Thermodynamic and kinetic analysis of isothermal microcalorimetric data: Applications to consecutive reaction schemes

Article abstract of DOI:10.1016/S0040-6031(98)00622-4

Recent developments have led to a general procedure that allows the analysis of isothermal microcalorimetric data to determine both kinetic and thermodynamic information. Such an analysis means that isothermal microcalorimetry is a powerful technique with which to gain information on a wide range of reactions. Previously, the method of analysis has, principally, been applied to simple, two-state, solution phase reactions. It is the purpose of the data presented here to show how such an analysis may be applied to solution phase reactions that follow more complex, consecutive reaction pathways using, as a model and example, the acid catalyzed hydrolysis of potassium hydroxylamine trisulfonate.

Full text of DOI:10.1016/S0040-6031(98)00622-4

Thermochimica Acta 328 (1999) 39±45
Thermodynamic and kinetic analysis of isothermal microcalorimetric data:
applications to consecutive reaction schemes
Simon Gaisford1, Andrew K. Hills, Anthony E. Beezer*, John C. Mitchell
Experimental Thermodynamics Group, School of Physical Sciences, The University, Canterbury, Kent CT2 7NH, UK
Accepted 17 November 1998
Abstract
Recent developments have led to a general procedure that allows the analysis of isothermal microcalorimetric data to
determine both kinetic and thermodynamic information. Such an analysis means that isothermal microcalorimetry is a
powerful technique with which to gain information on a wide range of reactions. Previously, the method of analysis has,
principally, been applied to simple, two-state, solution phase reactions. It is the purpose of the data presented here to show how
such an analysis may be applied to solution phase reactions that follow more complex, consecutive reaction pathways using, as
a model and example, the acid catalyzed hydrolysis of potassium hydroxylamine trisulfonate. # 1999 Elsevier Science B.V.
All rights reserved.
Keywords: Microcalorimetric data; Solution phase reactions; Thermodynamic analysis
1. Introduction
for example [1]). Knowledge of the rate constant
allows the prediction of a shelf-life if the reaction
order is known.
In this work we show how it is possible to analyze
microcalorimetric data using modi®ed kinetic
equations. The equations presented herein apply to
solution phase, consecutive reaction schemes. The
analysis of data deriving from other reaction schemes
will be discussed in a future publication. Solution
phase reactions have been well characterized and
often follow speci®c reaction pathways. More
often than not, it is possible to express such pathways
using relatively simple kinetic expressions (see,
The output from a heat conduction, isothermal
microcalorimeter is power versus time, and it follows
that analysis of these data allows information to be
obtained on both the kinetics and thermodynamics of
the reaction under investigation. In the absence of data
obtained via other chemical assay techniques (for
example, HPLC), thermodynamic data do not allow
molecular interpretation, but kinetic data form the
basis of mechanistic investigations. Many systems
that are of interest pharmaceutically, including
solid-state or heterogeneous reactions and biological
processes, are complex, and analyses of such data
often present considerable dif®culties. The general
nature of the microcalorimetric technique may offer
*Corresponding author.
1Present address: The School of Pharmacy, University of
London, 29/39 Brunswick Sq. London, WC1N 1AX, UK.
0040-6031/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved.
P I I : S 0 0 4 0 - 6 0 3 1 ( 9 8 ) 0 0 6 2 2 - 4

40
S. Gaisford et al. / Thermochimica Acta 328 (1999) 39±45
some advantages for the study of these complex
reactions, without the need for further analytical
investigation.
Previous work from our group has resulted in a
general method of analysis of microcalorimetric
data that allows the recovery of both kinetic and
thermodynamic parameters [2]. The method involves
water. Sample mixtures were prepared by mixing
aliquots of the potassium hydroxylamine trisulfonate
and perchloric acid standards. The time of mixing
the solutions was noted. Reference mixtures were
prepared using an equivalent quantity of perchloric
acid, as present in the sample, diluted in deionized
water.
