Dissociation kinetics of macrocyclic trivalent lanthanide complexes of 1,4,7,10-tetraazacyclododecane-1,7-diacetic acid (DO2A)

Article abstract of DOI:10.1039/c0dt01440k

The [H⁺]-catalyzed dissociation rate constants of several trivalent lanthanide (Ln) complexes of 1,4,7,10-tetraazacyclododecane-1,7- diacetic acid (LnDO2A⁺, Ln = La, Pr, Eu, Er and Lu) have been determined in two pH ranges: 3.73-5.11

Full text of DOI:10.1039/c0dt01440k

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PAPER
Dissociation kinetics of macrocyclic trivalent lanthanide complexes of
1,4,7,10-tetraazacyclododecane-1,7-diacetic acid (DO2A)†
Chih-Cheng Lin,a,b Chia-Ling Chen,a Kuan-Yu Liua,c and C. Allen Chang*a,c
Received 21st October 2010, Accepted 12th January 2011
DOI: 10.1039/c0dt01440k
The [H+]-catalyzed dissociation rate constants of several trivalent lanthanide (Ln) complexes of
1,4,7,10-tetraazacyclododecane-1,7-diacetic acid (LnDO2A+, Ln = La, Pr, Eu, Er and Lu) have been
determined in two pH ranges: 3.73–5.11 and 1.75–2.65 at four different temperatures (19–41.0 ◦C) in
aqueous media at a constant ionic strength of 0.1 mol dm-3 (LiClO4). For the study in the higher pH
range, i.e. pH 3.73–5.11, copper(II) ion was used as the scavenger for the free ligand DO2A in
acetate/acetic acid buffer medium. The rates of Ln(III) complex dissociation have been found to be
independent of [Cu2+] and all the Ln(III) complexes studied show [H+]-dependence at low acid
concentrations but become [H+]-independent at high acid concentrations. Influence of the acetate ion
content in the buffer on the dissociation rate has also been investigated and all the complexes exhibit a
first-order dependence on [Acetate]. The dissociation reactions follow the rate law: kobs = kAc[Acetate] +
K¢klim[H+]/(1 + K¢[H+]) where kAC is the dissociation rate constant for the [Acetate]-dependent pathway,
klim is the limiting rate constant, and K¢ is the equilibrium constant for the reaction LnDO2A+ + H+ ¤
LnDO2AH2+. In the lower pH range, i.e. pH 1.75–2.65, the dye indicator, cresol red, was used to
monitor the dissociation rate, and all the Ln(III) complexes also show [H+]-dependence dissociation
pathways but without the rate saturation observed at higher pH range. The dissociation reactions
follow the simple rate law: kobs = kH[H+], where kH is the dissociation rate constant for the pathway
involving monoprotonated species. The absence of an [H+]-independent pathway in both pH ranges
indicates that LnDO2A+ complexes are kinetically rather inert. The obtained kAC values follow the
order: LaDO2A+ > PrDO2A+ > EuDO2A+ > ErDO2A+ > LuDO2A+, whereas the klim and kH values
follow the order: LaDO2A+ > PrDO2A+ > ErDO2A+ > EuDO2A+ > LuDO2A+, mostly consistent
with their thermodynamic stability order, i.e. the more thermodynamically stable the more kinetically
inert. In both pH ranges, activation parameters, DH*, DS* and DG*, for both acetate-dependent and
proton-catalyzed dissociation pathways have been obtained for most of the La(III), Pr(III), Eu(III),
Er(III) and Lu(III) complexes, from the temperature dependence measurements of the rate constants in
the 19–41 ◦C range. An isokinetic (linear) relationship is found between DH* and DS* values, which
supports a common reaction mechanism.
macrocyclic complexes for applications as magnetic resonance
imaging contrast agents,5 nuclear medicine,6 luminescence probes,7
solvent extraction agents,8 and artificial nucleases.9 Previously,
we have reported the results of our studies on the formation
stability, dissociation kinetics and structures of a number of Ln(III)
complexes of macrocyclic ligands, including K21DA (dapda = 1,7-
diaza-4,10,13-trioxacyclopentadecane-N,N¢-diacetic acid1b,c,2a),
K22DA (dacda = 1,10-diaza-4,7,13,16-tetraoxacyclooctadecane-
N,N¢-diacetic acid1a,2b), DO3A (1,4,7,10-tetraazacyclododecane-
1,4,7-triacetic acid2c,3a), DOTA (1,4,7,10-tetraazacyclododecane-
1,4,7,10-tetraacetic acid2d,e), and their structural analogs.3a,4a,c,e
The cavity sizes, basicities, steric factors and conformations of
the macrocyclic ligands all affect the physicochemical properties
of their Ln(III) complexes, in particular, the formation stabilities
and dissociation rates.10
Introduction
We have been interested in the basic understanding of the
coordination chemistry such as thermodynamic,1 kinetic,2
structural,3 and spectroscopic properties4 as well as the design,
synthesis and characterization of trivalent lanthanide (Ln)
aDepartment of Biological Science and Technology, National Chiao Tung
University, 75 Po-Ai Street, Hsinchu, Taiwan, 30039, R.O C
bDepartment of Cosmetics Application, Chin Min Institute of Technology,
Taiwan
cCurrent address: Department of Biomedical Imaging and Radiological Sci-
ences, National Yang-Ming University, No. 155, Sec. 2, Li-Nong St., Beitou,
Taipei, Taiwan, 112, R.O.C. E-mail: cachang@ym.edu.tw; Fax: +886-2-
28201093; Tel: +886-2-28201091
† Electronic supplementary information (ESI) available: See DOI:
10.1039/c0dt01440k
6268 | Dalton Trans., 2011, 40, 6268–6277
This journal is
The Royal Society of Chemistry 2011
©

Free lanthanide ions are not biologically essential and are toxic.
