&d which has a high kz,l/kl ratio can be illustrated by using dopamine as an example. From literature data,2 @ k ~= 8.90, pkz = 10.06, pk2.1 = 9.44 and pk1.z = 10,60), the p M versus plot can he constructed (Fig. 5). It is evident that extrapolation to a = 1 involves a high degree of uncertainty. Figure 6 shows plots of eqn. (12) using estimated p k ~values of 8.88, 8.90, and 8.92 (curves A, B, and C, respectively). It is again apparent that when the correct estimate of microconstant is used (curve B), the linear plot technique performs as predicted by theory. When slightly wrong esti-
mates of the micro-constant are used (curves A and C), the linear plot technique will only be useful when a particular portion of the plot is utilized. In this case, for example, data points which have M values between 0.4-0.7 are not severely affected by an incorrect estimate of kl. Thus, they may he used to construct linear plots in the determination of micro-constants. Thus, in certain cases, the linear plots developed in this communication may be useful as supplementary methods in the unambiguous determination of ionization constants, especially when spectrophotometric data are measured
with a high degree of precision and when computer curve fitting facilities are not readily available. I t is hoped that from this report the student can gain some insight into the various theoretical and practical aspects in the determination of micro-constants of polyionic compounds. In particular, the student is made aware of the unpleasant reality which occurs when seemingly valid equations, based on theoretical grounds, are of only restricted practical utility because of experimental limitations.
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