writing
a
kinetic equation that describes the
The calorimeter employed for these studies was an
LKB 2277 Thermal Activity Monitor (TAM, Thermo-
reaction under study, converting it to a calorimetric
form and then ®tting the calorimetric data using a
process of iteration. It has been shown previously
how such an analysis may be applied to reactions
that are perceived as being fast to medium term
in duration [2,3], and to reactions that occur in
the solid state [4]. It is the purpose of this paper
to show how the analysis may be extended to
allow the study of solution phase reactions that
follow consecutive pathways, using both real and
simulated data. For this work, we chose to study
the acid catalyzed hydrolysis of potassium hydroxy-
lamine trisulfonate, a reaction that proceeds via a
three-step, consecutive ®rst-order mechanism. We
show how the calorimetric data recorded can be
®tted to a suitable model to determine values
for the rate constants and enthalpies for this reaction,
and since the reaction has been the subject of pre-
vious investigations [5,6], compare our values with
literature data. We also present a general protocol
that should be adopted when using the Willson [2]
method to ®t calorimetric data.
È È
metric AB, Jarfalla, Sweden), which was housed in a
temperature controlled environment (21Æ0.18C),
allowing a baseline stability of Æ0.1 mW over 24 h
to be attained. The calorimeter was calibrated peri-
odically using an electrical substitution method.
Experiments were performed at 258C, 308C and
358C, in glass ampoules. Ampoules were sealed
with crimped aluminum caps, the caps being ®tted
with rubber sealing disks. All solutions used were
pre-equilibrated at the temperature of the particular
experiment, and ampoules were allowed to equili-
brate in the TAM for 5 min prior to the onset of
data capture. Such a short equilibration time, while
not being ideal, was necessary to ensure that the
maximum number of data were collected, the reaction
occurring relatively quickly. The heat-¯ow in, or
out, of the sample ampoule was recorded using the
dedicated Digitam 4.1 software. Data analysis was
performed using the software package ORIGINTM
(Microcal Software, MA, USA) and the mathe-
matical worksheet package Mathcad 6.0 (Mathsoft
Europe, UK).
2. Materials
4. Results
Potassium hydroxylamine trisulfonate is readily
prepared from potassium hydroxylamine-NN-disulfo-
nate [5,6]. Potassium hydroxylamine-NN-disulfonate
is not, however, commercially available, and was
prepared by the method outlined by Palmer [7].
Integration of the heat-¯ow (power, dq/dt (È), in
Watts) versus time (t, in seconds) plot obtained from
an isothermal microcalorimeter gives the heat output
(q, in Joules) for a particular reaction. If a suitable
kinetic equation can be written that describes the
reaction under investigation, then if either power is
plotted versus time or q is plotted versus power, it is
possible, using a suitable graphics software package,
to determine the constants of the kinetic expression by
a process of iteration [2±4].
3. Experimental
Stock solutions of potassium hydroxylamine
trisulfonate (0.01 M) and perchloric acid (0.2 M)
were prepared in deionized water, and were stored
at 258C. Mixtures were prepared for experiments
by dilution of the stock solutions with deionized