With the use of gadolinium(III) complexes as magnetic resonance
imaging (MRI) contrast agents as well as other potentially in
vivo/ex vivo applications using trivalent lanthanide complexes
such as luminescence probes for molecular imaging and artificial
nucleases, it is important to understand the formation stabilities
and dissociation kinetics of these lanthanide complexes. However,
the majority of the studies on lanthanide complexes are focused
on their thermodynamic, spectroscopic and structural aspects,
fewer studies are on the kinetic aspects. It is our belief that high
thermodynamic stability and low kinetic lability are preferred
when trivalent lanthanide complexes are to be used as safe
agents for medical and biological applications. For example, the
European Medicines Agency has classified the gadolinium(III)-
containing MRI contrast agents in three groups:5i (1) least
likely (safest) to release free gadolinium ions Gd3+ in the body
have a cyclical (macrocyclic) structure: Dotarem, Gadovist and
ProHance; (2) intermediate have an ionic linear structure: Mag-
nevist, MultiHance, Primovist (Eovist in the U.S.) and Vasovist;
and (3) most likely to release Gd3+ have a linear non-ionic
structure: Omniscan and OptiMARK. Among them, Omniscan,
the gadolinium(III) complex of DTPA-BMA (diethylenetriamine-
pentaacetic acid bis(methylamide)) causes the majority of cases of
a rare and serious syndrome termed nephrogenic systemic fibrosis
(NSF)5j and is probably related to its relatively low thermodynamic
stability and high kinetic lability.
Lately, we have been designing macrocyclic Ln(III) complexes
for use as artificial nucleases. A number of Ln(III) complexes
of DO2A (1,4,7,10-tetraazacyclododecane-1,7-diacetic acid) as
well as others have been studied to evaluate their efficiencies to
catalyze phosphodiester bond hydrolysis of a model compound,
BNPP (sodium bis(4-nitrophenyl)phosphate).9 DO2A forms quite
stable Ln(III) complexes and is kinetically rather inert. However,
two different sets of LnDO2A+ stability constants have been
reported. The one reported by Sherry et al. used an out-of-cell
potentiometric EDTA competition approach at pH 9–12 and
the logarithmic stability constants are in the range 16.6–20.6
for La(III)–Yb(III) DO2A complexes.11 The other set reported by
us using capillary electrophoresis measurements of equilibrium
concentrations of reaction species at lower pH indicated that the
constants are smaller and are in the range 10.94–13.16 for the
La(III)–Lu(III) DO2A complexes.4f The discrepancies remain to be
explained and further studies, particularly complex dissociation
kinetics, would be desirable because the complex formation
stabilities are inversely related to their dissociation reaction rates.
In this paper we report the results of our study on the acid-
catalyzed dissociation reaction kinetics of some Ln(III) complexes
of DO2A. The structural formulae of ligands DO2A, K21DA and
K22DA are shown in Scheme 1.
Scheme 1 Structural formulae of the ligands DO2A, K21DA and
K22DA.
Because the stability of CuDO2A (log Kf = 18.9–21.99)12–14 is much
greater than those of the corresponding Ln(III) complexes (log Kf =
10.94–13.16),4f the exchange reaction (eqn (1)) will be complete
under the experimental conditions (20-fold excess of Cu2+). This
has been confirmed by reacting LaDO2A+ (5.95 ¥ 10-5 mol dm-3)
and EuDO2A+ (5.0 ¥ 10-5 mol dm-3) separately with (1.0–5.0)
mmol dm-3 Cu2+ salt solutions at pH 4.50, [Acetate] = 5.0 mmol
dm-3. The final average absorbance values were 0.222 0.002 and
0.183 0.002, respectively, which were in consistence with that
if all DO2A ligand reacted with Cu2+ ions, i.e. 0.219 and 0.184,
respectively (eCuDO2A,280 nm = 3680). Thus, the experimental data fit
pseudo-first-order kinetics with respect to the complex.
LnDO2A+ + Cu2+ → CuDO2A + Ln3+
(1)
Independence of kobs on [Cu2+]. The kobs values for the disso-
ciation of the LnDO2A+ (Ln = La, Eu, Lu) complexes show
zero-order dependence on copper(II) ion concentration (Table S1,
ESI†). Such a [Cu2+]-independence in the dissociation kinetics of
macrocyclic LnK21DA+,2a and LnK22DA+,2b complexes as well
as other Ln(III) complexes15–17 were also reported previously. Pre-
sumably, the copper(II) ion is unable to form a rate-determining,
meta-stable binuclear intermediate with the LnDO2A+ complex
due to steric constraints imposed by the macrocycle when a Ln(III)
ion is already present in its cavity. For the LnDO2A+ complexes,
the order of the dissociation rates is LaDO2A+ > EuDO2A+ >
LuDO2A+ which is consistent with the order of their formation
stabilities, i.e. log Kf values 10.94, 12.99 and 13.16, respectively,4f
and the greater the formation stability, the slower the dissociation
rate.
It is noted that only in a few cases, [Cu2+]-dependence were
observed for the macrocyclic Ln(III) complex dissociation
reactions, e.g. Ln(III) complexes of 1,4,10-triaza-7,13-dioxacyclo-
pentadecane-N,N¢,N¢¢-tripropionic acid (N-pr3[15]aneN3O2),18
1,7,13-triaza-4,10,16-trioxacyclooctadecane- N,N¢,N¢¢-triacetic
acid (N-ac3[18]aneN3O3),19,20 and 1,7,13-triaza-4,10,16-trioxa-
cyclooctadecane- N,N¢,N¢¢-trimethylacetic acid (N-ma3[18]-
aneN3O3).19,20 The first group of complexes has a 15-membered
Results and discussion
LnDO2A+ dissociation reactions in the pH range 3.73–5.11
DO2A forms relatively stable complexes with Ln(III) ions, al-
though different stability constants were reported by different
research groups.4f ,11 Calculations using the respective values of the
protonation constants and the lower stability constants published
previously indicate that ≥99% of Ln(III) ion exists in the complexed
form at pH ≥ 6.5 under the conditions employed in this study.4f
This journal is
The Royal Society of Chemistry 2011
Dalton Trans., 2011, 40, 6268–6277 | 6269
©

Table 1 Effect of acetate concentration on LnDO2A+ dissociation kineticsa
Ln(III)
AC /mol-1dm3s-1
La
Eu
Er
Lu
k
(1.12 0.05) ¥
10-1
(3.23 0.16) ¥ 10-3
(3.99 0.53) ¥ 10-6
(8.62 1.31) ¥ 10-4
(4.04 0.43) ¥ 10-6
(3.73 0.72) ¥
10-4
k0/s-1
(9.15 1.56) ¥
(1.38 0.23) ¥
10-5
10-6
r2
0.995
1.16
0.993
1.066
0.935
1.004
0.930
0.977
˚
Ionic radius/A
a pH = 4.61, [LnDO2A+] = 5.0 ¥ 10-5 mol dm-3, [Acetate] = (1.0–5.0)¥10-3 mol dm-3, [Cu2+] = 1 ¥ 10-3 mol dm-3, T = 25 ◦C, m = 0.1.
