A method for esterification reaction rate prediction of aliphatic monocarboxylic acids with primary alcohols in 1,4-dioxane based on two parametrical taft equation

Article abstract of DOI:10.1002/kin.20845

Esterification reaction rates of aliphatic monocarboxylic acids with primary alcohols in 1,4-dioxane as inert solvent were investigated. Acids were esterified with 1-propanol and alcohols with acetic acid as model reactants at a constant temperature of 60°C, at a fixed ionic strength and pH in a batch reactor with a constant volume. For evaluation of reaction rates, an exact kinetic equation for the equilibrium reaction was applied. Under these conditions and for low reactants, concentrations reaction rate depends only on the structure of reactants and, therefore, can be predicted by a correlation equation with two Taft coefficients (inductive and steric effects). From these equations, it is possible to estimate the esterification reaction rate constant for other acid-alcohol pairs. This methodology may also be suitable for other kinetic systems measured under comparable experimental conditions.

Full text of DOI:10.1002/kin.20845

A Method for Esterification
Reaction Rate Prediction
of Aliphatic Monocarboxylic
Acids with Primary Alcohols
in 1,4-Dioxane Based
on Two Parametrical Taft
Equation
´
ˇ ´
JAN VOJTKO, PETER TOMCIK
Department of Chemistry, Faculty of Education, Catholic University in Ru zˇ omberok, Hrabovsk a´ cesta 1, Ru zˇ omberok
SK-034 01, Slovak Republic
Received 4 December 2012; revised 16 November 2013; accepted 18 December 2013
DOI 10.1002/kin.20845
Published online 7 January 2014 in Wiley Online Library (wileyonlinelibrary.com).
ABSTRACT: Esterification reaction rates of aliphatic monocarboxylic acids with primary alcohols
in 1,4-dioxane as inert solvent were investigated. Acids were esterified with 1-propanol and
◦
alcohols with acetic acid as model reactants at a constant temperature of 60 C, at a fixed ionic
strength and pH in a batch reactor with a constant volume. For evaluation of reaction rates, an
exact kinetic equation for the equilibrium reaction was applied. Under these conditions and
for low reactants, concentrations reaction rate depends only on the structure of reactants and,
therefore, can be predicted by a correlation equation with two Taft coefficients (inductive and
steric effects). From these equations, it is possible to estimate the esterification reaction rate
constant for other acid-alcohol pairs. This methodology may also be suitable for other kinetic
systems measured under comparable experimental conditions. ꢀC 2014 Wiley Periodicals, Inc.
Int J Chem Kinet 46: 189–196, 2014
INTRODUCTION
contemporary chemical engineering and organic syn-
thesis. Esters such as cetylpalmitate play an impor-
tant role as natural waxes [1–3] in the chemical indus-
try. Esters are relatively inexpensive solvents (ethyl-,
methyl-, butyl-, or isobutylacetate etc.) and have found
novel electrochemical applications as co-solvents for
low-temperature electrical double layer capacitors [4].
Methyl esters of fatty acids (FAME) represent an
Esterification of aliphatic monocarboxylic acids with
primary alcohols is a significant chemical reaction in
Correspondence to: Peter Tomcik; e-mail: peter.tomcik@ku.sk.
Contract grant sponsor: Grant Agency of the Slovak Republic.
Contract grant number: VEGA 1/0008/12.
ꢀ
C
2014 Wiley Periodicals, Inc.

ˇ ´
VOJTKO AND TOMCIK
1
90
ecological biofuel for diesel engines improving their
combustion performance [5–8]. Production of FAME
has increased today together with the number of en-
gines suitable for biofuel. In addition, esters are the
major source of flavors and/or fragrance in flowers and
fruits. These nontoxic natural products are included
in the FEMA GRAS program and are produced artifi-
cially in large amounts, serving as food and cosmetic
additives [9–12].
