A Method for the Measurement of Site-Specific Tautomeric and Zwitterionic Microspecies Equilibrium Constants

Article abstract of DOI:10.1021/ac960922d

We describe a method for the individual measurement of simultaneously occurring, unimolecular, site-specific "microequilibrium" constants as in, for example, prototronic tautomerism and zwitterionic equilibria.Our method represents an elaboration of that of Nygren et al. (Anal.Chem. 1996, 68, 1706-10), which thereby becomes generalized and improves the accuracy.Specifically, by making spectral measurements as a function of temperature, we demonstrate the ability to determine site-specific microenthalpies unambiguously.Analysis proceeds via multivariate nonlinear regression modeling of, for example, the Gibbs-Helmholtz relation.Additional determinations of macroscopic equilibrium constants as a function of temperature, in combination with the previously determined microenthalpies, in turn enables the determination of all remaining site-specific thermodynamic parameters, i.e., microentropies, micro free energies, and/or microequilibrium constants, and moreover allows us to resolve and measure the spectra of tautomeric isocoulombers.To our knowledge, we have hereby devised the first such universally applicable and accurate measurement method on record.

Full text of DOI:10.1021/ac960922d

Anal. Chem. 1997, 69, 1642-1650
A Me t h o d fo r t h e Me a s u re m e n t o f S it e -S p e c ific
Ta u t o m e ric a n d Zw it t e rio n ic Mic ro s p e c ie s
Eq u ilib riu m Co n s t a n t s
J os e ph C. D’Ange lo a nd Tim othy W. Colle tte *
National Exposure Research Laboratory, U.S. Environmental Protection Agency, 960 College Station Road,
Athens, Georgia 30605
We describe a method for the individual measurement of
simultaneously occurring, unimolecular, site-specific “mi-
croequilibrium” constants as in, for example, prototropic
tautomerism and zwitterionic equilibria. Our method
represents an elaboration of that of Nygren et al. (An a l.
Ch em . 1 9 9 6 , 68, 1 7 0 6 -1 0 ), which thereby becomes
generalized and improves the accuracy. Specifically, by
making spectral measurements as a function of temper-
ature, we demonstrate the ability to determine site-specific
microenthalpies unambiguously. Analysis proceeds via
multivariate nonlinear regression modeling of, for ex-
ample, the Gibbs-Helmholtz relation. Additional deter-
minations of macroscopic equilibrium constants as a
function of temperature, in combination with the previ-
ously determined microenthalpies, in turn enables the
determination of all remaining site-specific thermody-
namic parameters, i.e., microentropies, micro free ener-
gies, and/ or microequilibrium constants, and moreover
allows us to resolve and measure the spectra of tautomeric
isocoulombers. To our knowledge, we have hereby de-
vised the first such universally applicable and accurate
measurement method on record.
a
The pK is of particular relevance to environmental process
modeling since it describes the degree of ionization of a com-
3
pound: a property that greatly affects its aqueous solubility and
largely determines its propensity to sorb to soil and sediment.2
a
While measurement of pK for ordinary amphiprotic compounds
is straightforward, the situation with regard to those compounds
that may simultaneously ionize to form an internal, i.e., zwitteri-
onic, salt is significantly more problematic. Specifically, there has
been a longstanding need for an unequivocally accurate and
general method for the experimental determination of tautomeric
and zwitterionic microspecies equilibrium constants, the lack of
which has precluded the acquisition of relevant data upon which
to conceptualize predictive models for this phenomenon and then
to evaluate their performance. EPA’s SPARC (SPARC Performs
Automatic Reasoning in Chemistry) physicochemical properties
4
algorithm has been developed to (predictively) calculate, inter
-7
alia, the ionization pK
a
’s for organic compounds5 using funda-
mental chemical structure theory. SPARC has undergone exten-
sive validity testing with unitary deprotonations;5 consequently,
ongoing development of SPARC with regard to prototropic
tautomerism will be closely coupled to application of this experi-
mental method.
-7
The specific equilibria with which, for the purpose of demon-
strating our method, we concern ourselves here (Figure 1), are
representative of zwitterion formation in general.8,9 The two
ionizable sites of 3-hydroxypyridine (CAS No.: 109-00-2), viz., the
aromatic hydroxyl and the pyridinium cation, have long been
recognized to deprotonate with an appreciable degree of simul-
taneity.10 The two cationic site-specific or “microscopic” dissocia-
Current process models for predicting the dispersal of organic
compounds released to the environment are based on the
assumption that these compounds exist exclusively as neutral
species with simple covalent attachments.2 Due to the chemical
complexity of many compounds of industrial importance, e.g.,
pesticides, dyes, etc., this assumption is untenable under a wide
variety of environmental circumstances. Indeed, failure to con-
sider speciation may constitute the largest source of uncertainty
in organic chemical exposure assessment because the processes
that determine chemical transport and transformation are greatly
altered when neutral chemicals ionize, tautomerize, or form
chemical complexes. Hence, to reduce this source of uncertainty,
we have set out to develop and apply analytical methodology that
will allow us to identify and quantify individual species of complex
chemicals in, for example, water as a function of pH. The work
reported herein provides the mathematical and methodological
framework for an accurate and generally applicable solution to
this problem.
tion reactions, k
combine to produce the aggregate or “macroscopic” cationic
deprotonation reaction, K . Similarly, the two electrically neutral
isocoulombers”, viz., the dipolar zwitterion and the tautomerically
1 2
and k (site designation numbers are arbitrary),
1
“
coupled neutral species, each separately deprotonate, with equi-
librium constants designated (according to a numerical sequence
(
(
(
(
(
3) Albert, A.; Serjeant, E. P. Ionization Constants of Acids and Bases; Butler &
Tanner Ltd.: Frome and London, 1962, Chapter 6, pp 106-12.
4) Karickhoff, S. W.; McDaniel, V. K.; Melton, C. M.; Vellino, A. N.; Note, D.
E.; Carreira, L. A. Environ. Toxicol. Chem. 1 9 9 1 , 10, 1405-16.