macrocyclic chelate ring and the stability constants of their
Ln(III) complexes are: log Kf = 10.94–11.66. The latter two have
18-membered macrocyclic chelate rings and the log Kf value of the
Gd(III)-N-ac3[18]aneN3O3 complex is 18.02.20 These three groups
of Ln(III) complexes have relatively much faster dissociation
reaction rates, which presumably may be related to either the less-
stable and more flexible N-propionate chelate ring formation or
the presence of several isomeric structures in aqueous solution21,22
without the advantages of ligand pre-organization for metal ion
complexation.2e
log kAc values against the ionic radii of the Ln3+ ions is linear which
may imply that the activation energy of the complex dissociation
reaction is lowered more for the lighter lanthanide ions (plot
not shown). This may also be related to the number of inner-
sphere coordinated water molecules on the Ln3+ ion, i.e. the
more the number of inner-sphere coordinated water molecules, the
greater the chance for acetate ion to replace them, and the more
lowered the activation energy. DO2A offers six coordination donor
sites, the remaining inner-sphere coordination water molecules
on the lanthanide ions are possibly four for LaDO2A+, three for
EuDO2A+,20 two to three for ErDO2A+ and two for LuDO2A+.9a
Among the four LnDO2A+ complexes studied, the extent
of acetate catalysis in complex dissociation is the greatest for
LaDO2A+. At [Acetate] = 5.0 mmol dm-3, acetate catalysis con-
stitutes an unprecedented 87% of the dissociation pathway of the
LaDO2A+ complex. Under the similar conditions, acetate catalysis
constitutes only 21% of the dissociation pathway of LaK21DA+
complex.2a The relative ease of acetate catalysis to enhance the
dissociation rate of the LaDO2A+ complex as compared to that
of the LaK21DA+ complex, may be related to a stronger acetate
coordination to the LaDO2A+ complex and the intrinsic slower
dissociation rate of the LaDO2A+ complex. Note that EuK21DA+
has seven coordination donor atoms from the ligand and two
inner-sphere coordinated water molecules.4a On the other hand,
EuK22DA+ with eight coordination donor atoms from the ligand
and only one inner-sphere coordinated water molecule4a exhibits
no anion-dependence for its dissociation reaction.2b The fact
Acetate catalysis. Fig. 1 shows the plots of the kobs values
for the dissociation of selected LnDO2A+ complexes against
acetate ion concentrations and straight lines are obtained with
appreciable slopes (full data are listed in Table S2, ESI†). Linear
least squares regression analyses of the data give slope values kAC
(acetate-dependent rate constant) and intercept values k0 (acetate-
independent rate constant) with r2 values close or equal to 1 (eqn
(2), Table 1).
kobs = k0 + kAC[Acetate]
(2)
that k0 is in the order LaDO2A+ > EuDO2A+~ErDO2A+
>
LuDO2A+, is roughly consistent with the trend of their formation
constants (log Kf, ErDO2A = 13.31)4f (vide infra).
+
Proton-independent and proton-dependent pathways. The kobs
values for the dissociation reactions (eqn (1)), the temperatures at
which they were measured, and the concentrations of hydrogen ion
are given in Table S3 (ESI†). When the observed rate constants are
plotted against [H+], saturation curves were obtained for all the
Ln(III) complexes studied, i.e. the observed rate constants show
[H+]-dependence at low acid concentrations but become [H+]-
independent at high acid concentrations (Fig. 2). All plots show
appreciable intercepts that correspond to the proton-independent
dissociation rate constants. The proposed mechanisms are shown
in Scheme 2 where Ln, L, LnL and LnHL are the Ln(III) ion,
ligand (DO2A), Ln(III) complex and protonated Ln(III) complex,
respectively.
Fig. 1 Plots of dissociation kobs (s-1) values vs. [Acetate] (mol dm-3) for
selected LnDO2A+ complexes; La (inset); Eu ( ); Er ( ); Lu (᭢).
᭹
᭺
The enhanced rates of dissociation of these complexes in the
presence of acetate ions can be attributed to the acetate ion
complexation. The parent complexes are positively charged, and
by the introduction of a negatively charged acetate ion into the
coordination sphere they become neutral and labilized with respect
to hydrogen ion attack. Under the experimental conditions, the
observed kAC value is the highest for LaDO2A+ and it decreases
with increasing atomic number of the Ln3+ ion. A plot of the
The rate law consistent with the proposed mechanisms is:
kobs = kd + kAC[Acetate] + K¢klim[H+]/(1 + K¢[H+])
This journal is The Royal Society of Chemistry 2011
(3)
6270 | Dalton Trans., 2011, 40, 6268–6277
©

Note that k0 in eqn (2) is equal to kd + K¢klim[H+]/(1 + K¢[H+])
in eqn (3). Because [Acetate] = 5.0 mmol dm-3, the term kd +
k
AC[Acetate] should not vary with [H+] and eqn (3) can be written
as
kobs = kd¢ + K¢klim[H+]/(1 + K¢[H+])
(5)
where kd¢ = kd + kAC[Acetate]. The values of kd¢, klim and K¢ are
resolved by an unweighted least-squares analysis and are listed in
Table 2.
It is further noted that if the kAC values obtained in Table 1
at 25 ◦C are used to calculate the kAC[Acetate] values and are
subtracted from the corresponding kd values, the resulted kd values
Scheme 2 The proposed mechanisms for the dissociation of LnDO2A+
in the pH range 3.73–5.11.
¢
are all negative. This indicates that the direct dissociation pathway
is negligible in the present system and the rate law can be simplified
as
kobs = kAC[Acetate] + K¢klim[H+]/(1 + K¢[H+])
(6)
¢
and kd ꢀ kAC[Acetate]. Thus, the kAC values for various LnL
complexes at different temperatures could be calculated by
¢
kd /[Acetate] using the data in Table 2, where [Acetate] = 0.005
mol dm-3.
The regular decrease in kd (i.e. kAC) values of the LnDO2A+
¢
complexes from La(III) to Er(III) have been discussed previously
(vide supra) which happens to be the same trend of their
thermodynamic stabilities. On the other hand, the limiting rate
constants (klim) are in a slightly different order La > Pr > Er >
Eu. The fact that all the log K¢ values are in the 2–3 range which
are similar to those found for the heavy lanthanide (Tb3+–Lu3+)
complexes of K21DA, also reflects similarities of the dissociation
mechanisms of the LnDO2A+ and the heavy LnK21DA+ (Ln =
Tb3+–Lu3+) complexes.