Because of the great diversity of esters and the fact
that esterification is an equilibrium reaction, it is nec-
essary to know the reaction rate for specific pairs of
alcohols and acids. The reaction rate is a crucial re-
action parameter, especially to determine the required
catalyst loading as well as for estimating the energy re-
quired to evaporate the water or esterification products
to shift reaction equilibrium toward ester formation. It
is usual to determine kinetic parameters for esterifi-
cation reactions experimentally on a case-by-case ba-
sis [13]. This process is time consuming and requires
additional energy.
In this paper we suggest a simple but effective
method for the prediction of esterification rates of
aliphatic monocarboxylic acids with primary alcohols
using protinic acid catalysis under homogeneous con-
ditions. Our procedure is based on Taft steric and po-
lar (inductive) coefficients; we show that the reaction-
rate constant is dependent only on the structure of the
acid or alcohol, allowing the prediction of esterifica-
tion rates under comparable reaction conditions for
reactants combinations not yet tested.
where kE is an esterification rate constant and kH is the
rate constant of ester hydrolysis.
The thermodynamic equilibrium constant K of es-
terification is defined as (assuming equal activity coef-
ficients of all species)
kE
kH
K =
(3)
and Eq. (2) can be rewritten as
ꢀ
ꢁ
1
voverall = kE cAcidcAlcohol −
cEstercWater
(4)
K
The differential equation, Eq. (4), can be integrated
see Appendix) and its general solution is
(
1
kE =
√
0
2
t.c
b − 4ac
Alcohol
ꢂ
ꢃ
√
2
2
c − b − b − 4ac x
×
ln
ꢂ
ꢃ
(5)
√
2
2
c − b + b − 4ac x
where
ꢀ
ꢁ
0
c
c
1
Acid
a =
1 −
(6)
(7)
(8)
0
K
Alcohol
0
Acid
0
0
0
c
+ c
c
+ c
Alcohol
Ester
Water
b =
+
0
0
c
K c
Alcohol
Alcohol
0
0
c
c
Ester Water
c = 1 −
0
0
THEORY
K c
c
Acid Alcohol
and x is the degree of conversion as a function of time.
Equation (5) together with imposed Eqs. (6), (7), and
The process of ester formation in organic chemistry is
well known. It can be described as
(8) allows the determination of the esterification rate
constant kE if the thermodynamic equilibrium constant
is known. In our experiments, we used only equimolar
mixtures of acids and alcohols without addition of ester
or water; hence,
Acid + Alcohol ↔ Ester + Water
(1)
As can be seen from Eq. (1), the overall chemical
reaction is an equilibrium process, esterification is the
forward reaction, and reverse reaction is ester hydrol-
ysis. The reaction rate is relatively slow in the absence
0
0
c
= c
= c
= 0
Ester
Water
+
0
0
−3
of H ions as catalysts. It may be autocatalyzed by
c
(1 mol dm
)
(9)
Acid
Alcohol
protons from dissociation of the carboxylic acid, how-
ever their dissociation is poor. Reaction rates are pro-
For these conditions, coefficient c from Eq. (8) is
equal to 1, coefficient b from Eq. (7) is equal to 2, and
+
portional to [H ] concentration up to approximately
−3
1
K
1
mol·dm
.
the coefficient from Eq. (6) a = 1 − ; and Eq. (5)
+
At constant concentration of [H ], the forward re-
takes the simpler form
action rate is governed by the second-order kinetic
equation.
ꢂ
ꢃ
K − 1 x −
ꢃ
√
√
√
√
√
K
K
K
t
kE =
ln ꢂ
(10)
2
voverall = kEcAcidcAlcohol − kHcEstercWater
(2)
K + 1 x −
International Journal of Chemical Kinetics DOI 10.1002/kin.20845

REACTION RATE PREDICTION OF ALIPHATIC MONOCARBOXYLIC ACIDS
191
If experimental parameters such as temperature, re-
actants concentrations, and the amount of catalyst re-
main the same, the esterification rate is dependent on
the structure of the reacting acids and alcohols, espe-
cially on their inductive, mesomeric, and steric effects.