The calorimetric equations that describe two-step,
three-step and four-step ®rst-order reaction schemes
are given by the following equations, respectively:

S. Gaisford et al. / Thermochimica Acta 328 (1999) 39±45
41
k1Át
k2Át
k2Át
dqobs
dt
e
e
e
k1Át
k1Át
ˆ k1 Á ÁH1 Á Ao Á e
‡ k1 Á k2 Á ÁH2 Á Ao
;
(1)
k2 k1
k1Át
dqobs
dt
e
ˆ k1 Á ÁH1 Á Ao Á e
‡ k1 Á k2 Á ÁH2 Á Ao
Ao Á k1
k2 k1
ꢀ
k2Át
k22 Á k1 Á Ao Á e
k1Át
k1Át
k2Át
‡ ÁH3 Ao Á k1 Á e
ꢀ k1 Á e
‡ k2 Á e
† ‡
k1 Á k2 k3 Á k2 ‡ k22 ‡ k3 Á k1
k2 k1
k2 Á k12 Á Ao Á e
k2 Á k1 Á Ao Á k3
k1Át
‡
k1 Á k2 ‡ k3 Á k2 k3 Á k1 ‡ k12
k1 Á k2 k3 Á k2 ‡ k22 ‡ k3 Á k1
ꢁ
k3Át
k2 Á k1 Á Ao Á k3 Á e
;
(2)
k1 Á k2 ‡ k3 Á k2 k3 Á k1 ‡ k12
k1Át
k2Át
e
dqobs
dt
e
k1Át
ˆ k1 Á ÁH1 Á Ao Á e
ꢀ
‡ k1 Á k2 Á ÁH2 Á Ao
Ao Á k1
k2 k1
k2Át
k22 Á k1 Á Ao Á e
k1Át
k1Át
k2Át
‡ ÁH3 Ao Á k1 Á e
ꢀ k1 Á e
‡ k2 Á e
† ‡
k1 Á k2 k3 Á k2 ‡ k22 ‡ k3 Á k1
k2 k1
k2 Á k12 Á Ao Á e
k2 Á k1 Á Ao Á k3
k1Át
‡
k1 Á k2 ‡ k3 Á k2 k3 Á k1 ‡ k12
k1 Á k2 k3 Á k2 ‡ k22 ‡ k3 Á k1
ꢁ
ꢂ
k3Át
k2 Á k1 Á Ao Á k3 Á e
k1
k1Át
k1Át
k2Át
‡ H4 Ao Á k1 Á e
Ao
ꢀ k1 Á e
‡ k2 Á e
†
k1 Á k2 ‡ k3 Á k2 k3 Á k1 ‡ k12
k2 k1
k22 Ák1 ÁAo Á e
k2 Á k12 Á Ao Á e
k2 Á k1 Á Ao Á k3 Á e
k2Át
k1Át
k3Át
‡
‡
k2 Á k1 k3 Á k2 ‡ k22 ‡ k3 Á k1
k2 Á k1 ‡ k3 Á k2 k3 Á k1 ‡ k12
k2 Á k1 k3 Á k2 ‡ k22 ‡ k3 Á k1
k3Át
k2 Á k1 Á Ao Á k3 Á e
k2 Á k1 ‡ k3 Á k2 ‡ k12 k3 Á k1
2
k2 Á k1 Á k32 Á Ao Á e
Á ꢀk1 k2†
k3Át
k1 Á k4 Á k2 ‡ k1 Á k4 Á k3 k1 Á k32 ‡ k2 Á k1 Á k3 k32 Á k2 ‡ k32 ‡ k4 Á k3 Á k2 k4 Á k32
k22 Á k1 Á k3 Á Ao Á e
Á ꢀk1 k2† k2 Á k12 Á k3 Á Ao Á e
Á ꢀk1 k2†
k2Át
k1Át
‡
k3 Á k2 k4 Á k2 ‡ k4 Á k3 ‡ k22
k12 k3 Á k1 k1 Á k4 ‡ k4 Á k3
k4Át
k2 Á k1 Á k3 Á Ao Á k4 Á e
Á ꢀk1 k2†
k3 Á k2 k4 Á k2 ‡ k4 Á k3 ‡ k22
2
k4Át
k1 Á k2 Á k3 Á k4 Á Ao Á e
Á ꢀk1 k2†
‡
‡
k1 Á k4 Á k2 ‡ k1 Á k4 Á k3 k1 Á k32 ‡ k2 Á k1 Á k3 k32 Á k2 ‡ k32 ‡ k4 Á k3 Á k2 k4 Á k32
ꢃ
k4Át
k1 Á k2 Á k3 Á k4 Á Ao Á e
Á ꢀk1 k2†
;
(3)
k12 k3 Á k1 k1 Á k4 ‡ k4 Á k3
where Ao is the number of moles of reactant available
for reaction, k1, k2 and k3 are the rate constants and
ÁH1, ÁH2 and ÁH3 are the enthalpies of the succes-
sive reaction steps, respectively.
The equations shown above may be used to generate
power±time data using a mathematical worksheet. The
ability to construct power±time data leads to many
advantages. Firstly, it is possible to study the effects of
the values of the reaction parameters on the shape of
power±time trace obtained. In this way, it is possible to
study, for example, the effects of the rate constant on a
particular scheme while holding the value of the
reaction enthalpy constant, or to investigate the dif-
ference in the values of the rate constants for a multi-

42
S. Gaisford et al. / Thermochimica Acta 328 (1999) 39±45
ꢀSO3†2N:O:SO33 ‡H2O k1!;ÁH1 SO3:NH:O:SO32 ‡HSO4
SO3:NH:O:SO23 ‡ H2O k2!;ÁH2 NH2:O:SO3 ‡ HSO4
NH2:O:SO3 ‡ H2O k3!;ÁH3 NH2:OH ‡ HSO4
The reaction has been studied previously using a
titration method [5,6], and a number of kinetic data are
available. The hydrolysis of the trisulfonate ion is the
fastest of the three steps, and produces the relatively
stable hydroxylamine-NO-disulfonate ion. Potassium
hydroxylamine-NO-disulfonate is stable enough to be
recrystallized from dilute acid solution. The disulfo-
nate is hydrolyzed slowly in dilute acid solution
forming the hydroxylamine-O-sulfonate ion and,
eventually, hydroxylamine and hydrogen sulfate.