Fig. 2 Plots of the dissociation kobs values vs. [H+] for the PrDO2A+
complex at different temperatures: 19 ◦C ( ); 25 ◦C ( ); 33 ◦C (᭢);
᭹
᭺
41 ◦C (ꢀ).
The klim values for the LnDO2A+ complexes are about two to
three orders of magnitude smaller than the corresponding ones
of the heavy LnK21DA+ complexes, indicating that the ligand
DO2A with a smaller macrocyclic ring size tends to form more
rigid Ln(III) complexes which dissociate more slowly than those
of the larger-sized and more flexible ligand, LnK21DA+. Indeed,
where kd is the proton-independent direct dissociation rate con-
stant, K¢ (= kH/kH¢) is the equilibrium constant for the formation
of the monoprotonated complex (eqn (4)), and klim is the limiting
rate constant for the protonated lanthanide complex.
LnL+ + H+ = LnLH2+
(4)
Table 2 Resolved rate and equilibrium constants for the dissociation reactions of LnDO2A+ complexes at various temperatures
Ln3+
La
T/◦C
kd¢/s-1
k
lim/s-1
K¢/mol-1 dm3
r2
19
25
33
41
(1.88 0.59) ¥ 10-4
(3.09 0.79) ¥ 10-4
(3.82 1.56) ¥ 10-4
(6.86 1.33) ¥ 10-4
(1.80 0.60) ¥ 10-3
(2.70 0.53) ¥ 10-3
(3.82 0.45) ¥ 10-3
(6.00 0.27) ¥ 10-3
(3.86 2.56) ¥ 10+3
(4.91 2.20) ¥ 10+3
(8.02 3.06) ¥ 10+3
(9.52 1.70) ¥ 10+3
0.983
0.991
0.990
0.998
Pr
Eu
Er
19
25
33
41
(1.55 0.65) ¥ 10-5
(1.94 0.40) ¥ 10-5
(3.24 1.43) ¥ 10-5
(7.79 2.04) ¥ 10-5
(1.97 0.64) ¥ 10-4
(2.29 0.11) ¥ 10-4
(3.11 0.32) ¥ 10-4
(6.32 0.77) ¥ 10-4
(3.90 2.57) ¥ 10+3
(8.06 1.30) ¥ 10+3
(9.24 3.49) ¥ 10+3
(6.84 2.40) ¥ 10+3
0.984
0.992
0.990
0.993
19
25
33
41
(2.20 1.89) ¥ 10-6
(7.86 2.63) ¥ 10-6
(1.15 0.60) ¥ 10-5
(2.03 1.12) ¥ 10-5
(5.71 1.47) ¥ 10-5
(1.14 0.15) ¥ 10-4
(1.98 0.51) ¥ 10-4
(3.56 0.75) ¥ 10-4
(4.49 2.51) ¥ 10+3
(5.46 1.73) ¥ 10+3
(4.28 2.31) ¥ 10+3
(4.89 2.36) ¥ 10+3
0.986
0.995
0.988
0.989
19
25
33
41
(1.07 0.41) ¥ 10-6
(3.03 0.92) ¥ 10-6
(3.93 0.94) ¥ 10-6
(8.54 12.5) ¥ 10-6
(9.24 1.22) ¥ 10-5
(1.80 0.44) ¥ 10-4
(2.09 0.12) ¥ 10-4
(5.38 1.63) ¥ 10-4
(2.09 0.42) ¥ 10+3
(1.58 0.55) ¥ 10+3
(3.33 0.36) ¥ 10+3
(3.34 1.88) ¥ 10+3
0.999
0.998
1.000
0.990
Lu
25
(4.99 1.34) ¥ 10-7
(3.20 0.33) ¥ 10-5
(2.31 0.38) ¥ 10+3
0.999
This journal is
The Royal Society of Chemistry 2011
Dalton Trans., 2011, 40, 6268–6277 | 6271
©

Table 3 Molar activation parameters for acetate-dependent dissociation and proton-dependent dissociation of selected LnDO2A+, LnK21DA+ and
LnK22DA+ complexes. DG* values are calculated at 25 ◦C
LnL
DHAC */kcal
DSAC */cal K-1
DGAC */kcal
DHlim*/kcal
DSlim*/cal K-1
DGlim*/kcal
LaDO2A+
PrDO2A+
EuDO2A+
ErDO2A+
9.4 1.4
12.7 2.2
16.5 3.9
15.2 3.0
-32.6 7.0
-26.8 9.9
-16.9 12.9
-22.9 11.9
19.1
20.0
21.1
22.0
9.1 0.5
8.8 2.0
14.2 1.0
12.7 2.7
-39.9 2.3
-45.5 10.9
-29.2 3.6
-33.4 10.6
21.0
22.4
22.9
22.7
DHd*/kcal
DSd*/cal K-1
DGd*/kcal
DHlim*/kcal
DSlim*/cal K-1
DGlim*/kcal
LaK21DA+
EuK21DA+
ErK21DA+
LuK21DA+
EuK22DA+
4.7
6.9
7.9
8.0
-49.5
19.5
21.4
20.8
20.1
22.1
n/a
n/a
14.0
15.6
n/a
-48.6
-43.4
-20.8
-12.6
20.2
19.4
-40.5
7.9 1.7
-47.5 9.4
the kobs values of the LnL+ complexes (L = K21DA and DO2A)
Table 3 lists the values of DH*, DS* and DG* (25 ◦C)
obtained from the Eyring plots for the acetate-dependent and
proton-dependent dissociation reaction pathways of LnDO2A+
(plots not shown). For comparison purposes, the data for selected
LnK21DA+ complexes are also included in Table 3. An interesting
isokinetic (linear) relationship is observed when the values of DH*
are plotted against those of DS* (Fig. 3).24a,25 This indicates that
the dissociation of all the LnDO2A+ complexes studied follow a
similar dissociation mechanism.
obtained in similar pH ranges are in the order LnK21DA+
LnDO2A+.