For acids and primary alcohols without mesomeric
effects, Taft suggested a two-parametric equation
before use with anhydrous CuSO4 (Lachema, Czech
Republic) to minimize water content in reaction mix-
tures. Inorganic chemicals such as H2SO4 (96%),
NaOH (titrimetric standard 5M), and CuSO4 anhy-
drous (Lachema, Czech Republic) were of analytical-
grade purity and were used as received.
Solutions of reactants were prepared by weighing
0.2 mol of corresponding acid together with 0.98 g
of anhydrous H2SO4. Weighed chemicals were quan-
titatively transferred into a 100 mL volumetric flask
(solution 1). Similarly, but without H2SO4, 0.2 mol
of corresponding alcohol was weighed and trans-
ferred into another 100 mL volumetric flask (solu-
tion 2). Finally, both solutions were filled to the mark
with pure dioxane. Equal volumes of the two solu-
tions were combined, giving a reaction mixture that
contained reactants with a starting concentration of
[Eq. (14)]. The Taft equation is a semiempirical linear
free energy relationship used in contemporary physi-
cal organic chemistry for the study of reaction mecha-
nisms [15–17] and for relative comparison of reaction
rates [18]. Usually the Taft equation has a general form
of
o
E
log kE = log k + ρ1 σ1 + δ ES
(11)
where σ1 is the inductive (polar substituent) constant.
It describes how a substituent will influence a reaction
rate with its inductive effect. Steric substituent con-
stant Es quantifies the influence of steric effects on the
reaction mechanism and its rate. Coefficients ρ1 and δ
are sensitivity factors for induction and steric effects.
Taft proposed that under identical experimental condi-
tions, the steric and inductive sensitivity factors would
remain constant and would not influence the ratio of
reaction rates, allowing comparison of reaction rates
−
3
1 mol·dm and H2SO4 as catalyst with a concentra-
−3
tion of 0.005 mol·dm
.
Residual carboxylic acid concentration was deter-
mined by titration. On the basis of these results, the
degree of conversion was calculated according to the
formula
cACIDstarting − cACIDdetermined
x =
(12)
cACIDstarting
0
E
relative to the standard value log k , that is, reaction-
rate constant of reference species having parameters
σ1 and ES equal to zero [19].
where cACIDstarting is the concentration of the acid at the
−3
beginning of esterification (in this work, 1 mol·dm
)
The aim of this study is an experimental deter-
mination of sensitivity factors ρ1 and δ in the Taft
equation. It reports the relative esterification rates
of aliphatic monocarboxylic acids with primary al-
cohols and demonstrates the calculation of reaction
rate constant for other esterification reactions under
the same experimental conditions. Parameters σ1 and
ES are known for substituents of any primary alcohol
and aliphatic monocarboxylic acid lacking mesomeric
effects.
and cACIDdetermined is the residual acid concentration at
a given time point.
For the determination of equilibrium constants,
10 mL of solution 1 and 10 mL of solution 2 were
pipetted into ampoules. The ampoules were sealed and
◦
left to stand in a bath thermostated at 60 C for 10 days.
Then, ampoules were opened and 1 mL was diluted for
titration with 30 mL water and this solution was titrated
−3
with 0.05 mol·dm NaOH with phenolphthalein as the
indicator; 0.2 mL was subtracted from the consump-
tion corresponding to catalyst content in the sample.
The measurements were repeated three times and the
degree of conversion was calculated.