5) Hilal, S. H.; Carreira, L. A.; Melton, C. M.; Baughman, G. L.; Karickhoff, S.
W. J. Phys. Org. Chem., 1 9 9 4 , 7, 122-41.
6) Hilal, S. H.; Karickhoff, S. W.; Carreira, L. A. Quant. Struct. Act. Relat. 19 95 ,
1
4, 348- 55.
7) Hilal, S. H.; Elshabrawy Y.; Carreira, L. A.; Karickhoff, S. W.; Toubar, S. S.;
(
(
1) Nygren, J.; Andrade, J. M.; Kubista, M. Anal. Chem. 1 9 9 6 , 68, 1706-10.
2) Kollig, H. P., Ed. Environmental Fate Constants for Organic Chemicals Under
Consideration for EPA’s Hazardous Waste Identification Projects, U.S.E.P.A.,
Athens, GA, Pub. No. EPA/ 600/ R-93/ 132, and references cited therein, 1993.
Rizk, M. Talanta 1 9 9 6 , 43, 607-19.
(8) Adams, E. Q. J. Am. Chem. Soc. 1 9 1 6 , 38, 1503-10.
(9) Edsall, J. T.; Blanchard, M. H. J. Am. Chem. Soc. 1 9 9 3 , 55, 2337-53.
(10) Albert, A.; Phillips, J. N. J. Chem. Soc. 1 9 5 6 , 1294-303.
1642 Analytical Chemistry, Vol. 69, No. 8, April 15, 1997
S0003-2700(96)00922-5 CCC: $14.00 © 1997 American Chemical Society

For example, accurate assessment of this parameter should be
useful in biochemistry for elucidating the energetics of both
protein folding and enzyme-substrate interactions.12 A specific
instance in which accurate computation of charge speciation is
important for analytical chemistry applications is in the estimation
of mobilities in capillary zone electrophoresis,13 where such
estimates are utilized for the selection of initial experimental
conditions. In addition, the methodology presented herein should
prove useful to thermodynamicists, in that all the important
parameters for describing equilibria (k, ∆H, ∆G, ∆S) can now be
obtained with site specificity for very complex, polyfunctional
molecules.
There have been a limited number of reports of zwitterionic
equilibrium studies (cf. Hilal6 and references cited therein).
However, these initial investigations all relied upon reasonable
and plausible ad hoc assumptions, which admittedly subjected the
results to systematic error. Wegschieder (cf. Edsall and Blan-
chard9 and references cited therein) was perhaps the first to
suggest the efficacy of estimating microconstant values for
particular site-specific reactions by measuring the macroconstant
for a chemically analogous reaction. Specifically, he suggested
that the deprotonation of an amino group in a given amino ester
could be used to estimate the corresponding microscopic depro-
tonation in the corresponding amino acid. Subsequently, Benesch
and Benesch14 were among the first to use a spectroscopic
method, wherein it is assumed that the extinction coefficient
maximum for absorption of a given moiety, e.g., the cysteine S-
group in the (ultraviolet) region between 2300 and 2400 Å, is
immutable to deprotonation of a tautomerically coupled site, e.g.,
the phenol moiety. Given the shift in the frequency of said
maximum with change in pH, Benesch and Benesch were well
aware of the limitations of this technique. Yet later, Wrathall, et
al.15 described a calorimetric method based on the assumption
that the enthalpy change of chemical analogues was retained by
the corresponding moieties of the compound (again cysteine)
under study.
Fig u re 1 . Reaction schematic of situation-specific zwitterionic
microreaction equilibria for 3- hydroxypyridine. We arbitrarily enumer-
ate the hydroxyl as site one and the pyridinium as site two. The
microequilibrium constant subscripts correspond to the enumerated
deprotonation sites, and the order of the subscripts from left to right
corresponds to the sequence in which the sites have, with increasing
pH, deprotonated. Hence, the last subscript on the right indicates the
particular site undergoing deprotonation for the given microequilibrium
constant. The quantitative aggregate of the two cationic (or anionic)
microequilibria combine to produce the titrimetrically observed mac-
roequilibrium constants K1 and K2.
1
1
scheme by Hill ) as k12 and k21, respectively, and also combine
to produce an aggregate macroscopic (anionic) deprotonation
reaction, K
2
. The relationships among the various equilibrium
9
constants are as follows (cf. Edsall and Blanchard and references
cited therein):
K ) k + k
(1)
(2)
(3)
1
1
2
1
1
1
)
+
K2 k12 k21
K K ) k k ) k2k
21
To date, what are possibly the only direct determinations of
zwitterionic microconstants have been those obtained via NMR.1
With the NMR technique, the relevant parameter for determining
the degree of ionization is the chemical shift δ. As stated by
Rabenstein,16 when exchange of the acidic proton between the
occupied and deprotonated forms is rapid on the NMR time scale,
the chemical shifts of the exchange-averaged resonances provide
a measure of the fractional ionization.19 As elucidated by Shrager
6-18
1
2
1
12
And since the site-specific deprotonations are tautomerically
coupled, the zwitterionic ratio is given by
k / k ) k ) k / k
21 12
(4)
1
2
zw
et al.,18 the observed chemical shift, δ
A
, of a particular site, A, is
the concentration weighted average of the chemical shift param-
eters at site A for tautomerically coupled microspecies.
As is the case for other isothermal spectroscopic techniques,
there is, in general, an insufficiency of information in NMR spectra
to unambiguously determine all of the necessary tautomerically
Given that there are three independent equations, eqs 1-3, in
the four unknowns k , k , k12, and k21, knowledge of any one of
1
2
the four microconstants is sufficient to specify the other three.9
It is unfortunately the case that previously devised methods of
microconstant measurement are usually predicated upon assump-
tions, vide supra, which render them insufficiently accurate for
subjection to theoretical modeling. Hence, our experimental
measurements will buttress the accuracy of algorithms, e.g.,
SPARC, developed for predicting the necessarily refined distinc-
tions between zwitterionic deprotonations and associated tauto-
meric equilibria.