>
Activation parameters. The activation parameters for the
proton-independent and proton-dependent pathways have been
obtained from the temperature dependence measurements of rate
constants. According to the Eyring equation (7),23
ln(k/T) = ln(kB/h) - (DH*/RT) + (DS*/R)
(7)
The EuDO2A+ complex has the greatest temperature variations
on both kAC and klim values leading to the highest DHAC* and
DHlim* values among all LnDO2A+ complexes studied. On the
other hand, due to its less unfavorable DS* barrier, the trend for the
overall DG* values is still LaDO2A+ < PrDO2A+ < EuDO2A+ £
ErDO2A+, for both the acetate-dependent and proton-dependent
pathways, which is in the same order as their formation stabilities.
(Note that the DGlim* values for the respective EuDO2A+ and
ErDO2A+ complexes are 22.9 and 22.7 kcal mol-1, which might
be considered similar within the experimental error limits.)
The DHAC* values of LnDO2A+ complexes are all greater
than those of the DHd* values of the corresponding LnK21DA+
complexes. On the other hand, the DSAC* values of LnDO2A+ are
all less negative than those of the DSd* values of the corresponding
LnK21DA+ complexes. These observations might be related to the
fact that DO2A is more rigid than K21DA and that the acetate-
dependent dissociation pathway involves charge neutralization
when the LnDO2A(Ac) ternary complex is formed,24b respectively.
the activation parameters of these dissociation reactions can be
obtained by plotting of ln(k/T) vs. 1/T, where k is the rate
constant, R is gas constant (1.987 cal mol-1 K-1—this unit is used
for convenient comparisons with previously published data), kB is
Boltzmann’s constant (1.381 ¥ 10-23 J K-1), h is Planck’s constant
(6.626 ¥ 10-34 J s) and T is absolute temperature (Fig. 3). The slope
and intercept values of these plots yield the activation enthalpy
(DH*) and entropy (DS*) data:
slope = (-DH*/R)
(8)
intercept = (DS*/R) + ln(kB/h)
(9)
The DGAC* values are all smaller than the corresponding DGlim
*
values at 25 ◦C, indicating that the acetate-dependent dissociation
pathways are quite competitive as compared to the proton-
dependent, limiting rate ones owing to less negative DS* values.
The DGlim* values at 25 ◦C for the proton-dependent pathways
are all greater than those of the LnK21DA+ complexes,2a reflecting
the higher thermodynamic stability of the LnDO2A+ complexes.
Comparing the DHd* and DSd* values of LnK21DA+ and
LnK22DA+ complexes, it is observed that the activation entropy
barrier is the dominant factor in controlling the direct dissociation
of LnK21DA+ and LnK22DA+ complexes, i.e. entropy-controlled
solvolytic dissociation kinetics,26 but it is relatively less important
in the acetate-dependent dissociation of LnDO2A+ complexes,
particularly for the heavy LnDO2A+ complexes. On the other
hand, activation enthalpy barrier is the dominant factor in
controlling the proton-dependent, limiting-rate dissociation path-
way for heavy LnK21DA+ and LnDO2A+ complexes (vide infra).
Fig. 3 Plots of values of DH* vs. DS*. Data are from Table 3: ( ) k
᭹
AC
pathway, pH range 3.73–5.11; ( ) klim pathway, pH range 3.73–5.11; (᭢)
᭺
kH pathway, pH range 1.75–2.65 (vide infra).
The free energy of activation, DG*, can be calculated according
to the following equation:
(10)
DG* = DH* - TDS*
6272 | Dalton Trans., 2011, 40, 6268–6277
This journal is
The Royal Society of Chemistry 2011
©

LnDO2A+ dissociation reactions in the pH range 1.75–2.65
An indicator method similar to the one used to study metal
ion–DOTA complex formation was chosen to study LnDO2A+
dissociation reactions (eqn (11)) in the pH range 1.75–2.65.27
LnDO2A+ + nH+ → Ln3+ + HnDO2A(n-2)+, n = 1–4
(11)
In this case, the pre-formed complex is subjected into an acid
solution with a dye indicator, cresol red. The cresol red has strong
absorption in the UV-Vis range and the changes of absorbance
values at 247 and 518 nm vary with [H+], in a linear relationship. As
the complex dissociates, the ligand is protonated and causes slight
incremental increases in solution pH which can be shown by the
indicator absorbance changes, and the rate could be monitored.
The experimental rate data of the LnDO2A+ dissociation reaction
monitored by the indicator method fit pseudo-first-order kinetics.
However, because the absorbance change is usually small, this
method could cause relatively larger uncertainties with still
acceptable results for the determination of the dissociation rates.
Fig. 4 Plots of the dissociation kobs values vs. [H+] for the EuDO2A+
complex at different temperatures: 19 ◦C ( ); 25 ◦C ( ); 33 ◦C (᭢);
᭹
᭺
41 ◦C (ꢀ).
The observed rate constants for the Ln(III) complexes obey the
rate law:
Dissociation kinetics. When LnDO2A+ dissociates, two equiv-
alents of protons would need to occupy the DO2A nitrogen
protonation sites with pKa values 10.94 and 9.55, respectively. The
third and fourth DO2A protonation sites are at the carboxylate
positions with the respective pKa values 3.85 and 2.55. At
the present [H+] range studied, i.e. 0.00288–0.0346 mol dm-3,
the third protonation site will also be occupied. The minimum
concentration of the strong acid (HClO4) required to induce ≥99%
dissociation of LnDO2A+ is 3 ¥ 5.0 ¥ 10-4 mol dm-3 = 1.5 ¥
10-3 mol dm-3 which is about 48% less than the lowest HClO4
concentration (0.00288 mol dm-3) employed for the current study.
Thus, the pseudo-first order condition required for [H+] is no
longer maintained. In cases like this, the initial rate method was
used to obtain the kobs values. Table S4 (ESI†) lists the kobs values
and plots of these values vs. [H+] at the four temperatures studied
display a first-order dependence, without the saturation behavior,
e.g. Fig. 4.
kobs = kd + kH[H+]
(12)
where kd and kH are the respective direct and proton-catalyzed
dissociation rate constants. Fitting the kobs data to eqn (12) gives
the intercept values kd and the slope values kH. However, due to
the relatively greater uncertainties of rate constant measurements
particularly at lower [H+] discussed above, the fits of kd values were
scattered around zero and the direct dissociation pathways were
assumed to be not significant. The kobs data were therefore fitted
to the simplified eqn (kobs = kH[H+] and the results are listed in
Table 4. Similar results were reported for rates of dissociation of
complexes formed with other macrocyclic ligands.28
The proton-catalyzed LnDO2A+ dissociation pathways go
through monoprotonated adducts. It is possible that the initial
proton attack occurs at a dissociated carboxylate oxygen. The
protonated species then dissociate directly. The rate-determining
step is probably associated with the distortion of the macrocycle
ring and/or the protonation of the first nitrogen site. At a given
temperature, both the kobs and the resolved kH values decrease in
the order: LaDO2A+ > PrDO2A+ > ErDO2A+ > EuDO2A+ >
LuDO2A+. This is roughly consistent with the fact that LnDO2A+
complex formation stability increases with increasing atomic
number (except that of Er and Eu), indicating that ionic interaction
between the Ln(III) ion and the macrocyclic ligand is probably the
major one. The absence of kd values indicates that these complexes
A general reaction scheme consistent with the observed behavior
for these complexes is given below (Scheme 3), where Ln, L, LnL
and LnHL are the Ln(III) ion, ligand (DO2A), Ln(III) complex
and protonated Ln(III) complex, respectively.
