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