EXPERIMENTAL
For the determination of esterification rate con-
3
Formic acid, acetic acid (glacial anhydrous 100%), 1-
propanoic (propionic) acid, 1-butanoic (butyric) acid,
stants, a 500 cm conical flask was immersed in a
◦
glass thermostat, stabilized a 60 ± 0.1 C. The reactor
2
3
-methylpropanoic (isobutyric) acid, 1-pentanoic acid,
-methylbutanoic acid, 2,2-dimethylpropanoic acid, 1-
was equipped with a magnetic stirrer and closed with a
plug, through which a long needle was introduced for
sampling the reaction mixture by syringe at chosen
time intervals to obtain 10–15 points of the kinetic
curve.
octanoic acid, 2-phenoxyacetic acid, 2-chloroacetic
acid, 2-bromacetic acid, 3-chloropropanoic acid, 2-
cyanoacetic acid propyl ester, methanol, ethanol, 1-
propanol, 1-butanol, 1-pentanol, 2-methyl-1-propanol,
The degree of conversion at each time point was de-
termined titrimetrically during kinetic measurements.
The first run for each esterification reaction pair was
provisional to determine the optimal sampling times
1-octanol, 2-phenoxyethanol, 2-chlorethanol, 2-
methoxyethanol, 1,2-dihydroxyethan, and 1,4-dioxane
were of synthesis grade (Merck, Germany) were dried
International Journal of Chemical Kinetics DOI 10.1002/kin.20845

ˇ ´
VOJTKO AND TOMCIK
1
92
Table I Thermodynamic Equilibrium Constant (K) and Esterification Rate Constant (kE) for Selected Aliphatic
Monocarboxylic Acids with 1-Propanol ES and σ1 Taken from Ref. [20]
5
5
kE·10
RSD ke
(%)
kE·10 Calculated
3
−1
3
−1 −1
Acid Name
ES
σ1
K
SDK
dm ·mol ·s−1
dm ·mol ·s
x∝
Formic acid
Acetic acid
+1.24 +0.02 4.19 0.80
44.0
1.97
1.60
0.81
0.65
0.84
0.24
0.067
0.67
1.44
1.28
1.13
0.33
0.59
0.35
5.0
4.6
3.1
3.7
4.6
3.6
4.2
4.5
4.5
2.8
3.9
3.5
3.0
3.4
2.9
32.7
1.97
1.67
0.87
0.68
0.81
0.24
0.06
0.79
1.15
1.45
1.36
0.32
0.69
0.28
0.67
0.66
0.66
0.66
0.66
0.66
0.65
0.67
0.68
0.63
0.55
0.54
0.63
0.58
0.66
0.0
0.0
3.85 0.70
1
1
2
1
3
2
1
2
2
2
2
2
3
-Propanoic acid
-Butanoic acid
–0.07
–0.36
–0.47
–0.39
–0.93
–1.54
–0.40
–0.01 3.70 0.70
–0.01 3.73 0.60
–0.01 3.74 0.50
–0.01 3.80 0.60
–0.01 3.53 0.50
–0.01 4.22 0.70
–0.01 4.35 0.80
-Methylpropanoic acid
-Pentanoic acid
-Methylbutanoic acid
,2-Dimethyl-pro-panoic acid
-Octanoic acid
-Phenoxyacetic acid
-Chloroacetic acid
-Bromoacetic acid
-Cyanoacetic acid
-Phenylacetic acid
-Chloropropanoic acid
–0.33 +0.40 2.83 0.05
–0.24 +0.47 1.47 0.03
–0.27 +0.47 1.34 0.03
–0.94 +0.57 3.01 0.05
–0.38 +0.12 1.94 0.04
–0.90 +0.17 3.67 0.06
Table II Thermodynamic Equilibrium Constant (K) and Esterification Rate Constant (kE) for Selected Primary
Alcohols with Acetic Acid ES and σ1 Taken from Ref. [20]
5
5
kE·10
RSD ke
(%)
kE·10 Calculated
3
−1
3
−1 −1
Alcohol Name
ES
σ1
K
SDK
dm ·mol ·s−1
dm ·mol ·s
x∝
Methanol
Ethanol
+1.24
0.0
+0.02
0.0
3.95
3.85
3.85
3.62
3.20
3.21
3.44
0.72
0.19
0.61
0.7
0.7
0.6
0.6
0.5
0.5
0.6
0.02
0.003
0.01
4.79
2.18
1.97
1.94
1.59
1.93
2.15
0.744
0.278
0.595
6.3
7.4
6.1
4.1
5.0
6.2
6.5
5.4
3.6
5.0
4.36
2.18
2.16
1.81
1.69
1.86
1.77
0.67
0.30
0.48
0.88
0.67
0.66
0.66
0.67
0.64
0.64
0.65
0.46
0.33
0.44
1
1
2
1
1
2
2
2
2
-Propanol
-Butanol
-Methyl-1-propanol
-Pentanol
-Octanol