Scheme 3 The proposed mechanisms for the dissociation of LnDO2A+
in the pH range 1.75–2.65.
Table 4 The kH values for the dissociation reactions of LnDO2A+ at 19, 25, 33 and 41 ◦C
kH/mol-1 dm3 s-1
T/◦C
La
Pr
Eu
Er
Lu
19
(2.36 0.05) ¥ 10-1
(3.66 0.06) ¥ 10-1
(8.65 1.08) ¥ 10-1
1.00 0.03
(1.17 0.13) ¥
10-1
(1.21 0.17) ¥ 10-1
(2.49 0.22) ¥ 10-1
(5.75 0.06) ¥ 10-1
1.15 0.06
(1.59 0.12) ¥ 10-2
(2.34 0.23) ¥ 10-2
(8.54 0.30) ¥ 10-2
(1.57 0.04) ¥ 10-1
25
2.23 0.13
(1.92 0.04) ¥
10-1
33
(3.14 0.07) ¥
10-1
41
(6.34 0.14) ¥
10-1
This journal is
The Royal Society of Chemistry 2011
Dalton Trans., 2011, 40, 6268–6277 | 6273
©

Table 5 Activation parameters for proton-catalyzed dissociation of selected LnDO2A+, LnK21DA+ and LnK22DA+ complexes in the pH range
◦
1.75–2.65; DGH* values are calculated at 25 Ca
LnDO2A+
LnK21DA+
LnK22DA+
DHH*/kcal
DSH*/cal K-1
DGH*/kcal
DHH*/kcal
DSH*/cal K-1
-23.7
DGH*/kcal
13.6
DHH*/kcal
DSH*/cal K-1
-17.4
DGH*/kcal
18.1
La
Pr
Eu
Er
Lu
n/a
6.5
n/a
7.4
n/a
n/a
12.9
n/a
6.5
n/a
7.7
12.2 2.3
13.1 0.8
18.1 0.6
19.6 2.4
-19.5 7.5
-18.0 2.6
-0.92 2.2
0.25 7.9
18.0
18.4
18.3
19.5
-26.9
15.4
-37.7
17.7
-31.7
17.2
a [H+] = 2.88 ¥ 10-3–2.31 ¥ 10-2 mol dm-3 for LnDO2A; [H+] = 8.4 ¥ 10-6–2.47 ¥ 10-4 mol dm-3 for LnK21DA; [H+] = 5 ¥ 10-4–7.5 ¥ 10-3 mol dm-3 for
LnK22DA.
are structurally rigid, probably due to that the ligand DO2A is
pre-organized. This is true for other cyclen-structurally related
lanthanide complexes such as LnDO3A2c and LnDOTA-.2d
and the Ln(III) ion, and the overall formation stability of the Ln(III)
macrocyclic complex. Fig. 5 shows the plots of the log Kf values
for the Ln(III) complexes of DO2A,4f K21DA1c and K22DA.1a It is
seen that DO2A with the smallest 12-membered macrocyclic cavity
size among the three ligands tends to form stronger complexes
with heavier Ln(III) ions due to mainly strong ionic interactions.
However, the fact that the stabilities for the LnDO2A+ complexes
from Sm(III) to Lu(III) do not very as much as those from La(III)
to Nd(III) indicate probably that the negative cavity size effect
tends to be gradually important in going from the smaller, heavier
Ln(III) ions to the larger, lighter Ln(III) ions. On the other hand,
K22DA with a 18-membered macrocyclic ring forms stronger
complexes with the larger, lighter Ln(III) ions due to its better cavity
size to Ln(III) ion diameter fit, allowing perhaps close-to-normal
Ln(III) ion–ligand ionic interactions without too much ligand
stereochemical constraints (i.e. structural “dislocation”).29 The
stability trend of the LnK21DA+ complexes with 15-membered
macrocycles could be considered as the compromise of that
discussed above for the LnDO2A+ and LnK22DA+ complexes.
The following two simple postulates are tentatively proposed to
account for the general activation parameter observations for the
Ln(III) macrocyclic complexes discussed above:
1. Complexes with greater formation stabilities tend to have
greater DHH* barriers relative to others of the same macrocyclic
ligand.
2. The more macrocyclic ligand conformational flexibility, the
greater the relative DSH* barrier toward DGH*.
It is known that the formation of Ln(III)-18-crown-6 complexes
are entropy driven.30 Formation of Ln(III) complexes with N-
donor ligands is also entropy-driven but has high activation
enthalpies.31 It was also reported that the activation parameters
for the formation and dissociation reactions of dioxouranium(VI)–
diphosphonic acid complexes are comparable, and although the ki-
netic activation enthalpy barriers are greater, the thermodynamic
stabilities are driven by favorable entropy change.32 Note that there
are still subtle variations remained to be explained to test the above
postulates, e.g. why the activation enthalpy barrier of LuK22DA+
is greater that of EuK22DA+.
Activation parameters. The activation parameters for the
proton-dependent pathways have been obtained from the Eyring
plots using the temperature-dependent rate constants and are
listed in Table 5 (plots not shown), together with those of
LnK21DA+ and LnK22DA+ complexes for comparison purposes.
The DGH* values of the LnDO2A+ complexes are, as expected,
smaller than both DGAC* and DGlim* values obtained at higher pH
range. On the other hand, these DGH* values are larger than those
of the corresponding LnK21DA+ and LnK22DA+ complexes,
reflecting the greater thermodynamic stability and kinetic inertness
of the LnDO2A+ complexes. The isokinetic (linear) relationship
observed for the higher pH range is also observed in this lower pH
range when the values of DH* are plotted against those of DS*
(Fig. 3).