(
(
12) Jaenikcke, R., Ed. Protein Folding; Elsevier: Amsterdam, 1980.
13) Weinberger, R. Practical Capillary Electrophoresis; Academic Press, Inc.:
New York, 1993; Chapters 2 and 3, pp 17-80.
(
(
14) Benesch, R. E.; Benesch, R. J. Am. Chem. Soc., 1 9 5 5 , 77, 5877-81.
15) Wrathall, D. P.; Izatt, R. M.; Christensen, J. J. J. Am. Chem. Soc. 1 9 6 4 , 86,
4779-83.
(16) Rabenstein, D. L. J. Am. Chem. Soc. 1 9 7 3 , 95, 2797-803.
(
(
(
17) Griffith, R., Jr.; Pillai, L.; Greenberg, M. S. J. Solution Chem., 1 9 7 9 , 8, 601-
3.
18) Shrager, R. I.; Cohen, J. S.; Heller, S. R.; Sachs, D. H.; Schechter, A. N.
Biochemistry, 1 9 7 2 , 11, 541-7.
19) Grunwald, E.; Loewenstein, A.; Meiboom, S. J. Chem. Phys. 1 9 5 7 , 27, 641-
1
Aside from its value to environmental modeling, the zwitterion
ratio is a more generally important physicochemical parameter.
(
11) Hill, T. L. J. Phys. Chem. 1 9 4 4 , 48, 101-11.
2.
Analytical Chemistry, Vol. 69, No. 8, April 15, 1997 1643

coupled parameters.18 Nevertheless, there may be specific
at each of the various pH’s, and where the spectra are sampled at
w discrete frequencies; E is a matrix consisting of columns of
molar absorptivity coefficient spectra for each of the s microspecies
at the same discrete frequencies w as in A, and C is a matrix
consisting of p columns of microspecies concentrations at each
of the sampled pH’s, with each row representing a different
microspecies concentration profile. For our example in this
scheme, species 1 represents the cation, species 2 represents the
zwitterion, species 3 represents the neutral molecule, and species
4 represents the anion. The concentrations of the cation and anion
both vary with pH in a manner typical of any diprotic acid. The
concentration of the isocoulombers is given as
cases1
6-18
where one ionization site is sufficiently removed in
distance such that the protonation state of that site does not affect
the chemical shift of the site A used for analysis. In such cases,
the microspecies concentrations can be determined uniquely.16,17
This situation is analogous to the assumption of absorbance
immutability of Benesch and Benesch14 but possibly with, in such
cases, significantly greater accuracy.
We will show herein that the key to accurate determination of
microconstants is to introduce further dimensionality into the
experiment. Although isocoulombers exist in direct proportion
to each other, and are thus linearly dependent upon each other,
throughout an entire span of pH, their relative proportions change
with temperature according to, for example, the Gibbs-Helmholtz
k1Hπ
C2π ) ca
(7)
2
0
D(Hπ)
relation. In fact, temperature variation (and other more recent
2
1
methods of extending dimensionality ) has long been employed
in the the similar problem of estimating conformer populations
in solution. Thus, where titrimetry alone is inadequate, one may
similarly produce the requisite resolution of tautomers by extend-
for the zwitterion and
k2Hπ
ing isothermal global spectroscopic analysis of pH-dependent acid/
base equilibria22 into the temperature domain. Such a procedure
C3π ) ca
(8)
D(Hπ)
is, however, predicated upon sufficient temperature invariance of
the relevant species’ spectra, and this prerequisite may not hold
in all cases.23 Recently, Nygren et al.1 applied the van’t Hoff
relation, a linearized transformation of Gibbs-Helmholtz, to
characterize monomer/ dimer equilibria using temperature varia-
tions. Their method, however, was dependent on acquiring the
isolated monomer spectra of their compounds by using ap-
propriately limiting dilute solutions. Neither chemical nor physical
methods are, in and of themselves, capable of resolving the spectra
of isocoulombers and therefore of obtaining microconstants,
because, vide supra, it is impossible to know with assurance a
priori when or if species isolation has in fact occurred.
for the neutral, where c
a
is the analytical concentration of
3
-hydroxypyridine, and where
2
D(H ) ) H + K H + K K
(9)
π
π
1
π
1
2
It is of particular importance to note that the isocoulombers
exist in direct proportion to each other throughout the entire pH
range. They are therefore mathematically linearly dependent
upon one another, and there is, in general, no isothermal
experiment, e.g., titration, spectroscopic, etc., that can determine
the concentration of either species uniquely. In fact, the number
of possible hypothetical microconstant solution pairs consistent
with any temperature-invariant titration experiment is in fact
infinite.
We wish therefore to obtain a direct and unequivocally accurate
measurement of microconstants and their associated zwitterionic
equilibria. This may be accomplished by employing the Gibbs-
Helmholtz relation, which provides an experimental relation for
the variance of an equilibrium constant as a function of temper-
ature, viz.,
MATHEMATICAL DESCRIPTION
Given Beer’s law, A ) ꢀbc, it is important for our purposes
here to recall that the molar absorptivity (extinction coefficient),
ꢀ, is a characteristic constant of the material proportional to the
absorbing cross-sectional area of the molecule24 and is therefore
largely, although not completely, independent of temperature.23
We thus, with no loss of generality, incorporate the cell’s path
length, b, into the absorptivity coefficient, e ) ꢀb. For the four
absorbing species represented by Figure 1, we then have for the
absorbance, A at any particular pH, π, and wavelength, λ.
