Modified Marcelin-de Donder equations for the kinetics of one-and two-step reversible chemical reactions in nonideal solutions

Article abstract of DOI:10.1134/S0036024406110355

An approach to a unified description of the kinetics of reversible chemical reactions on both sides of the equilibrium state was suggested. The description was based on the dynamic equilibrium principle, the isotherm-isobar equation for a chemical reaction, and the volume densities of the thermodynamic activities of the reagents (normalized by the pure component states). The equations obtained were applied to describe our own and literature data on the kinetics of acid-catalyzed transesterification reaction between butyl acetate and methanol at 298 K and esterification of ethylene glycol with acetic acid at 313 K. It was shown that the observed rate constant could be represented in the form of the sum of partial contributions describing the reactions in the presence of a large excess of one of the reagents. Nauka/Interperiodica 2006.

Full text of DOI:10.1134/S0036024406110355

ISSN 0036-0244, Russian Journal of Physical Chemistry, 2006, Vol. 80, No. 11, pp. 1874–1879. © Pleiades Publishing, Inc., 2006.
Original Russian Text © V.G. Povarov, O.B. Sokolova, I.L. Karpova, 2006, published in Zhurnal Fizicheskoi Khimii, 2006, Vol. 80, No. 11, pp. 2103–2108.
GENERAL PROBLEMS
OF CHEMICAL THERMODYNAMICS
Modified Marcelin–de Donder Equations for the Kinetics
of One- and Two-Step Reversible Chemical Reactions
in Nonideal Solutions
V. G. Povarova, O. B. Sokolovab, and I. L. Karpovaa
a St. Petersburg State University (Petrodvorets Branch), Universitetskii pr. 26, Petrodvorets, 198504 Russia
b Gertsen State Pedagogical University, St. Petersburg, 191186 Russia
E-mail: povarovvg@rambler.ru
Abstract—An approach to a unified description of the kinetics of reversible chemical reactions on both sides
of the equilibrium state was suggested. The description was based on the dynamic equilibrium principle, the
isotherm–isobar equation for a chemical reaction, and the volume densities of the thermodynamic activities of
the reagents (normalized by the pure component states). The equations obtained were applied to describe our
own and literature data on the kinetics of acid-catalyzed transesterification reaction between butyl acetate and
methanol at 298 K and esterification of ethylene glycol with acetic acid at 313 K. It was shown that the observed
rate constant could be represented in the form of the sum of partial contributions describing the reactions in the
presence of a large excess of one of the reagents.
DOI: 10.1134/S0036024406110355
The problem of a theoretical description of the According to de Donder, the rate V of the process can
kinetics of reversible chemical reactions in nonideal be written as
solutions has been studied for about 100 years [1]. The
V = V+ – V– = k+aAaB – k–aCaD.
(1)
traditional approach based on the law of mass action
takes into account the influence of solvent properties
through the equilibrium constants of formation of reac-
tive reagent forms [2]. Most often, this allows the
experimental data on a separate kinetic curve to be
described fairly accurately.
Here, k+ and k– are the rate constants for the forward
and back reactions and ai is the thermodynamic activity
of component i. This equation is consistent with the
thermodynamic equilibrium condition, if we assume
that the equation
A good description of separate kinetic curves unfor-
tunately does not mean that the kinetics as a whole is
described satisfactorily. In particular, rate constants
change jumpwise in moving along the reaction line
through the equilibrium point, which contradicts the
dynamic equilibrium principle. Still larger constant
variations are observed in the passage from one reac-
tion line to another [3]. Naturally, the predictive ability
of such models is fairly limited. In 1920–1930, de
Donder suggested, as a continuation of Marcelin’s stud-
ies, that reagent concentrations in the law of mass
action should be replaced by their thermodynamic
activities or fugacities to extend the range of the appli-
cability of the law of mass action to nonideal systems
[4, 5]. This suggestion, logical from the point of view
of thermodynamics, did not find wide use in practical
kinetic studies but initiated long-term discussion of
the role played by thermodynamic activities in chem-
ical kinetics and, generally, interrelation between
kinetics and thermodynamics.