-Methoxy-ethanol
-Chloroethanol
-Phenoxy- ethanol
-Dihydroxy-ethane
–0.07
–0.36
–0.47
–0.39
–0.40
–0.19
–0.24
–0.33
+0.03
–0.01
–0.01
–0.01
–0.01
–0.01
+0.30
+0.47
+0.40
+0.25
a
0.966
a
1
x
1−x
Calculated from dynamic measurements according to the formula k =
at short times of the reaction when it is possible to consider
2
t
the direct reaction of second order.
corresponding to significant changes in the degree of
conversion. Generally, samples were taken every 5 min,
and then the time sampling interval was extended to
The calculated degree of conversion was used di-
rectly for the calculation of the equilibrium constants
according to the formula
2
0 min. Each point on the kinetic curve was used to
[ester][water]
acid][alcohol]
x2
(1 − x)
calculate the corresponding rate constant.
K = [
=
(13)
2
RESULTS AND DISCUSSION
The results are summarized in Tables I and II. Rel-
ative standard deviations do not exceed 8% for each
measurement.
Thermodynamic equilibrium constant values were
used to calculate esterification rate constants accord-
ing to Eq. (10). It has been found that a discontinuous
isothermal reactor equipped with a stirrer is very con-
venient for creating a homogeneous environment for
First, thermodynamic equilibrium esterification con-
stants of 15 aliphatic monocarboxylic acids with
propanol and 11 primary alcohols with acetic acid were
determined. 1,4-Dioxane was used as the solvent be-
cause it is inert and dissolves all components of the
reaction system to give a homogeneous solution.
International Journal of Chemical Kinetics DOI 10.1002/kin.20845

REACTION RATE PREDICTION OF ALIPHATIC MONOCARBOXYLIC ACIDS
193
−
−
−
6
5
4
−
−
−
4.5
5.0
5.5
−
5.5
−5.0
−4.5
−
3
−4
−5
−6
log kE(experimental)
log kE(experimental)
Figure 2 Correlation between the experimentally obtained
Figure 1 Correlation between the experimentally obtained
values of the esterification rate constants of acetic acid with
experimental
values of the esterification rate constants of 1-propanol with
selected primary alcohols (log
kE) and the values
experimental
calculated
selected aliphatic monocarboxylic acids (log
and the values log
kE)
log
kE calculated from Eq. (15).
calculated
kE calculated from Eq. (14).
primary alcohols and Taft equation for this reaction can
be expressed as (with ethylacetate is as the reference
compound)
both polar (water) and less polar (esters) compounds
in 1,4-dioxane as a solvent.
1-Propanol was used as a model alcohol to deter-
mine the dependence of the esterification rate on the
structure of the 15 aliphatic monocarboxylic acids. The
0experimental
log kE = log kEethylacetate − 1.55 σ1 + 0.27 ES (15)
standard least squares method was used to determine
0
experimental
−5
experimental
where log k
is equal to 2.18 × 10
the dependence log k
versus σ1 and ES (with
Eethylacetate
−1 −1
. A plot of log kE(calculated) versus log
3
dm mol
s
parameter value from Anslyn and Dougherty [20]); the
results can be described by Eq. (14) (with propyl ac-
etate as the reference compound):
kE(experimental) was constructed (Fig. 2) and found to be
linear with a slope of 0.96523 (SDSLOPE = 0.063, R =
2
0.981).