∆
Ha
1
1
Aλπ ) e c + e c + e c + e c
(5)
λ1 1π
λ2 2π
λ3 3π
λ4 4π
kaT ) ka0 exp
-
(10)
[
R
(T0 T)
]
In matrix terms, for the general case of w wavelengths, s number
of species, and p number of pH measurements, we have
where ∆H
a
is the “microenthalpy” specific to the deprotonation
sequence a, T is an arbitrary absolute temperature, ka0 is the
equilibrium microconstant for the deprotonation sequence a at
Awp ) EwsCsp
(6)
0
the defined absolute base temperature, T , and kaT is the micro-
where A is a matrix consisting of p columns of absorbance spectra
constant for the deprotonation sequence a at temperature T. This
particular form assumes only that the temperature variation is
limited enough for the microenthalpy to be considered effectively
constant. Since at any temperature, K1T ) k1T + k2T, we have for
the temperature dependence of the macroconstant
(
20) Abraham, R. J.; Bretschneider, E. In Internal Rotation in Molecules; Orville-
Thomas, W. F., Ed.; Wiley-Interscience: New York, 1974; Chapter 13, pp
4
81-584.
21) Saltiel, J.; Sears, D. F., Jr.; Choi, J.-O.; Sun, Y.-P.; Eaker, D. W. J. Phys. Chem.
9 9 4 , 98, 35-46.
(
1
(
(
22) Frans, S. D.; Harris, J. M. Anal. Chem. 1 9 8 4 , 56, 466-70.
23) Saltiel, J.; Choi, J-O.; Sears, D. F., Jr.; Eaker, D. W.; Mallory, F. B.; Mallory,
C. W. J. Phys. Chem. 1 9 9 4 , 98, 13162-70, and references cited therein.
24) Hughes, H. K.; et al. Anal. Chem. 1 9 5 2 , 24, 1349-54.
∆
H1
∆H2
1
1
1
1
K1T ) k10 exp
-
+ k20 exp
-
[
R
(T T)
]
[
R
(T T)
]
0
0
(
11)
(
1644 Analytical Chemistry, Vol. 69, No. 8, April 15, 1997

where k10 and k20 are used to indicate the microconstants (k
) extant at the defined base temperature.
To demonstrate this contention, and in order to simplify the
coming discussion, we define the “reduced” microenthalpy, h
1
and
To eliminate one of the four microconstant parameters, we
introduce temperature-specific values of the zwitterion constant,
i.e., analogous to eq 4, we have
k
2
i
)
k10
k210
k120
∆H
i
/ R, for i ) 1, 2, and also define
)
kzw0
)
(17)
k20
1
1
t )
-
(12)
(13)
(T T)
0
Accordingly, we may use the quotient k10/ kzw0 in place of k20 and
the product k120 zw0 in place of k210
k
.
or equivalently,
In order to eliminate one of the microenthalpy terms, we define
the parameter η, such that
T
1
0
t )
(1 - T
)
T0
h21 ) h + η
(18)
1
Note that t is a monotonic increasing function of T. In terms of
these definitions, eq 11 becomes
i.e., we must in general allow for perturbation of the enthalpy
change of a given microscopic deprotonation, e.g., 3-hydroxypy-
ridinium to 3-hydroxypyridine, due to the protonation state of the
tautomerically coupled site, i.e., 3-hydroxylate-pyridinium to 3-
hydroxylate-pyridine. Using the Gibbs-Helmholtz expansion of
eq 4, with eqs 17 and 18, we have
K1T ) k10 exp(h t) + k exp(h t)
(14)
1
20
2
Using these notational simplifications, we were then able to
mathematically prove25 that it is possible, in principle, to determine
all of the Gibbs free energy thermodynamic parameters, including
in particular the two microconstants, for the two cationic microre-
k10 exp(h t)
k120kzw exp[(h + η)t]
1
1
)
(19)
k / k exp(h t)
k120 exp(h t)
12
10 zw
2
actions of Figure 1 (k
1
) k10 and k ) k20 at the designated base
2
temperature) by measurement of the cationic macroconstant at
four different temperatures, K10, K11, K12, and K13. Thus, four
parameters are necessary and sufficient when fitting the cationic
macroconstants alone.
which after simplification and rearrangement leads to
h12 ) h + η
(20)
2
In order to simultaneously incorporate the anionic reaction path
in the analysis, we merely note that if we include measurement
which forces us to conclude that said enthalpic perturbation is
identical for both pairs of the tautomeric couple.
2
of the anionic macroconstant, K , at the base temperature (K20),
simple rearrangements of eq 3 yield both k12 and k21 at the base
temperature (k120 and k210) as functions of the previously utilized
macroconstants, i.e.,
Accordingly, the final set of four Gibbs-Helmholtz microre-
action equations in six parameters reads
k ) k exp(h t)
1t
10
1
K10
k10
K10
k20
k120
)
K20
k210
)
K20
(15)
k ) k / k exp(h t)
2t
10 zw0
2
k12t ) k120 exp[(h + η)t]
2
Accordingly, we have determined the necessity for addition of one
single further parameter.
k
21t ) k120kzw0 exp[(h
+ η)t]
(21)
1
Finally, to determine the parameterization required for char-
acterizing the temperature variance of the anionic microreactions,
and the temperature-dependent macroconstant equations become
we measure the anionic macroconstant at temperature T, i.e., K21
,
and note that, analogous to eq 15 we have
K ) k exp(h t) + (k / k ) exp(h t)
(22)
1t
10
1
10 zw0
2
K11
K11
k2T
k12T
)
K21
k21T
)
K21
(16)
k1T
and
K2t- ) (k120 exp[(h + η)t])
1
-1
+
1
and where k1T ) k10 exp(h
1
2
t) and k2T ) k20 exp(h t). Therefore,
-
1
for complete thermodynamic characterization of all microreactions
for the two macroscopic deprotonation reactions of Figure 1, we
require six and only six parameters. Accordingly, of the eight
parameters produced from a naive expansion of the four sequence-
(k120kzw0 exp[(h + η)t]) (23)
2
for the cationic and anionic macroconstants, respectively. Finally,
the tautomeric equilibrium is
kzwt ) kzw0 exp[(h - h )t]
(24)
specific deprotonation microreactions of Figure 1, i.e., k
1
, h
1 2
, k ,
1
2
h
2
, k12, h12, k21, and h21, two are superfluous. Consequently, we
may readily eliminate one of the four microconstants and one of
the four reduced enthalpies.