+
–
°
°
°
°
k /k = exp[(µA + µB – µC – µD)/RT]
(2)
r
°
°
= exp(–∆G /RT) = exp(A /RT) = K
°
is fulfilled at equilibrium. Here, µi is the standard
chemical potential of component i, –∆G° = A° is the
standard chemical affinity of the reaction, Kr is the ther-
modynamic equilibrium constant, R is the gas constant,
and T is the absolute temperature. Clearly, rate constant
calculations according to de Donder require knowledge
of component activity coefficients along the reaction
line and the numerical differentiation of the experimen-
tal kinetic curve. The procedure is obviously laborious,
which can only be balanced by the possibility of calcu-
lating the rate of the reaction at any composition point
from several (ideally, single) experimental kinetic
curves. This is what could not have been done. The
problem is therefore the absence of any improvement in
kinetic description if the obligatory requirement of con-
sistency of thermodynamic and kinetic descriptions is
Let us consider the problem for the example of a introduced. More recently, as the thermodynamics of
reversible bimolecular reaction of the type A + B = C + D. nonequilibrium processes has been developing, it
1874

MODIFIED MARCELIN–DE DONDER EQUATIONS
1875
V+ × 104, V– × 104, mol/(l s)
became clear that (1) is not the only equation compati-
ble with thermodynamics. In 1953, van Risselberg for-
mulated a more general restriction imposed by thermo-
dynamics on the rates of forward and back stages of a
reversible chemical process [6]. This restriction had the
form
1
2
3
4
2
–∆G = A = RTln(V+/V–).
(3)
1
0
Here, ∆G and A are already current (actual at the given
time moment) rather than standard Gibbs energy and
chemical affinity. It is easy to see that (3) is consistent
with (1) if condition (2) is met. It, however, does not
require (and does not presume) that k+ and k– should
each be constant and allows activities in (1) to be
replaced by affinities multiplied or divided by any val-
ues that are cancelled in calculating the V+/V– ratio. It is
reasonable to introduce volume densities of thermody-
namic activities for reactions of the type under consid-
eration, namely, [ai] = ai/U, where U is the molar vol-
ume of the reaction mixture [3]. Equation (1) then takes
the form
0
0.1
0.2
0.3
0.4
0.5
x(BuOH)
0.6
+
Fig. 1. Butanol mole fraction dependences of (1, 2) V and
–
(3, 4) V for transesterification at 298 K; (1, 3) description
according to the law of mass action for a second-order
reversible reaction and (2, 4) calculations by (3) and (5).
V = V+ – V– = k+aA/UaB/U – k–aC/UaD/U
results of the corresponding calculations are shown in
Fig. 1. The V+ and V– values calculated by (3) are also
shown. In order to obtain them, we first interpolated the
experimental kinetic curves using polynomials third-
order in time and then found the derivatives of concen-
tration with respect to time at the corresponding points.
The coefficients of the polynomials are listed in Table
1. After differentiation, the rate of the reaction was
always divided by 10 and the analytic concentration of
the catalyst, p-toluenesulfonic acid (p-TSA), to com-
pare rates at equal p-TSA concentrations (0.1 M). It
was proved in special experiments [7] that the rate of
the reaction was proportional to the concentration of p-
TSA. The chemical affinity was calculated as
(4)
= k+[aA][aB] – k–[aC][aD].
The most important advantage of the volume densi-
ties of activities over activities as such is their transfor-
mation into usual molar concentrations in ideal sys-
tems. Equation (4) is an example of a modification of
the classic Marcelin–de Donder equation acceptable
from the point of view of thermodynamics. In this
work, we inquire into its applicability to the description
of the kinetics of reversible processes.
We do this for the example of acid-catalyzed esteri-
fication and transesterification reactions. The necessary
conditions for kinetic curve measurements are com-
bined in these reactions with the possibility of correctly
calculating reagent activity coefficients. The experi-
mental data on the kinetics of transesterification of
n-butyl acetate with methanol were obtained by Panov
and Garipova in 1990–1995 at the Faculty of Chemistry
of St. Petersburg State University [7–10]. The kinetics
of esterification of ethylene glycol with acetic acid was
studied by the present authors. The procedure for chro-
matographically analyzing reaction mixtures was also
described in [7–10], and the activity coefficients were
calculated using the UNIFAC model and standard
equations [11]. The geometric parameters of groups
and intergroup interaction parameters for liquid-vapor
equilibria were also taken from [11]. Molar density
changes along reaction lines were ignored.
A = –∆G = –RTln(KxPγ) + RTlnPa.
(5)
Here, Kx is the concentration equilibrium constant and
Pγ and Pa are the reaction products of the activity coef-
ficients of the reagents in the equilibrium mixture and
of reagent thermodynamic activities at the point of
interest to us, respectively. Suppose that the V+ and V–
values, activity coefficients, and thermodynamic equi-
librium constants are known. We can then calculate the
k+ and k– values by the modified Marcelin–de Donder
equation (Eq. (4)), provided condition (2) is met. We
found that these values decreased linearly along the
reaction line. The approach that we used, however, did
not presuppose them to be constant.
