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A
Methodology: The Determination of
Kinetic Parameters for Single Independent Chemical Reaction Sequences
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Kenneth M.
Maloney*
and Ana Clare Frye
Chemistry and Physics Division, Baton Rouge Community
College, 201 Community College Drive,
Baton Rouge, LA 70806
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Given the basic equation, WF
= ¡·trp·exp(-SL·tr) [WF is the proportion reacted, ¡
is the scale reaction factor, tr is the
reduced time, p and SL are the curvature and reaction terminus
factors respectively], an essential consistency test requirement, for a
given chemical reaction to be a true independent sequence, is that the
curvature reaction factor, p, and the terminus reaction factor, SL,
must be equivalent(i.e., p = SL). The single replacement Mg(s) +
2HCl(aq) ® MgCl2(aq) + H2(g)
reaction is an irreversible reaction with a well defined and easily
determined endpoint; therefore, it was used to test the validity of this
important requirement. The
experimental results confirmed the boundary condition requirement that p =
SL for this single sequence reaction.
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Introduction
Ideally, chemical reactions are
performed in well-defined, fixed reaction volumes where the initial
concentrations of the reactants, at the initiation of the reaction
sequences, are equivalent to the initial concentration measurements. Practically, however, many reactions
necessarily occur in instantaneously generated reaction volumes where the
volumetric capacities differ substantially from the original containment
volumes of the reactants prior to the initiation of reaction. Realization of a methodology, in which
these circumstances are simply treated as added dimensions to the analysis
and determination of kinetic parameters, would be highly desirable.
It has been found that such a
methodology is inherently a derivative of the consistency tests and basic
equation previously outlined.1
In this paper, this methodology is detailed and the utility of the approach demonstrated by an
evaluation and analysis of the experimental data obtained from a study of
the single replacement Mg(s) + 2HCl(aq)
® MgCl2(aq) + H2(g)
irreversible reaction system.
Basic Considerations
For the reaction A + B + C….etc ® Products, we
have the simple n-order rate law
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Rate = - d[A]/dt
= k×[A]n
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(1)
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for the case of equal
initial concentrations of the individual reactants(i.e., [A] = [B] = [C] =
…etc.) where k is the specific rate constant. Expressed in terms of the proportion
reacted(WF)1 and the initial concentration([A0])
, one has
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Integration of eq(1) and
substitution into eq(2) gives
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WF = 1 – {1
- [(1 - n)×k×t /[A]0 (1 - n) ] } 1/(1-n)
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(3)
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where the units of k are
conc-(n-1) sec-1.
Eq(3) becomes
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where z
= (n – 1)× k/[A0]
(1 – n), k = k×t’, tr = t/t’,
and e
= 1/(1-n). tr is the corresponding reduced independent variable
where the quantity t’ represents the value used to reduce the independent
variable(t) sequence. 0n the
occasion that one simply wishes to determine the order of the reaction, eqs
3 and 4 reduces to
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WF = 1 - {1
- [(1 - n)×t/tC] }1/(1-n)
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(5)
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where tC = k-1
∙[A]0 (1 - n). For the singularity at n = 1(i.e.,
first-order),
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Figure 1 illustrates the
proportion reacted vs time for the conventional definition of reaction
order-n(i.e., eq 1).
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Figure 1. Graphical
representation of the proportion reacted vs reduced time for values of the
conventional reaction order factor over a range of n = 0 – 3.
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Many important chemical reactions,
however, proceed through a number of complex pathways. For the vast majority of cases, the
reactants A, B, C,….. etc are not present in equal concentrations. Therefore, the rate law is, routinely,
much more complicated. Fortunately,
many of these complex systems are amenable to meaningful
simplifications. For example,
Lavabre, et. al.2 examined seven general kinetic schemes
encompassing reversible, mixed first- and second-order as well as
autocatalytic reactions and found them all to be governed by a single rate
law. A single dimensionless variable
Y(t), which represents the characteristics of the course of the reaction,
was introduced. Y(t) is given by the
expression
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Y(t) = (1 – m)/[exp(a×t)
– m]
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(7)
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where the parameter m
characterizes the geometrical shape and the value of the a parameter
depends on the particular reaction scheme under consideration2. In effect, Y(t) is the normalized
amplitude of the kinetics. Thus,
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Separately1, it was shown
that a particular reaction sequence maybe uniquely represented by
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WF(t) = ¡r ×trp×Exp(-SL×tr)
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(9)
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The independent variable, tr, can be time, concentration or any other independent
variable with the same functional dependence relative to WF. ¡r is a scale
factor, p is a geometrical shape factor, and SL is a reaction
terminus factor. Together, the parameters ¡r, p, and SL
define a unique set of reaction field factors(RxnF-factors) for the
particular reaction sequence of interest.
Given that any reaction sequence can be expressed in terms of the
proportion reacted versus an appropriate independent variable, the
order/curvature factors n, m, and p factors(hereinafter call
geometry factors) can be systematically determined1 by the
nonlinear regression of the measured experimental data with eqs 5,6, 8, and
9. Illustrated in Fig 2 is the
functional relationship between n, m, and p.