Spectroscopic Determination of Enthalpies. As will be
demonstrated later, it will often be necessary to use absorption
(
25) See Supplementary Information.
spectroscopy to determine the reduced microenthalpy terms h
1
Analytical Chemistry, Vol. 69, No. 8, April 15, 1997 1645

and h
determinations.
2
.
We will now show the possibility of making these
where the M matrix consists of the terms (constants) in paren-
theses in eq 29 and C consists of the exponential microenthalpy
terms.
Equation 29 reveals that molar absorptivity coefficients and
microconstants always occur together as products in the elements
of matrix M and hence may not be partitioned. Nonetheless it is
If we choose to vary temperature and make (constant pH)
spectral measurements at τ different temperatures we have
(
analogous to eq 6)
Awτ ) E C
(25)
ws sτ
π
apparent that, given measurements of D(H ,t) for all temperatures
of Swt, one may determine microenthalpies via nonlinear regres-
sion modeling using spectroscopy. Although one may model the
system of eq 30 directly, there are various advantages to using
multivariate methods, and we have chosen to use SVD modeling,
as exemplified by Hug and Sulzberger.27
where τ takes the place of p. It is of course possible to determine
the total (sum) concentration of the isocoulombic pair, as well as
the concentrations of the ionic species, via titration at constant
temperature.26 One may therefore readily determine the molar
absorptivity coefficient spectra for the ions separately, i.e., the
column vectors Ew1 and Ew4, as well as that for the sum of the
isocoulombic pair. We reiterate that the isocoulombic pair may
not be resolved, one from the other since they are linearly
codependent.
EXPERIMENTAL SECTION
Titrations were performed using an Orion expandable ion
analyzer EA920 equipped with automatic temperature compensa-
tion and a digital thermometer. Titrations were performed with
2
a N purge and temperatures were controlled to within 0.1 °C.
Consequently, we may with no loss of generality restrict our
attention to the isocoulombic pair spectrum, Owt, where
The pH electrode, thermometer, and purge line were all inserted
through a large rubber stopper which sealed a 100 mL beaker.
Titrant was delivered with a (200-1000 µL) Finnpipette by
removing a small rubber stopper from the main beaker stopper
and quickly inserting/ removing the pipet. 3-Hydroxypyridine was
acquired from Aldrich and used as received. The pH electrode
was calibrated prior to each titration using Fisher pH 4 and 10
buffers to two decimal digits, e.g., 10.00. The pH measurements,
however, were recorded to three decimal digits (10.000). Two
titrations, cationic and anionic, were performed at each temper-
ature using 10 mM solutions, excluding 60 °C, where the cationic
titration was not performed due to equipment failure.
Owτ ) Awτ - Ew1C1τ - Ew4C4τ ) EwêCêτ
(26)
and where ê ) 2, 3, i.e., Owt is the product of the isocoulombic
submatrices of Ews and Cst in eq 25. Now for any given wavelength
λ, and temperature t, we have
cak exp(h t)Hπ
cak20 exp(h t)H
10
1
2
π
O ) e
+ eλ3
(27)
λt
λ2
D(H ,t)
D(H ,t)
π
π
The self-dissociation constant of water, K
, was fit to a Gibbs-
w
Helmholtz expression, assuming a linear change in enthalpy, over
where it should be noted that D(H
eq 9) solely of macroconstants and H
determined by means of potentiometric titrimetry. In order to
use singular value decomposition (SVD) modeling (vide supra),
one must separate out all components that are not explicit
functions of temperature. As one may readily ascertain from eq
π
,t), consisting as it does (cf.
, can be experimentally
3
the temperature range 0-60 °C (cf. Albert and Serjeant, Appendix
π
2
8
II). The dielectric constant of water from 25 to 200 °C was
modeled with a simple exponential decay function using data at
1
0 MP and was then corrected to atmospheric pressure. This
value was used to compute the (temperature-dependent) Debye-
Huckel limiting law coefficient29 A (A ) 0.509 at 25 °C) at the
various temperatures.
Equilibrium constants were calculated using a full elaboration
of the proton condition and were corrected for activity using the
extended Debye-Huckel limiting law: assuming zero contribution
to ionic strength from the zwitterion. Ion size parameters were
taken from Kielland30 with that for 3-hydroxypyridine assumed
equal to 6.0.
27, it is possible to do so by factoring out all multiplicative constant
terms into one expression separate from the exponential mi-
croenthalpy functions. If we take
e c k H
e c k H
λ3 a 20 π
λ2
a
10
π
O )
exp(h t) +
exp(h t) (28)
λt
1
2
(
)
(
)
D(H ,t)
D(H ,t)
π
π
UV spectroscopy was used to obtain macroconstants at room
temperature, i.e., there was no thermostating of the sample cells.
Solutions were 100 µM in 3-hydroxypyridine and were pH buffered
and prepared at a constant ionic strength of 0.01 M. Solutions
were run in pseudorandom order. A thermostating fixture was
acquired later and used to control cell temperature to within 0.1
then the denominator function, D(H
π
,t), above must be measured
or otherwise estimated a priori and then be properly incorporated
into either the isocoulombic matrix or the microenthalpy matrix.
We may then take each column vector of Owt, i.e., the absorbance
spectrum at each temperature, and multiply it by the denominator
function, D(H
π
,t) ≈ K1t
π
H , operative at that temperature to
°
C using circulated water from a constant-temperature water bath.
UV cell temperature was measured with a hand-held digital
thermocouple) thermometer. Thermostated solutions were not
buffered but were simply 100 µM in 3-hydroxypyridine.
construct the “denominator-corrected” isocoulombic matrix Swt
:
(
S ) (e c k ) exp(h t) + (e c k ) exp(h t) (29)
λt
λ2
a
10
1
λ3
a
20
2
(
(
(
27) Hug. S. J.; Sulzberger, B. Langmuir 1 9 9 4 , 10, 3587-97, and references
cited therein.