The rate constants for transesterification of butyl
The authors of [7–10] experimentally studied the
acetate were calculated in [7–10] from experimental kinetics of the reaction over the whole central region of
data using the kinetic equation for reversible second- the tetrahedron of methanol–methyl acetate–butanol–
order reactions. The rate constants for the MeOH + butyl acetate system compositions. Proceeding as
BuAc stage at 298 K and [H+] = 1 M calculated from described above, we calculated 87 k+ values and ana-
kinetic curves to the left and right of equilibrium were lyzed the composition dependence of this parameter.
found to be 4.3 × 10–6 and 2.2 × 10–6 l/mol/s, respec- This dependence was found to be linear everywhere
tively, which gave a difference of about two times. The and can, at T = 298 K and [p-TSA] = 0.1 M, be
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 11 2006

1876
POVAROV et al.
Table 1. Coefficients of interpolation polynomials for the kinetics of transesterification of methyl acetate and butyl acetate
at 298 K, Kx = [BuOH][MeOAc]/[MeOH][BuOAc] = 1.56 (calculated from the data obtained in [7–10])
B × 10–4,
C × 10–9,
D × 10–14
,
α0
A, mol/l
[p-TSA], mol/l
Rcorr
mol/(l s)
mol/(l s2)
mol/(l s3)
BuOH + MeOAc
1.28
3 : 1
1 : 1
1 : 3
2.90
5.97
8.99
–0.820
–0.867
–0.823
–0.631
4.88
0.124
0.108
0.119
0.999
0.999
0.999
–0.385
1.30
–0.705
MeOH + BuOAc
–5.56
3 : 1
1 : 1
1 : 3
0.145
–0.24
1.81
2.54
0.642
7.60
17.7
0.0497
0.098
0.055
0.999
0.997
0.999
–10.2
–0.003
–0.854
0.238
Note: Time is in seconds and concentrations in mol/l; α is the initial alcohol : ester ratio.
0
described by the equation (correlation coefficient 1.53 × 10–6, A2 = 2.05 × 10–6 + A4 = 3.16 × 10–6, and
A3 = 1.94 × 10–6 + A4 = 3.05 × 10–6.
0.953) [3]
k+ = 1.11 × 10–6 + 4.24 × 10–7xBuOH
+ 2.05 × 10–6xBuAc + 1.94 × 10–6xMeOH
The Ai coefficients are simply the partial contribu-
tions of separate reagents to the rate constant. In its
physical meaning, Ai is the rate constant for the reaction
under consideration in the vicinity of the corresponding
vertex of the tetrahedron of compositions. Note that
direct rate measurements when one of the components
is present in a large excess is a difficult experimental
task. Another circumstance is also of importance. The
result obtained is directly related to the proper selection
of standard states for calculating activity coefficients.
In the UNIFAC model, these states are pure substances,
that is, the vertices of the composition tetrahedron
themselves. It is the rate constants near them that enter
into (6). This is no mere chance. Thermodynamics does
not consider the mechanisms of reactions, and nonide-
ality parameters should therefore be included with
respect to such system states in which the reaction
occurs by the same mechanism. In our problem, this
requirement is met. If we selected a state in which the
reaction does not occur at all as the standard state (for
instance, the ideal gas state), the activity coefficients
would contain contributions from processes that are not
related directly to the kinetics of the reaction. Possibly,
this is the reason why the earlier attempts at the intro-
duction of activity coefficients into kinetic equations
were unsuccessful.
(6)
.
Here, xi is the mole fraction of reagent i. Using this
dependence and Eq. (4), we can easily calculate both
the V+ and V– values and their difference. The results
presented in Fig. 2 show close agreement with the
experimental data. The physical meaning of the numer-
ical coefficients of (6) becomes clear when this equa-
tion is rewritten in the more traditional form
k+ = A1xBuOH + A2xBuAc + A3xMeOH + A4xMeAc
,
where xi is the mole fractions of the reagents (Σxi = 1).
We then have A4 = 1.11 × 10–6, A1 = 4.24 × 10–7 + A4 =
V × 104, mol/(l s)
1
2
3
2
4
1
Next, let us check the generality of the suggested
approach for the example of reversible esterification of
ethylene glycol with acetic acid. This system was
selected because, first, this reaction occurs without
stratification over a wide range of compositions includ-
ing equilibrium and, secondly, reagent activity coeffi-
cients in alcohol–acid mixtures differ from one to a
much greater extent than in the alcohol–ester system.