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Figure 2. Graphical
representation of the functional relationship between the geometry factors n, m, and p.
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An important requirement of the
consistency tests, for reaction rate data, is that p = SL at the
terminus of each independent reaction sequence1. In this paper, in order to test this
important requirement, a frequently used reaction, in the general chemistry
laboratory, is employed as the reaction characterization tool. Namely, the single replacement reaction
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Mg(s) + 2 HCl(aq)
® MgCl2(aq) + H2(g)
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(I)
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The occurrence of the
distinct phase changes, in this type(I) of reaction, enables the progress
of the reaction to be followed visually and the endpoint or reaction
terminus to, also, be definitively observed. In fact, it is, in part, for many of
these characteristics that reaction(I) is routinely investigated
gravimetrically. Therefore, important reaction parameters can be easily
monitored by equipment/instruments common to the general chemistry
laboratory.
Experimental
Section
The
samples of magnesium particulates were derived from turnings obtained from
Malinckrodt Co., and the purity was reported to be 99.868%. The HCl solutions were prepared from
concentrated(99.99% purity,12.1 N) HCl obtained from Fisher Scientific Co.
Both materials were used as received.
A dual purpose Hanna Laboratories model HI 9025C digital
temperature/pH meter and a model 600L Olympus digital camera were used to
obtain the data during the course of each run. The buret-beaker apparatus and the
temperature/pH meter combination(Fig 3) were secured on a metal rack at a
predetermined focal distance from the digital camera.
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Figure 3. Apparatus
used to measure the volume of accumulated hydrogen gas in the
magnesium-hydrochloric acid reaction.
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The data for all of the runs
were recorded at 1-minute intervals for the entire course of each reaction.
Prior to each run, a weighed
particulate sample of Mg(s) was placed in the approximate center of a watch
glass previously positioned at the bottom of the temperature controlled
bath(2000 mL beaker of distilled H2O). Five(5) mL of 6M HCl was first added to an empty 50 mL buret
followed by the careful addition of 49.3 mL of distilled water. Each run was then initiated by carefully
and rapidly inverting the filled buret onto the surface of the submerged
watch glass. Subsequently, the buret
was clamped into position forming a simple pressure contact with the curved
surface of the watch glass. Thus,
irrespective of the random motion of the Mg(s) particulate reactant, which
occurs as a result of the bubbling action of the H2 release, the
Mg(s) reactant was always completely secured within the inner circumference
of the buret. This configuration
ensured the complete capture of the evolved H2 within the volume
of the buret as well as a metered displacement of the liquid contents out
of the buret into the unstirred beaker of distilled water. In effect, this simple set-up is a simple
reaction-controlled flow unstirred reactor (RCFUR) where the pH is periodically
measured, within the bath, at a distance from the reaction center.
Results and
Dissussion
Determination of tL. The consistency requirement1
of
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SL =
p for t = tL(i.e.,
where t' = tL)
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must be uniquely fulfilled for
each single reaction sequence where tL is the terminal value of
the independent variable t. In
order to accommodate the inevitable vagaries attendant to experimental
data, Eq 9 can be expressed in a more robust form as the following
translation
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WF = d×WF' = dס×trd×p× Exp(- d×p×tr)
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(10)
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where d represents a nonzero
constant. Since all values of d ¹
1 are constrained to yield graphs that are identical to those obtained when
d = 1, one has the fact that WF and WF’ are graphically equivalent. For a given translation, tL =
t'/d; thus,
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WF = ¡×tp× Exp[- p×t]
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were t = t/tL.
Alternatively, if the nonzero constant, d, is constrained to d =1,
the value of t' = tL. The
terminal boundary condition is WF = 1, t = 1 and ¡ = ep.
Therefore, one has
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WF = ¡×tp× Exp[- p×t]
= tp×
exp[p(1 - t)]
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(11)
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Hence, the values of the reaction factors, reported herein,
are the minimum values that preserve the invariance in the nonlinear
regression results of the experimental data.
In the
consideration of reaction(I) as a conventional titration system - if upon
initiation of the reaction, the reactants are instantaneously combined,
then the equivalence point of the titration(teq) and the
reaction terminus(tL) must be coincident with the accumulated
volume of H2(g){V(H2)} because Mg(s) is the
predetermined limiting reactant(i.e., teq = tL). The indicator in this system is, thus,
simply {V(H2)}. Expressed
in the form of the proportion reacted, one has
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WF = {V(H2)}t/{V(H2)}max
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(12)
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where {V(H2)}t and {V(H2)}max
are the accumulated volumes at the elapsed times t and tL respectively. The configuration of the experiment is
such that the corresponding increase in acidity, due to the release of
unreacted HCl(aq) and MgCl2(aq) into the outer containment bath,
is less than it would otherwise be in the absence of the limiting Mg(s)
reactant. This moderated reduction
in the pH (illustrated in Figures 4-7) was measured and, in part, employed
as a tool to monitor the relative performance characteristics of each
experiment.