28) CRC Handbook of Chemistry and Physics, 63rd ed.; CRC Press, Inc.: Boca
Raton, FL, 1982; Table E-57.
29) Castellan, G. W. Physical Chemistry, 2nd ed.; Addison-Wesley Publishing
Co.: Reading, MA, 1971: Chapter 16.
or in matrix terms,
Swτ ) Mwê Xêτ
(30)
(
26) Kubista, M.; Sjoback, R.; Albinsson, B. Anal. Chem. 1 9 9 3 , 65, 994-8.
646 Analytical Chemistry, Vol. 69, No. 8, April 15, 1997
(30) Kielland, J. J. Am. Chem. Soc. 1 9 3 7 , 59, 1675-8.
1

The data collected during this phase of the experiment was
subjected to SVD with subsequent nonlinear regression modeling
in order to determine the macroconstants, an estimate for the
number of species, and the ion species spectra. Activity coefficient
corrections were neglected here. The SVD algorithm decomposes
T
an arbitrary data matrix, A, into three matrices, A ) USV , such
that the fraction of the total variance in A, which is accounted for
by the orthogonal columnar basis vectors in U, is provided by
the square of the diagonal elements of S.27 The first four variance
coefficients obtained by this procedure for our isothermal spectral
data matrix, Awp, were as follows: 0.8774, 0.0744, 0.0478, and
0
.0004. Hence, the last (fourth) term accounts for a full 2 orders
of magnitude less variance than the third term, and from this we
may conclude that, as expected, there are only three linearly
independent species observable using isothermal spectra. The
Fig u re 2 . UV absorbance intensities for 100 µM 3-hydroxypyridine
as a function of pH for four specific wavelengths: 240 (solid), 242
(dash-dot), 244 (dash-dot-dot), and 246 (dash) nm.
a
macro-pK ’s obtained in this fashion were 4.871 and 8.681, in good
agreement with the titrimetric values despite the 100-fold differ-
ence in analyte concentration.
Computations were performed on a Dell PC equipped with a
Having proven the theoretical possibility of establishing mi-
croconstants and associated site-specific thermodynamic param-
eters via temperature-variant (potentiometric) titrations, we then
attempted to do so experimentally. What was observed was a
fairly wide variation in estimated site-specfic thermodynamic
parameters and intersection temperatures. For example, we took
the full, i.e., cationic and anionic, set of experimental data and
performed the five-parameter nonlinear regressions resulting from
having first specified the microconstant ratio parameter: kzw0, for
1
20-MHz pentium processor. Singular value decompositions with
concomitant nonlinear regression were performed using MATLAB
to procure macroconstants and enthalpies using UV data. Non-
linear regression was performed on titration data to procure
microconstant parameters using the Solver algorithm of Excel 4.0.
The optimization dependent variable utilized was the simultaneous
variation coefficient, CV, computed from both macroconstants
together, i.e.,
n
2
various values of kzw0. As kzw0 varies from 1.00, representing
Kc - Kx
i
i
CV )
microconstant pair equality, to 54.6, exp(4), the variance coefficient
only varies from 0.0090 to 0.0087, a statistically negligible decrease
of only 3%. This indicates that there is (experimentally) an excess
of at least one degree of freedom in parameterization. Hence,
accuracy would require that we measure at least one, and possibly
more, of the six parameters by means of a method independent
of titrimetry. This we were able to do via spectroscopic determi-
nation of the two site-specific microenthalpies.
∑
(
)
x
i)1
Kxi
where Kc is the computed value for either macroconstant, Kx is
the corresponding experimental value, and the index i runs over
all temperatures for both macroconstants: K
1
and K
2
. Symbolic
algebra was performed using Mathcad 5.0.
As shown above, it is possible to determine the microenthalpy
terms via measurement of the isocoulombic pair spectra across a
range of temperature. We first made spectroscopic measurements
at temperatures close to those measured titrimetrically. Next we
used the full six-parameter optimized equation set (eq 22) to
compute appropriately interpolated values of the K1t macrocon-
stants at the temperatures achieved for spectroscopic measure-
ment. We then multiplied the spectrum at each temperature by
the appropriate K1t macroconstant (eq 29) to obtain the denomina-
tor-corrected isocoulomber matrix, Swτ. Figure 3A shows four row
vectors from the isocoulombic pair matrix, Owt, for the wave-
lengths representing the four peak maxima observed in the
isocoulombic pair spectra. Note that three of the peaks decrease
with increasing temperature, while one, 276 nm, increases with
temperature. Since the row vectors are the weighted sums of
the isocoulombic species concentrations (cf. eqs 26 and 27), Figure
3A provides a qualitative indication that there are at least two
species extant in these spectra. Figure 3B demonstrates the
RESULTS AND DISCUSSION
We initially set out to duplicate the procedure of Benesch and
Benesch14 (and others), who used the assumption of spectral
invariance to coupled site protonation state. Figure 2 provides a
visual indication of the approximate nature of this assumption in
the case of 3-hydroxypyridine. What one hopes to find (experi-
mentally) is a frequency region in the spectrum where change in
pH produces little or no peak shift. By this we mean that the
various moments describing a given peak, mode, mean, variance,
skew, etc., are invariant to change in pH and that the only change
is a monotonic (increasing or decreasing) variation in intensity.