0
experimental
Correlation equations (14) and (15) are signifi-
log kE = log k
+ 0.22 σ1 + 0.98 ES
Epropylacetate
cantly different. The most significant in esterification
of 1-propanol with different acids is the steric effect,
connected with formation of protonated intermediate,
whereas in the process of esterification of acetic acid
with primary alcohols, the crucial influence comes
from inductive effects of the corresponding alcohol.
Very negative induction sensitivity factor ρ1 = –1.55 in
Eq. (15) indicates that the reaction of acetic acid with
primary alcohols is accelerated by electron-donating
substituents that promote nucleophilic attack of the al-
cohol on the protonated intermediate.
Using 1,4-dioxane as reaction medium enables esti-
mates of the reaction-rate constant because it depends
only on the structure of the reactants. Previous mea-
surements of structural influence on the esterification
rates have always been carried out in different reac-
tion environments or at a high concentration of one
(14)
0
experimental
−5
where log k
is equal to 1.97 × 10
Epropylacetate
3
−1 −1
dm mol
s . The induction sensitivity factor of
ρ1 = 0.22 shows that the reaction of 1-propanol with
aliphatic monocarboxylic acids is mildly sensitive to
induction effects, whereas δ = 0.98 confirms a strong
steric influence on acid deprotonation. According to
the Eq. (14), values of log kE(calculated) were calculated
for each propylester formation. A plot log kE(calculated)
versus log kE(experimental) was found to be linear
(
Fig. 1) with a slope of 0.96755 (SDSLOPE = 0.035,
2
R = 0.984) showing good correlation between exper-
imental and calculated values of log kE.
Similarly, acetic acid was selected as model acid
for the measurements of the esterification rates with 11
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ˇ ´
VOJTKO AND TOMCIK
1
94
Table III Predicted and Calculated Esterification Rate Constants for Another Ester to Verify Proposed Procedure.
Experimental Data Obtained from Second-Order Kinetic Equation at Low Degree of Conversion
5
5
5
kE(calc)·10
kE(exp)·10 (K = 1)
kE(exp)·10 (K = 4)
3
−1
3
−1 −1
3
−1 −1
]
Ester Name
-Butylphenyacetate
Formula
[dm ·mol ·s−1
]
[dm ·mol ·s
]
[dm ·mol ·s
1
C12H16O2
C4H7O2Br
C10H20O2
C7H13O2Cl
0.45
0.31
0.67
0.17
0.42
0.35
0.65
0.14
0.42
0.35
0.65
0.14
Ethylbromacetate
1
-Penthyvalerate
2-Methyl-1-propyl-3-chlorpropionate
component, where the esterification-rate equation was
reduced to pseudo-first-order or the second-order equa-
tion was used but reactions were not followed to equi-
librium.
Knowing sensitivity factors for the two Taft param-
eters for both the acid and the alcohol allows one to
estimate the esterification rate for any primary alcohol
reacting with any aliphatic monocarboxylic acid under
the same conditions at which rate constant of reference
species and their sensitivity factors were measured.
Because the reaction center is the same for any acid
and alcohol, the rate of esterification for any ester XY
calculated esterification reaction rate value is 1.97 ×
6 3
−
−1 −1
10 dm ·mol ·s . This value is in relatively good
−6
3
−1 −1
agreement with a value of 1.65 × 10 dm ·mol ·s
obtained experimentally. Similar to this, esterification
rate constants of next four esters were predicted and
verified experimentally at low degrees of conversion.
The results are summarized in Table III. The good
agreement between predicted and experimental values
of rate constant was achieved.