In general, the dissociation has higher activation DHH* enthalpy
barrier for the LnDO2A+ complexes than those of LnK21DA+ and
LnK22DA+ complexes, except possibly LaDO2A+. Among the
LnDO2A+ complexes studied, heavier LnDO2A+ complexes tend
to have greater DHH* barriers than those of the lighter complexes.
On the other hand, the activation entropy barriers become
the controlling factor for the dissociation of the LnK21DA+
and LnK22DA+ complexes, except for LaK21DA+. The relative
activation entropy barrier toward DGH* is in the order LnDO2A+
< LnK21DA+ < LnK22DA+. One possible way to explain these
observations is to consider the ligand cavity size-Ln(III) ion
diameter fit, the strength of ionic interaction between the ligand
Comparisons of the dissociation kinetic data of LnDO2A+ with
other Ln(III) macrocyclic complexes
LnDO2A+ (LnL+) complexes are suitable candidates for the study
of dissociation kinetics in a wide pH range; the fast equilibrium
formation of the metastable intermediate (LnLH2+) at higher pH is
Fig. 5 Plots of the log K values for the Ln(III) complexes of DO2A (
,
᭹
f
ref. 4f ), K21DA ( , ref. 1c) and K22DA (᭢, ref. 1a).
᭺
6274 | Dalton Trans., 2011, 40, 6268–6277
This journal is
The Royal Society of Chemistry 2011
©

Table 6 Various kinetic rate and equilibrium constants of selected macrocyclic Ln(III) complexes at 25 ◦C
Ln complex
kd/s-1
kH/mol-1 dm3 s-1
k
lim/s-1
K¢/mol-1 dm3
LaDO2A+,a
PrDO2A+,a
(3.09 0.79) ¥ 10-4
(1.94 0.40) ¥ 10-5
2.23 0.13
(2.70 0.53) ¥ 10-3
(2.29 0.11) ¥ 10-4
(4.91 2.20) ¥ 10+3
(8.06 1.30) ¥ 10+3
(3.66 0.06) ¥
10-1
EuDO2A+,a
ErDO2A+,a
LuDO2A+,a
(7.83 2.68) ¥ 10-6
(3.05 0.94) ¥ 10-6
(4.99 1.34) ¥ 10-7
(1.92 0.04) ¥
10-1
(1.14 0.15) ¥ 10-4
(1.80 0.46) ¥ 10-4
(3.20 0.33) ¥ 10-5
(5.48 1.76) ¥ 10+3
(1.58 0.56) ¥ 10+3
(2.31 0.38) ¥ 10+3
(2.49 0.22) ¥
10-1
(2.34 0.23) ¥
10-2
CeDO3Ab
(1.8 0.8) ¥ 10-3
(4.4 0.1) ¥ 10-4
~10-7–5 ¥ 10-10
56.7
GdDO3Ab
GdDOTA-,c
CeTETA-,d
LaK21DA+,e
PrK21DA+,e
EuK21DA+,e
ErK21DA+,e
LuK21DA+,e
LaK22DA+,f
PrK22DA+,f
0.101
(4.69 0.15) ¥ 10-2
(4.14 0.20) ¥ 10-3
(1.76 0.06) ¥ 10-3
(3.32 0.13) ¥ 10-3
(1.08 0.06) ¥ 10-2
(3.97 0.20) ¥ 10-4
(9.14 1.40) ¥ 10-5
(7.18 0.11) ¥ 102
(8.19 0.15) ¥ 101
(3.11 0.04) ¥ 101
(1.04 0.12) ¥ 10-2
(4.11 0.36) ¥ 10-2
(6.64 0.85) ¥ 10+3
(1.22 0.11) ¥ 10+4
1.21 0.02
(5.36 0.09) ¥
10-1
EuK22DA+,f
(4.31 1.47) ¥ 10-4
(5.70 0.96) ¥
10-1
LuK22DA+,f
EuAc3[15]N3O2
EuAc3[18]N3O3
(5.58 0.87) ¥ 10-4
(1.62 0.03) ¥ 10-5
(1.32 0.03) ¥ 10-1
1.64 0.08
1.38 0.06
(9.76 0.12) ¥ 103
g
h
a This work; kd = kAC [Acetate = 0.005 mol dm-3]. b Ref. 2c. c Ref. 10b, 28. d Ref. 2d. e Ref. 2a. f Ref. 2b. g Ref. 18. h Ref. 19.
no longer stable at lower pH and the proton-dependent saturation
behavior at higher pH becomes linear dependent at lower pH.
The kinetic data obtained for the LnDO2A+ complexes in two
pH ranges allow their comparisons with the more stable cyclen-
based complexes (e.g. LnDOTA- and LnDO3A) and complexes
with larger macrocyclic rings (e.g. LnK22DA+ > LnK21DA+).
The formation stability constants for a number of cyclen-
based macrocyclic Ln(III) complexes are in the order LnDOTA-
> LnDO3A > LnDO2A+. For macrocyclic ligands with 12, 15
pre-organized for metal ion complexation2e or have a number of
isomeric structures with much more flexible ring conformations,21
which lead to faster dissociation rates.
Concerning the formation constants of LnDO2A+ and summary
In a previous paper we have used an extra-thermodynamic
relationship of a good linear correlation between the log KCaL
values and log KGdL values (r2 = 0.95) for a number of selected
macrocyclic and linear aminopolycarboxylate ligands to estimate
and 18-membered rings respectively, the order is LnDO2A+
>
LnK22DA+ > LnK21DA+, with a few exceptions (e.g. LnDO2A+
where Ln = La, Ce and Pr). The larger macrocycles, K22DA
and K21DA are more flexible than the smaller 12-membered ring
DO2A. The basicity of DO2A is also greater than K21DA and
K22DA.
+
that the log KGdDO2A value should be in the range 12–13, and
is consistent with what we obtained.4f From the present kinetic
studies, we found that the kH values for the LnDO2A+ com-
plexes (log K range = 10.94–13.16) are closer to those of the
LnK22DA+ complexes (log K range = 12.21–10.84). For example,
the LaDO2A+ complex with a log K value of 10.94 has a kH value
of 2.23 mol-1 dm3 s-1, which is greater than that of the LaK22DA+
complex (1.21 mol-1 dm3 s-1) and is consistent with a higher log K
value of 12.21 for the LaK22DA+ complex. Similarly, EuDO2A+
(log K = 12.99) has a kH value of 0.192 mol-1 dm3 s-1 and is smaller
than that of EuK22DA+ (kH = 0.570 mol-1 dm3 s-1, log K 12.02).