The published kinetic studies were based on titration,
using 35S labeled compounds. The ®rst hydrolytic step
was observed to proceed at a much faster rate than the
second and third hydrolyses, the ®nal two hydrolyses
proceeding at similar, slow, rates. Kinetic data for the
three steps are represented in Table 1.
It is notable that the authors of the above work were
not able to conduct the assay titrations at 258C for the
second hydrolytic step, since the rate of reaction was
too slow, and that a very high concentration of acid
was required to enable the third hydrolytic step to be
studied at 258C; that is, the reaction conditions had to
be varied extensively to allow the investigation of the
second and third hydrolytic steps. Using the micro-
calorimeter, it was possible to study the reaction
directly at 258C, 308C and 358C.
Fig. 1. Power±time data for a three-step, consecutive, first-order
reaction scheme constructed using MathcadTM having entered the
1
following reaction parameters: Aˆ0.00001 moles, k1 ˆ3Â10 5 s
,
5
1
6
1
1
k 2 ˆ5Â1 0
s
,
k 3 ˆ6 Â10
s
,
ÁH1 ˆ3 0 k J m o l
,
ÁH2ˆ45 kJ mol 1, and ÁH3ˆ40 kJ mol 1. The power±time data
for each of the three reaction steps are shown, as well as the overall
power±time data that would be observed calorimetrically.
step reaction that is required to allow the separation of
the trace into its component parts. It is also possible to
model the power±time data for each of the individual
steps of a particular scheme, as well as to investigate
the overall power±time data that would be observed
for a reaction.
Some simulated data for a three-step consecutive
reaction are shown in Fig. 1. These data were created
using the mathematical worksheet, having entered the
values shown for the constants. Data are shown for the
reaction steps A!B, B!C and C!P, as well as for
the overall signal that would be observed calorime-
trically. These data may then be exported into a
mathematical ®tting program, and treated as if they
were real data. It is possible to ®t the data to a
suitable model and recover the values of the reaction
parameters. Since the values of the parameters are
known absolutely, because they were used to cons-
truct the original data, it is possible to know precisely
if the ®tting procedure is accurate. In this way,
it is possible to gain experience in the techniques
of ®tting and to understand how a particular model
behaves.
The microcalorimetric data were ®tted to (2), in
order to recover the values of the reaction parameters.
The study reaction undergoes both endo- and exother-
Table 1
Kinetic data for the acid catalyzed hydrolysis of potassium
hydroxylamine trisulfonate
Reaction
Temperature
[HClO4]
(M)
104k
I (M)
1
(8C)
(s
)
Step 1b
Step 2b
Step 3c
Step 3c
25.0
75.0
45.0
25.0
0.0005
0.025
0.0708
1.01
0.089
1.80
0.0011a
0.025
1.00
0.042
0.085
To apply the models to real data, we chose to study
the acid catalyzed hydrolysis of potassium hydroxyl-
amine trisulfonate, which reacts according to the
following model:
1.05
aIonic strength adjusted using NaClO4.
bData reproduced from [5]: [trisulfonate]ˆ0.01 M.
cData reproduced from [6]: [monosulfonate]ˆ0.04 M.

S. Gaisford et al. / Thermochimica Acta 328 (1999) 39±45
43
mic processes and, consequently, the reaction para-
meters must include both positive and negative ÁH
values. Since there are no data available in the litera-
ture for these constants, the initial estimates for the
enthalpy values entered into the ®tting package were
all positive, and the iterative process determined the
signs of the ®nal constants. The value of Ao was
calculated to equal 3Â10 6 moles, and was set as a
constant. It was observed that the model gave an
excellent ®t to the data.