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Figure 4. Apparatus used
to The proportion reacted vs reduced time for the magnesium-hydrochloric
acid reaction at 22 °C [experiment - (o);
regression - ()].
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Figure 5. The
proportion reacted vs reduced time for the magnesium-hydrochloric acid
reaction at 32.7 °C [experiment - (o);
regression - ()].
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Figure 6. The
proportion reacted vs reduced time for the magnesium-hydrochloric acid
reaction at 50.4 °C [experiment - (o);
regression - ()].
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Figure 7. The proportion
reacted vs reduced time for the magnesium-hydrochloric acid reaction at
60.5 °C
[experiment - (o); regression - ()].
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Determination
of the Overall Rates, Effective Reaction Volumes, and the Specific Rate
Constants. In the case of reaction(I), the overall
rate equation will take the form of:
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R = k*× [Mg]a[HCl]n
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(13)
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Now, since the concentration of magnesium is constant
(and so [Mg]a) then it becomes part of k as
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R = k×[HCl]n
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(14)
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where k = k* [Mg]a. It is important to note that, because the
HCl reactant is in excess, the form of eq 14, as applied to reaction (I),
can also be understood in an alternate manner. In effect, the reactant, Mg, is the
limiting reagent. Regression of the
WF vs reduced time plots, illustrated in Figures 4 - 7, consistently yield
values of p = 1 which is equivalent to n = 0(see Figure 2). The overall
reaction is, therefore, pseudo-zero order.
In essence, since the WF vs t plots express the progress of reaction
(I) in terms of H2 production, the rate of the reaction is,
thus, independent of the Mg reactant.
The specific rate follows
directly from eq 11 as
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WF = ¡×tp× Exp(-k*×t ) = tp× exp[p(1 - t)]
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(15)
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where k* = p/tL. Because of the CFUR configuration, the
reaction progress of each experiment, due to the progressive change in the
HCl concentration, are manifested as a pH vs reduced time profile (Figures
4 - 7). Inasmuch as the reduced time
axis is directly proportional to the corresponding concentration values,
Figures 4 -7 represent titration curves.
Figures 4 -7 are, in fact, clearly analogous to the curves which
result from the titration of a weak base by a strong acid. Additionally, however, in each case, the
equivalence point also establishes the value of tL for each
experiment. The order of the HCl
reactant was determined by the nonlinear regression of the progressive
increases in the deviations from the initial baseline(eq 5). Nonlinear regression of the experimental
data yields a reaction order of n = 2.
Thus,
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R = k×[HCl]2
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(16)
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where, given the
stoichiometry defined by the reaction (i.e., eq I), one has
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R' = + d[H2]/dt = - (1/2)×d[HCl]/dt
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where R = 2×R’. Finally, the relative rates of
disappearance of reactants and formation of products yields
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k = 2×R'/[HCl]2
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The specific rate constants
and a comparison of the visually estimated values of tL vs the
corresponding regression values of tL are given in the Table
below.
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Table : Reaction
Parameters for Mg(s) + 2 HCl(aq) ® MgCl2(aq) + H2(g)
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Temperature
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k
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tLa(est)
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tLb(reg)
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°C
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L mol-1×s-1 s
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s
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22.0
32.7
50.4
60.5
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0.0268
0.0420
0.0971
0.142
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780
660
540
420
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734
636
549
404
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aVisual estimate; bNonlinear
regression
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The Arrhenius activation
energy was found to be 29.6 kJ/mol(Fig 8).
This is in excellent agreement with other published results.3
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Figure 8. Plot of
log k versus 1/T for the single replacement Mg(s) + 2 HCl(aq) ® MgCl2(aq)
+ H2(g) reaction(data from table).
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Conclusions
The
experimental results, presented herein, confirmed the boundary condition
requirement that p = SL at the end of a single independent
reaction sequence. In the
application of this tool, the relationship between the order of a reaction
and the geometry/curvature of the reaction progress curve is readily
apparent. Removal of the
singularity, inherent to the first order case for the conventional rate
equation, is shown to be an algebraic reformulation in which the singular
point is simply redefined as the boundary condition tL = finite
for p = 1. In effect, the first
order singularity can be restated, simply, as a point in a continuum. The fact that the progress of many other
chemical reactions can be expressed in terms of a single dimensionless
variable enhances the utility of this approach as an additional tool in the
analysis of reaction rate data.
References
1. Maloney, K.
M. http://kennethmaloney.com/Publications/Article1BasicEqn2002July16.htm
2003.
2. Lavabre,
D.; Pimienta, V; Levy, G; Micheau, J.C. J. Phys. Chem. 1993, 97,
5321-5326.
3. Birk, J.
P.; Walters, D. L. J. Chem. Educ. 1993, 70, 587-589.
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© • Kenneth M. Maloney, Ph.D.
SCIENCE
and TECHNOLOGY • All rights reserved
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