In summary, LnDO2A+ (LnL+) complexes are suitable candi-
dates for the study of dissociation kinetics in a wide pH range;
the metastable intermediate (LnLH2+) at higher pH is no longer
stable at lower pH and the acid-dependent saturation behavior at
higher pH becomes linear dependence at lower pH. The absence
of an [H+]-independent pathway in both pH ranges indicates that
LnDO2A+ complexes are kinetically rather inert. The obtained
Table 6 lists various kinetic rate and equilibrium constants of
selected macrocyclic lanthanide complexes at 25 ◦C. The overall
dissociation rate constant kH should be in the following order
and are found as expected: LnDOTA- < LnDO3A < LnDO2A+
< LnK22DA+ < LnK21DA+. For example, for GdDOTA-,
GdDO3A, EuDO2A+, EuK22DA+, EuK21DA+ complexes at
25 ◦C, the measured kH values are: ~0 (not observed), 4.4
¥
10-4, 1.92 ¥ 10-1, 0.57 and 31.1, respectively. (The data of Gd(III)
complexes are used when Eu(III) data were not available).
The complexes CeTETA- (log Kf
=
13.12, TETA is
1,4,8,11-tetraazacyclotetradecane-1,4,8,11-tetraacetic
acid)4d
and EuAc3[18]N3O3 (log Kf = 17.26, Ac3[18]N3O3 is 1,7,13-
trioxa-4,10,16-triazacyclooctadecane-N,N¢N¢¢-triacetic
acid)21
k
AC values follow the order: LaDO2A+ > PrDO2A+ > EuDO2A+
> ErDO2A+ > LuDO2A+, whereas the klim and kH values follow
the order: LaDO2A+ > PrDO2A+ > ErDO2A+ > EuDO2A+ >
have much faster dissociation kinetic rates as compared to
their structural analogues with similar complex formation
stabilities. This is presumably because both ligands are either not
This journal is
The Royal Society of Chemistry 2011
Dalton Trans., 2011, 40, 6268–6277 | 6275
©

LuDO2A+, mostly consistent with their thermodynamic stability
order, i.e. the more thermodynamically stable the more kinetically
inert. Of some interest is that acetate catalysis constitutes an
unprecedented 87% of the dissociation pathway of the LaDO2A+
complex at [Acetate] = 5.0 mmol dm-3, 25 ◦C.
initial and final absorbance values, respectively. In a few cases,
particularly in the lower pH range studies involving low [H+]
and the faster dissociating LaDO2A+ complex, the initial rate
method was employed to obtain the kobs values by fitting the data
to the equation: DA/Dt/(bel) = kobs[ML]0, where DA is the change
of absorbance in the time period Dt, b = 1.0, el is the molar
absorptivity of cresol red at the wavelength l, and [ML]0 is the
initial LnDO2A+ concentration. Sigma plots were used for curve
fitting. Each value of kobs reported represents the average of three
replicate runs.
Experimental
Materials and standard solutions
Reagent grade lithium acetate (Aldrich), lithium perchlo-
rate (Aldrich), lanthanide (La, Ce, Pr, Eu, Er, Lu) nitrates
(Aldrich/Alfa), copper nitrate (MCB), acetic acid (MCB), per-
chloric acid (Fisher) and other reagents were used as re-
ceived. The ligand DO2A was synthesized by following a
procedure published elsewhere9c and was found to be analytically
and spectroscopically pure. All solutions were made in deion-
ized water. Aqueous solutions of the ligand were prepared by
weight and standardized by potentiometric and complexometric
titrations with standardized NaOH solution and cerium nitrate
solutions, respectively.1a,c Standard solutions (~0.01 mol dm-3) of
lanthanide nitrates were standardized by EDTA titrations with
xylenol orange as indicator.
Complexes were made in situ by mixing appropriate amounts
of lanthanide nitrate and ligand solutions (molar ratio ~ 1.02 : 1.0)
and adjusting the pH to ca. 6.5–7.0 with (CH3)4NOH. For studies
at high pH range, the complex concentration in the reaction
mixtures was kept at 5.0 ¥ 10-5 mol dm-3 and the buffer solutions
were made by using a constant acetate ion concentration (e.g.
Acknowledgements
The authors wish to thank the National Science Council of the
Republic of China (Taiwan) for financial support (grant number
NSC-98-2113-M-010-001-MY3).
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5
¥ 10-3 mol dm-3) and varying [Acetic acid]. For studies at
lower pH, the pH of each solution was adjusted by HClO4
and the final concentration of reagents were [LnDO2A+]
=
5.0 ¥ 10-4 mol dm-3, [cresol red] = 4.17 ¥ 10-5 mol dm-3.
The ionic strength was adjusted to 0.1 mol dm-3 with LiClO4
for studies at both pH ranges. Hydrogen ion concentrations were
calculated from the pH measurements by the expression -log[H+] =
pH - 0.11.2a
Kinetic measurements
Spectra and kinetic runs were made with a Hewlett Packard 8453
diode-array spectrophotometer equipped with a thermostat cell
holder with a constant temperature circulating bath (FIRSTEK
SCIENTIFIC B403). The solution temperature was controlled
to within 0.1 ◦C. For studies at higher pH range, as the
Ln(III) complexes do not show appreciable absorption in the near-
ultraviolet region, copper(II) ion (1.0 ¥ 10-3 mol dm-3, 20 fold) was
used as the scavenger of the free ligand and the reaction kinetics
were followed by monitoring the growth in absorbance due to the
CuDO2A complex formation at 280 nm. For studies at lower pH,
the dye, cresol red, was used to monitor the dissociation rate due
to the fact that complex dissociation results in slight pH increase
and therefore the cresol red absorbance changes.27 The absorbance
changes of cresol red at 247 and 518 nm in the range of 0.02–0.1
unit were measured.
In most cases, pseudo-first order rate constants (kobs) were
calculated using the integral rate method by fitting the absorbance
(A) vs. time (t) data for at least three half-lives to the equation: At =
t
A0 + A•(1 - e-k ), where At, Ao and A• are the instantaneous,
obs
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The Royal Society of Chemistry 2011
©

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©