Such behavior would provide some degree of confidence for
supposing spectral invariance. In fact, without exception, each
UV peak of 3-hydroxypyridine undergoes a pronounced shift in
frequency with change in pH. Therefore, without further cor-
roborating evidence, the only conclusion one can reliably infer
from Figure 2 is that there are two distinct macroscopic depro-
tonation steps. Figure 2 also provides an indication of the degree
of subjectivity involved in assignment of microconstants by means
of this procedure. Where the assumption of spectral invariance
is accurate, the zwitterionic ratio, kzw, is proportional to the ratio
of the magnitudes of the two sigmoidal transitions occurring in
each of the four isofrequency curves. Thus, the range of potential
values for kzw using this conjecture is quite large.
corresponding row vectors from Swτ. Finally, we used SVD
nonlinear regression modeling on Swτ to validate the hypothetical
number (two) of species and then also to procure the two site-
specific reduced microenthalpies, h
spectra variance terms derived from Swτ, vide infra, were 0.9954,
0.0040, and 0.0005. The microenthalpy values, ∆H and ∆H
1 2
and h . The first three basis
1
2
,
Analytical Chemistry, Vol. 69, No. 8, April 15, 1997 1647

Fig u re 4 . Experimental and computed macro-pKa’s, and computed
micro-pKa’s for 3-hydroxypyridine as a function of temperature. (A)
Cationic: circles, experimental values for pK1; stars, calculated values
for pK1; dotted, calculated values for pk1; dashed, calculated values
for pk2. (B) Anionic: circles, experimental values for pK2; stars,
calculated values for pK2; dotted, calculated values for pk12; dashed,
calculated values for pk21. Cationic calculations are via eq 22, and
anionic calculations are via eq 23.
Fig u re 3 . UV absorbance intensities for 100 µM 3-hydroxypyridine
as a function of temperature for four specific wavelengths: 210 (dash),
246 (dash-dot), 276 (solid), and 314 nm (dot). (A) Original (unmodi-
fied) absorbance intensities. (B) Product of absorbance intensities
with the cationic macroconstant K1, extant at the corresponding
temperature.
Ta b le 1 . Op t im ize d Mic ro re a c t io n P a ra m e t e rs fo r
3
-Hyd ro x yp yrid in e a t 2 0 °C
predominant microconstants (dotted lines) for both the cationic
(Figure 4A) and anionic (Figure 4B) deprotonations, Figure 4
gives a visual indication of the error to be made by using the
parameter
value ( std dev
944 ( 281
h1 (K)
k1
h2 (K)
kzw
k12
η (K)
-
-
5
9
1
(1.164 ( 0.061) × 10
technique of Nygren et al. for this particular case of 3-hydroxy-
5250 ( 1210
9.8 ( 3.2
pyridine. Even at 5 °C, such error is readily discernible.
Particularly given our interest in measurement of zwitterionic
compounds whose kzw is close to unity at room temperature, it
may well often be the case that such error could be considerably
more pronounced. Be that as it may, we contend that the
procedure described herein is both more general and more
accurate.
Qualitative Identification. Accurate measurement of the
microconstants allowed us to compute the concentration of both
isocoulombers as a function of temperature. This in turn allowed
us to determine the molar absorptivity spectra for both isocou-
lombers, which we show in Figure 5B. The cation and anion
spectra are included in Figure 5A and C, respectively. An internal
comparison of all four species-specific spectra and analysis of
related reference spectra are consistent with assigning the
spectrum of species 2 to that of the zwitterion and the spectrum
of species 3 to that of the neutral isocoulomber.25 We assert that
the extracted spectra are so readily rationalized that they cor-
roborate the efficacy of our technique. Hence, we substantiate
the deduction of Metzler and Snell,31 that the zwitterion predomi-
nates at room temperature. One notes from examination of these
spectra that in the region of 240 nm, and again in the region of
(1.751 ( 0.003) × 10
-3315 ( 8
measured by this procedure were 1.88 and 10.4 kcal mol-1 (7.87
and 43.5 kJ mol-1), respectively.
Having procured reliable estimates for two of the six param-
eters, we next inserted these microenthalpy values as constants
into the two macroconstant expressions (eqs 22 and 23) and then
proceeded to optimize the remaining four parameters, k10, kzw0
k
,
120, and h, using, as before, nonlinear regression on the titri-
metrically measured macroconstants. In marked contrast to the
previous five-parameter case, CVs for the three-parameter opti-
mization resulting from having utilized the microenthalpy values
measured above, with various values of the zwitterion ratio
parameter, kzw, considered as an independent variable, clearly relax
to a reliable minimum (Figure S1, Supporting Information). The
final values (using a base temperature of 20 °C) for all six
parameters are given in Table 1, and the temperature dependence
of the various equilibria are graphically portrayed in Figure 4. The
1 2
variance coefficient was 4.7% for K and was 3.0% for K .
By examining the difference between the calculated values for
the macroconstants (closed diamonds, stars) and the quantitatively
(31) Metzler, D. E.; Snell, E. E. J. Am. Chem. Soc. 1 9 5 5 , 77, 2431-7.
1648 Analytical Chemistry, Vol. 69, No. 8, April 15, 1997

Fig u re 5 . Molar absorptivity spectra of 3-hydroxypyridine. (A) Cation. (B) The two isocoulombic species, zwitterion (dotted) and unpolarized
neutral (dashed). (C) Anion.
Ta b le 2 . Th e rm o d yn a m ic Mic ro re a c t io n P a ra m e t e rs (J m o l- ) fo r 3 -Hyd ro x yp yrid in e a t 2 5 °Ca
1
1
2
12
21
-
5
-6
-9
-8
k
1.23 ( 0.06 (0.17) × 10
28 000 ( 130 (330)
7800 ( 2300 (4000)
-68.8 ( 7.9 (20.2)
1.2 ( 0.2 (0.5) × 10
2.15 ( 0.05 (0.10) × 10
49 476 ( 4 (9)
2.1 ( 0.4 (0.9) × 10
∆
∆
∆
G
H
S
33 100 ( 610 (1560)
44 000 ( 10000 (22 000)
36 ( 34 (87)
44 400 ( 800 (2100)
-6000 ( 200 (5700)
-169 ( 8 (21)
30 000 ( 10000 (25 000)
-66 (34 (83)
a
Uncertainties are standard deviations and (0.95 confidence interval). Each column represents a different microreaction sequence.
310 nm, that the only responsive spectrum is that of the zwitterion.
Ta b le 3 . Co m p a ris o n o f Ex p e rim e n t a l a n d (S P ARC)
Close examination of these peaks indicates that any temperature
variance of the zwitterion spectrum is relatively minor and thus
provides additional confidence, post hoc, in the reliability of the
microenthalpy values as determined by SVD nonlinear regression
modeling. Moreover, examination of Figure 5 emphatically
illustrates the difficulty in selecting, a priori, a frequency region
where moiety spectral invariance holds true.