Our aim was also to study the effect of the appearance
of the second reversible stage.
0
0
0.1
0.2
0.3
0.4
x(BuOH)
0.5
+
–
Fig. 2. Butanol mole fraction dependences of V and V for
transesterification at 298 K; rate constant and rate calcula-
tions by (6) and (4), respectively; see Fig. 1 and Table 1 for
conditions.
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MODIFIED MARCELIN–DE DONDER EQUATIONS
1877
Table 2. Coefficients of interpolation functions for esterification–hydrolysis at 313 K
Substance
c0, M
t0, s
A1, M
t1, s
A2, M
t2, s
Hydrolysis, [H2SO4] = 0.1 M
Water
5.41902
2.14113
2530.134
3250.0
2.4608
1.4405
12612.4715
10637.2323
1.2970
0.6908
15276.7125
16659.8430
Ethylene glycol diacetate
Esterification, [H2SO4] = 0.094 M
Acetic acid
5.43968
1.27151
0.0
0.0
2.6325
2.4479
1361.08331
1429.04068
2.2407
0.9252
7829.68691
7277.01354
Ethylene glycol
The reaction under consideration involves two was minimized. Here, the external summation was over
reversible stages,
all five mixture components and the internal summation
over all time moments at which measurements were
taken. The k1 and k2 values were varied. The situation
was found to be fully analogous to that with the trans-
esterification reaction, we obtained an excellent
description of all kinetic curves taken separately and a
jumpwise change in the rate constant in passing through
the equilibrium state. For instance, k1 = 2.9 × 10–5 and
k2 = 5.7 × 10–6 l/(mol s) for esterification at 313 K and
catalyst concentration 0.1 M. The same values calculated
from the kinetic curves obtained under the same condi-
tions were k1 = 1.35 × 10–5 and k2 = 3.9 × 10–6 l/(mol s).
HO(CH2)2OH + AcOH = HO(CH2)2OAc + H2O,
HO(CH2)2OAc + AcOH = AcO(CH2)2OAc + H2O.
The reaction was studied at an acetic acid–ethylene gly-
col or water–diethylene glycol molar ratio of 2 : 1 at
298–323 K in the presence of sulfuric acid as a catalyst.
Ethylene glycol was purified by vacuum distillation at
110°C and 10 torr, and acetic acid was preliminarily
distilled with a dephlegmator (30 cm) and frozen. Eth-
ylene glycol diacetate was synthesized by the standard
procedure for the preparation of esters described in [12]
and purified by vacuum distillation with a dephlegma-
tor (30 cm) at 95°C and 10 torr. The reaction mixtures
were prepared gravimetrically on a VLR-200 balance
and held at the temperature of measurements in a UTU-4
water thermostat. Reaction mixture composition was
determined by gas chromatography on an LKhM-80
chromatograph with a heat conductivity detector. The
conditions were a packed Teflon column (4 mm × 5 m),
10% carbowax M on inerton, hydrogen as a carrier gas,
30 ml/min, column temperature 165°C, detector tem-
perature 200°C, vaporizer temperature 200°C, and
detector current 100 mA. The reagents left the column
in the order water (W), acid (A), diacetate (D), ethylene
glycol (E) + monoacetate (M).
Calculations of the rates V+ and V– and rate con-
stants by (3) and (4) require experimental kinetic curves
to be preliminarily differentiated. To do this, the exper-
imental ci(t) dependences were interpolated using the
Origin 5.0 program as
ci = c0i + A1 exp(–(t – t0)/t1) + A2 exp(–(t – t0)/t2).
For esterification, we interpolated the time depen-
dences of the concentrations of acetic acid and ethylene
glycol, and, for hydrolysis, those of water and diace-
tate. The interpolation coefficients are listed in Table 2.