CONCLUSIONS
(
index X is related to alcohol and Y to acid) can be
A novel, simple, and reliable procedure for estimation
of esterification rates was developed. The esterification
rate constants of aliphatic monocarboxylic acids with
expressed as
log kXY = log kXA + 0.98EY + 0.22 σ1Y (16)
where
1-propanol and of primary alcohols with acetic acid
were measured under conditions of a nearly ideal so-
lution. It was observed that 1,4-dioxane is a suitable
solvent for these measurements because it is inert and
readily dissolves polar (acids, alcohols, and water) and
less polar (esters) species present in reaction mixture.
For the quantification of relationship between struc-
ture and reaction rate, a two-parameter Taft equation
was used, including inductive and steric effects. Be-
cause all rate measurements were made under equiv-
alent experimental conditions (inert solvent, relatively
low concentrations of reactants, constant concentra-
tion of hydrogen ions, and constant temperature and
volume), calculations using the second-order kinetic
equation provided consistent reaction-rate constants.
On the basis of these results, it is possible to estimate
reaction-rate constants that were not experimentally
measured using only the known Taft steric and in-
duction constants for each reactant. This methodology
should also be applicable to other types of bimolecular
chemical reactions.
log kXA = log kEA + 0.27EX − 1.55 σ1X (17)
where kEA is reaction-rate constant of ethylacetate for-
mation and kXA is the reaction-rate constant of the
corresponding acetate formation. Values of log kPA
(reaction-rate constant of propylacetate formation) can
also be used for calculation but it should be recalcu-
lated from value log kEA according to the equation
log kPA = log kEA +0.27ESPropanol −1.55 σ1 Propanol
=
log kEA − 0.0034
(18)
By combination of Eqs. (16), (17), and (18), a final
equation for the calculation of rate constant is obtained:
log kXY = (log kPA + 0.0034) + 0.98ESY
+
0.22 σ1Y + 0.27ESX − 1.55 σ1X
(19)
APPENDIX: INTEGRATION OF KINETIC
EQUATION
The validity of this approach was verified on amyl
isovalerate as an example. Amylisovalerate is an es-
ter of amyl alcohol (1-pentanol, ES = –0.39, σ1 =
Consider the general equilibrium chemical reaction
A + B ↔ R + S
–
0.01) and isovaleric acid (3-methylbutanoic acid,
ES = –0.93, σ1 = –0.01). According to Eq. (19), the
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REACTION RATE PREDICTION OF ALIPHATIC MONOCARBOXYLIC ACIDS
195
Definitions:
kE: reaction rate constant of forward reaction,
kH: reaction rate constant of backward reaction.
= kEcB,0dt
1
dx = kEcB,0dt
2
ax − bx + c
decomposition on partial fractions, assuming D =
cA,0 − cA
cA,0
x =
− degree of conversion
2–