Values for the reaction parameters returned from the
®tting process for each data set are represented in
Tables 2 and 3. Note that, in all cases, the iterative
procedure returned values for ÁH1 that were positive
and values for ÁH3 that were negative, but the sign of
ÁH2 varied with temperature. For reactions studied
under identical conditions, the ®tting process returned
values that were generally reproducible (the one
exception is the values returned for ÁH1 at 308C).
It should be borne in mind that, when ®tting data to
an equation which is quite complex, it is possible to
obtain a large number of answer sets, each of which
appears to be equally valid. It was observed that, when
®tting the power±time data from each of the study
reactions, many answer sets were obtained, each giv-
ing what appeared to be an excellent ®t line to the data.
The values of the enthalpy parameters varied widely in
each answer set, and it was necessary to choose the
answer set giving the lowest ꢀ2 (the program's statis-
tical measure of ®t) value. This problem has been
noted previously [7,8] for consecutive reaction
schemes. It follows that, if any literature values for
any of the enthalpy parameters were available, it
would be possible to compare the answer sets with
these values, to aid the ®tting process, but this was not
possible.
However, Candlin and Wilkins [5,6] did publish
values for the rate constants for the individual hydro-
lytic steps, and although the published data were not
recorded under the same conditions as the experimen-
tal data presented above, some comparisons can be
made. At 258C, the published value for k1 is
8.9Â10 6 s 1, compared with the value determined
from the ®tting procedure of 4.2Â10 4 s 1, a 100-fold
difference in reaction rate. The only other value
Table 2
ÁH values for the acid catalyzed hydrolysis of potassium
hydroxylamine trisulfonate (0.001 M), determined by fitting
power±time data to Eq. (2) (reactions were conducted using an
acid concentration of 0.0005 M)
Temperature
(8C)
ÁH1
(kJ mol
ÁH2
(kJ mol
ÁH3
(kJ mol )
1
1
1
)
)
determined at 258C is k3, determined to equal
6
8.5Â10
by Candlin and Wilkins [5,6] and
35
134.0
147.0
140.5
135.3
56.37
95.84
28.30
104.8
141.0
122.9
31.75
35.21
33.48
82.23
167.7
82.36
125.0
37.44
36.23
36.84
47.39
6.7Â10 6 s 1 by ®tting microcalorimetric data. How-
ever, the published value was determined with an
added acid concentration of 1.01 M, far in excess of
the 0.0005 M acid used in the calorimetric experiment.
It was noted above that, when choosing an equation
to which to ®t some microcalorimetric data, the equa-
tion with the fewer or fewest variables, but which
gives an acceptable ®t line, should be selected. This
applies to reactions where the exact mechanism of
reaction is unknown. In the case of the hydrolysis of
potassium hydroxylamine trisulfonate it is known that
the reaction follows a three-step consecutive reaction
scheme, and that was the model chosen for ®tting.
However, it might seem reasonable to assume that, at
such a low added concentration of acid, the third
hydrolytic step might not occur during the lifetime
of the experiment. If that were the case, then the
reaction would follow a two-step consecutive reaction
scheme, and following the advice given above, should
be ®tted to a model with fewer variables. Eq. (1)
35
35 (Average)
30
30
30 (Average)
25
Table 3
Rate constant values for the acid catalyzed hydrolysis of potassium
hydroxylamine trisulfonate (0.001 M), determined by fitting
power±time data to Eq. (2) (reactions were conducted using an
acid concentration of 0.0005 M)
1
)
1
)
1
k3 (s )
Temperature (8C)
k1 (s
k2 (s
4
4
4
4
4
4
4
5
5
5
5
5
5
6
4
4
4
5
5
5
6
35
5.5Â10
4.7Â10
5.1Â10
4.4Â10
4.4Â10
4.4Â10
4.2Â10
4.3Â10
5.0Â10
4.8Â10
2.9Â10
2.6Â10
2.8Â10
4.6Â10
2.6Â10
2.1Â10
2.3Â10
5.8Â10
6.6Â10
6.2Â10
6.7Â10
35
35 (Average)
30
30
30 (Average)
25

44
S. Gaisford et al. / Thermochimica Acta 328 (1999) 39±45
Table 4
Activation energies determined via an Arrhenius analysis for the
three reaction steps of the hydrolysis of potassium hydroxylamine
trisulfonate (0.001 M)
1
)
Concentration
of acid (M)
Activation energy (kJ mol
Step 1
14.74
Step 2
179.3
Step 3
270.7
0.0005
Fig. 2. Power±time data for the hydrolysis of potassium hydro-
xylamine trisulfonate (0.001 M) at 358C, with 0.0005 M acid
added, and the fit line (dotted) obtained after the data were fitted to
Eq. (1).