Co m p u t e d p k a ’s a t 2 5 °C
experimentala
SPARC
pk1
pk2
pk12
pk21
4.91
5.90
8.668
7.67
5.0
5.8
8.5
7.7
a
Significant digits only.
Regression Statistics. In order to produce statistical infer-
ences about the (nonlinear regression) parameter set, we esti-
factored into these computations. The variance-covariance
matrix for the pair of parameters k120 and h was obtained using
macroconstant K2t data only, again without factoring in variability
in the other four parameters. Microspecies thermodynamic
parameters at 25 °C along with their variance estimates are given
in Table 2. Measured values for the microconstants (25 °C) are
compared to those computed by the SPARC algorithm in Table
2
mated the approximate variance-covariance matrix: s (D), of the
3
2
regression coefficients as
2
-1
s (D) ) MSE(D′D)
(31)
where D is the matrix of partial derivatives, i.e., the Jacobian, of
the temperature-dependent macroconstant (K1t) with respect to
the microconstants parameters, i.e.,
3. Deviations of the SPARC computed microconstants from those
measured here are well within previously established error
estimates for the algorithm (vide infra).
∂
K1t ∂K1t ∂K1t ∂K1t ∂K1t ∂K1t
,
,
,
,
,
∂
k10 ∂kzw0 ∂h1
∂h2 ∂k120
∂η
CONCLUSION
The only assumptions required of this method are quantitative
and not qualitative in nature and contribute only to the extent of
random, as opposed to systematic, error. These assumptions are,
first, that both microenthalpies and species spectra are effectively
constant throughout the temperature range under consideration.
(We are currently investigating the prevalence and significance
evaluated for each temperature of Swτ at the final nonlinear
regression estimate of those parameters. Using the five (inclu-
sive) wavelengths between 210 and 218 nm, i.e., 22 degrees of
freedom, we estimate the 95% confidence interval (CI) for the
microenthalpies as 960 (4050 J) and 5200 cal (21800 J), respec-
tively. These SEM values correspond to 0.95 CI variation
coefficients of 0.51 and 0.50; hence, we have eliminated the
possibility of nonsingular codependence of the two microenthal-
pies.
of the temperature-induced UV spectral shift phenomenon of
Saltiel et al.23), second, that the magnitudes of the two mi-
croenthalpies as well as the degree of resolution between the
spectra of the two isocoulombers, are sufficient for the change in
relative proportion of said isocoulombers to be discernible at
several temperatures. This last is a reasonable assumption in the
UV domain and is, given sufficient sensitivity, an excellent
assumption using more qualitatively informative spectra, as are
those of NMR or vibrational spectroscopy; and, finally, that one
We then obtained the variance-covariance matrix for the pair
of parameters k10 and kzw0, using macroconstant K1t data only.
1 2
Variability in the microenthalpy parameters ∆H and ∆H was not
(
32) Neter, J.; Wasserman, W.; Kutner, M. H. Applied Linear Statistical Models,
nd ed.; Richard D. Irwin, Inc.: Homewood, IL, 1985; Chapter 14.
2
Analytical Chemistry, Vol. 69, No. 8, April 15, 1997 1649

may explicitly account for the presence of conformers when
necessary.
ACKNOWLEDGMENT
This research was supported in part by an appointment (J.C.D.)
to the Postgraduate Research Participation Program at the
National Exposure Research Laboratory (Athens, GA) adminis-
tered by the Oak Ridge Institute for Science and Education
through an interagency agreement between the U.S. Department
of Energy and the U.S. Environmental Protection Agency. The
authors also thank Drs. Sam Karickhoff (EPA), Lionel Carreirra
Accordingly, we have hereby produced the first general self-
checking method for accurate experimental measurement of
tautomeric acidity equilibrium (micro)constants, the first resolved
UV spectra of isocoulombers, species that are linearly codependent
throughout any isothermal pH domain, and the first verification
of a SPARC microconstant parameter set. While we here
measured macroconstants by means of potentiometric titrimetry,
it is in general more accurate to do so by means of spectroscopy.3
When one includes the necessity, as here, to produce spectra as
a function of temperature in order both to independently assess
microenthalpies and to provide qualitative species identification,
then information-rich spectroscopic measurement becomes rela-
tively more efficient than titrimetry as well. Hence, due to its
compatibility with aqueous samples, and given recent advances
in the sensitivity of its instrumentation, viz., owing primarily to
the exceptionally high quantum efficiencies of modern charge
coupled device (CCD) detectors, and to the high spectrometer
throughput which results from employing notch filters that
efficiently remove Rayleigh scattering, Raman spectroscopy ap-
pears to be an optimum tool to apply in this arena. We are
therefore in the process of constructing a refined and definitive
microconstant measurement device, a notch-filtered, CCD-based,
imaging Raman spectrometer that is fiber-optically coupled to an
autotitration apparatus, for simultaneous automated measurement
of all necessary observables.
(
UGA), and Said Hilal (NRC) for their helpful and lively discus-
sions. We also thank Mr. George D. Yager (EPA) for his various
technical contributions. Note: Mention of trade names or
commercial products does not constitute endorsement or recom-
mendation for use by the U.S. Environmental Protection Agency.
SUPPORTING INFORMATION AVAILABLE
(
1) A formal mathematical proof of our contention that the four
microconstant parameters of eqs 11, 14, and 22, may be deter-
mined from four cationic macroconstants extant at four different
specified temperatures; (2) interpretation and rationalization of
the spectra of Figure 5; (3) Figure S1 demonstrating the statistical
reliability of our procedure (8 pages). Ordering and access
information is given on any current journal masthead page.
Received for review September 11, 1996. Accepted
X
February 10, 1997.
AC960922D
X
Abstract published in Advance ACS Abstracts, March 15, 1997.
1650 Analytical Chemistry, Vol. 69, No. 8, April 15, 1997