b 4ac > 0,
cA = cA,0 (1 − x)
1
1
=
√
2
2
ax − bx + c
b − 4ac
cB = cB,0 − cA,0x
cR = cR,0 + cA,0x
cS = cS,0 + cA,0x
ꢊ
ꢋ
1
1
×
√
−
√
2
2
b
b −4ac
2a
b
2a
b −4ac
x − 2a
−
x −
+
2a
x
ꢌ
1
1
1
√
√
dx − √
2
2
b
2a
b −4ac
2a
2
b − 4ac
x −
−
b − 4ac
kE
K =
kH
0
x
t
ꢌ
ꢌ
1
×
√
√
dx = cB,0kE tdt
The overall reaction rate can be expressed as
2
b
b −4ac
2a
x − 2a
+
0
0
ꢀ
ꢁ
√
2
1
b
b −4ac
1
x −
−
+
voverall = kEcAcB − kHcRcS = kE cAcB −
cRcS
2a
2a
ln
√
const = cB,0kEt
K
2
2
b
b −4ac
b − 4ac x − 2a
for x(t = 0) = 0
2a
assuming the following about the mentioned defini-
tions:
√
2
b
b −4ac
dx
dt
ꢄ
ꢅ
x − 2
−
+
a
2a
cA,0
= kE cA.0(1 − x)(cB,0 − cA,0x)
√
const = 1
2
b
b −4ac
x −
ꢆ
ꢇ
2a
2a
1
×
−
(cR,0 + cA,0x)(cS,0 + cA,0x)
K
therefore
√
cA,0 dx
2
b − b − 4ac
1
cA.0(1 − x)(cB,0 − cA,0x) − K (cR,0 + cA,0x)(cS,0 + cA,0x)
const =
√
2
b + b − 4ac
=
kEdt
for kE can be derived as
cA,0 dx
ꢈ
ꢉ
ꢈ
ꢉ
x2
c
2
−
1
K
c
2
+ x −cA,0cB,0 − c
2
√
√
b2−4ac
2a
A,0
A,0
A,0
b
a
1
2
x − 2
−
+
kE =
√
ln
cA,0 dx
2
b
2a
b −4ac
×
tcB,0 b − 4ac x −
1
K
1
K
2a
−
(cA,0cS,0 + cA,0cR,0) + cA,0cB,0 −
cR,0cS,0
ꢊ
ꢋ
√
√
2
=
kEdt
b − b − 4ac
b + b − 4ac
×
2
dx
ꢈ
ꢉ
ꢄ
ꢅ
2
1
K
1
K
x cA,0 1 −
− x cB,0 + cA,0 + (cS,0 + cR,0)
respectively.
dx
1
2
×
1
cR,0cS,0
cA,0
k
E
=
√
+
kEdt
cB,0 − K
tcB,0 b − 4ac
ꢆ
ꢆ
√
ꢇ
ꢇ
ꢈ
√
ꢉ
2−4ac
ꢈ
√
ꢉ
ꢉ
b+
b−
b
b
=
2
2
x b − b − 4ac −
ꢈ
b − b − 4ac
2
a
×
ln
√
√
ꢉ
2−4ac
ꢈ
√
dx
2
2
x b + b − 4ac −
b + b − 4ac
ꢂ
ꢃ
2a
ꢈ
ꢉ
cA,0
cB,0
1
K
cB,0+cA,0
1 cS,0+cR,0
x2
1 −
− x
+
ꢂ
ꢃ
cB,0
K
cB,0
√
2
x b − b − 4ac − 2c
1
2
dx
kE =
√
ln
ꢂ
ꢃ
√
×
1
cR,0cS,0
tcB,0 b − 4ac
2
+
1 −
x b + b − 4ac − 2c
K cA,0cB,0
International Journal of Chemical Kinetics DOI 10.1002/kin.20845

ˇ ´
VOJTKO AND TOMCIK
1
96
finally,
2. Isbell, T. A.; Carlson, K. D.; Abbott, T. P.; Phillips,
B. S.; Erhan, S. M.; Kleiman, R. Ind Crops Prod 1996,
ꢂ
ꢃ
ꢃ
√ 2
5, 239–243.
2
c − b − b − 4ac
x
x
1
2
3
4
5
6
7
8
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S. Food Chem 2007, 101, 1552–1557.
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kE =
√
ln
ꢂ
√
tcB,0 b − 4ac
2
2
c − b + b − 4ac
295.
for our experiments,
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ꢀ
cA,0
ꢁ
1
K − 1
a =
1 −
=
cB,0
K
K
cA,0 + cB,0
cB,0
cR,0 + cS,0
KcB,0
1 + 1
b =
+
=
+ 0 = 2
1
2009, 88, 2011–2016.
cR,0cS,0
KcA,0cB,0
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c = 1 −
= 1
ꢂ
ꢍ
ꢃ
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0. de Sousa Galv a˜ o, M.; Narain, N.; do Socorro Porto dos
K−1
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− 2 − 4 − 4 K
x
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1926.
K−1
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√
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International Journal of Chemical Kinetics DOI 10.1002/kin.20845