Note that the concentrations of the other reagents can
be calculated from reaction stoichiometries and mate-
rial balance equations if the initial composition is
known. The calculation results are shown in Fig. 3. We
see that an interesting phenomenon is observed for the
first stage, namely, the rates of both forward and back
reactions decrease as equilibrium is approached. No
such effect was observed for transesterification. It can
be explained by a decrease in the concentration of ace-
tic acid, which is both a reagent and a catalyst. Natu-
rally, (6) cannot describe this phenomenon. We there-
fore suggested a more complex relation. The partial rate
constant was assumed to depend linearly on the mole
fraction of acetic acid,
In order to estimate the applicability of the law of
mass action to a separate kinetic curve, we numerically
integrated the system of equations
d[W]/dt = V1 + V2,
d[D]/dt = V2,
where V1 = k1[E][A] – k–1[M][W] and V2 = k2[M][A] –
k−2[D][W] are the rates of the first and second reversible
reactions. The k1, k–1, k2, and k–2 values are related as
k1/k–1 = Kr1 , and k2/k–2 = K2r , where Kr1 = 2.67 and
Kr2 = 0.8 are the concentration equilibrium constants at
ki = k0i + kiAxA.
(7)
313 K of the first and second stages also measured
experimentally. Calculations were performed by mini-
mizing the total deviation of the theoretical curve from
experimental. The function
The overall rate constant can then be written as
k = Σ(k0i + kiAxA)xi,
(8)
where the summation is over all five system compo-
nents. Determining the mole fraction of monoacetate
F = Σ(Σ|ciexp – citheor |).
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 11 2006

1878
POVAROV et al.
ki × 104, l/(mol s)
Vi × 103, mol/(l s)
16
4
3
2
1
1
2
12
8
1
2
4
0
0.1
0.2
0.3
0.4
0.5
xw
0
0.1
0.2
0.3
0.4
0.5
xw
+
–
Fig. 4. Water mole fraction dependences of k and k for the
first stage of esterification at 313 K: (1) calculations by
(3) and (4) and (2) interpolation according to (10); see
Table 2 for conditions.
+
Fig. 3. Water mole fraction dependences of (1) V and
–
(2) V at the first stage of esterification at 313 K calculated
by (3) and (5).
from the relation xW + xA + xE + xM + xD = 1 and substi- fied. The problems mentioned above will be studied in
tuting it into (8) yields
future experiments.
In conclusion, recall the main stages of the sug-
gested procedure:
k = k0M + Σ(k0i – k0M + kiAδiA)xi + xAΣ(kiA – kMA)xi.(9)
(1) Measurement of ci(t) kinetic curves at several
Here, δiA = 1 at i = A and δij = 0 otherwise. The coeffi-
cients of (9) can be found by the method of least
squares through solving the system of linear equations
obtained from nine conditions of the form ∂E2/∂Hk = 0,
reagent ratios on both sides of equilibrium.
(2) The determination of the concentration equilib-
rium constant Kx for at least one pair of mutually revers-
ible reaction mixtures.
(3) The interpolation of kinetic curves by suitable
ci(t) = Fi(t) functions and calculation of the Vi(ti) =
dFi(t)/dt rate at t = ti.
(4) The calculation of reagent activity coefficients at
experimental points with respect to the pure substances.
(5) The calculation of the thermodynamic equilib-
rium constant kp = KxPγ.
where E2 = Σ(kexp – ktheor)2 is the sum of the squares of
the deviations of the experimental rate constant values
from those calculated by (9) and Hk denotes the sought
coefficients (k0i − kM0 + kiAδiA, kiA – kMA, or k0M ). The
calculation results are shown in Fig. 4, and the coeffi-
cient values are given in the equation
k+1 = – 2.6 × 10–9 + 1.3 × 10–3xW + 1.5 × 10–3xA
+ 7.9 × 10–3xE – 4.5 × 10–4xD – 1.9 × 10–2xWxA
(6) The separation of the rate of every reversible
stage into V+ and V– according to the equation ∆G =
–RTln(V+/V–).
(10)
+ 1.1 × 10–2xA2 – 3.8 × 10–2xAxE + 1.3 × 10–2xAxD.
(7) Rate constant calculation by the equation V+/–
k+/–P[ai]νi .
=
In spite of a good description of the composition depen-
dence of k+ for both stages, the coefficients of (10) do
not quite satisfy the physical meaning of the problem.
Indeed, the closeness of k0M to zero is hardly probable,
and the negative k0D – kM0 values are unlikely. This may
(8) The interpolation of the composition depen-
dences of k+/– and rate calculations at arbitrary compo-
sitions.
be caused by the poor choice of interpolation equation (8)
or the performance of calculations along one reaction
line only. The validity of the latter remark is favored by
the observation that both these values are not only neg-
ative but also close to zero in magnitude (compared
with their analogues). In any event, the effectiveness of
the suggested approach to the kinetics of reversible
reactions in nonideal solutions was on the whole veri-
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