describes power±time data for a reaction undergoing a
two-step consecutive mechanism and this was used to
®t the microcalorimetric data. A typical ®t line is
shown in Fig. 2. It can be seen that, using the two-
step consecutive model, it was not possible to obtain a
good ®t line to the microcalorimetric data. The ®t is
good over the initial section of data, which is to be
expected since, over this region, the ®rst two reaction
steps predominate and the reaction is essentially a
two-step process. As the reaction progresses, the third
step begins to contribute to the observed signal, and
the model no longer describes the overall reaction.
This is a good evidence that the data observed derive
from a three-step process, and that Eq. (2) was the best
choice model for ®tting the data.
Since the microcalorimetric data were recorded at
three temperatures, it is possible to plot ln k versus 1/T
and obtain estimates for the values of the activation
energies for each step. These values are listed in
Table 4. It can be seen from the values given in
Table 4 that the activation energy of the ®rst step is
considerably lower than those of the second and third
steps.
Fig. 3. Deconvolution of the observed power±time data obtained
for the hydrolysis of potassium hydroxylamine trisulfonate
(0.001 M) at 308C using the mathematical worksheet.
individual hydrolytic steps are constructed. An exam-
ple is shown in Fig. 3. The ability to separate complex
microcalorimetric signals into their component parts
using speci®c models is a signi®cant advance in the
analysis of microcalorimetric data and, in a future
publication, we will show how more complex parallel,
consecutive/parallel and higher order reaction
schemes may be analyzed in a similar way.
5. Summary
An additional bene®t of using the mathematical
worksheet is that, using the data presented in Tables 2
and 3, it is possible to deconvolute the power±time
curves for the individual reactions. The process is,
essentially, the reverse of the way microcalorimetric
data were simulated earlier. The reaction parameters,
obtained from the ®tting process, are entered into the
worksheet and the power±time curves for the three
The Willson method [4] is based on ®tting calori-
metric data to a speci®c model. The work presented
here shows how such an analysis may be extended to
allow the analysis of calorimetric data derived from
reactions undergoing consecutive, solution phase
reactions. The use of simulated data and the ability
to model accurately speci®c reaction schemes have

S. Gaisford et al. / Thermochimica Acta 328 (1999) 39±45
45
many advantages. It is now possible to model many
complex reactions, and to explore the behavior of
those reactions, without the need to obtain real micro-
calorimetric data. In this way, it is possible to examine
the effects of varying the reaction parameters, such as
the rate constants or reaction enthalpy, on the power±
time signal that would be obtained, and to learn how a
reaction behaves before starting a series of experi-
ments. Having constructed some data, it is possible to
import those data into a mathematical ®tting program,
and treat them as if they are real microcalorimetric
data. It is then possible to learn how to ®t those data to
a model. It is possible to explore the effects of ®tting
different equations to the data, to see if the mechanism
of reaction can be deduced on the basis of the ®t lines
obtained.
reaction intermediates will be present at maximum
concentration and a chemical assay technique, such as
HPLC, can be employed. In this way, it is possible to
gain both mechanistic and molecular information.
This is a signi®cant step forward in the analysis of
microcalorimetric data.
Acknowledgements
The authors express their gratitude to the late
Richard Lipscombe, without whose considerable tech-
nical expertise much of this work would not have been
possible.
The study of the hydrolysis of potassium hydro-
xylamine trisulfonate has proved the ability to ®t
microcalorimetric data, derived from a reaction under-
going a complex reaction mechanism, to a model, and
to use the reaction parameters obtained to deconvolute
those data.
It follows that it is possible to follow a protocol for
analyzing microcalorimetric data for reactions where
the mechanism is unknown. The data should be ®tted
to a series of models based on rate equations to
determine which scheme gives the best ®t. Using
the parameters obtained, it is possible to determine
at what time, following initiation of the reaction, any
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