Download Determination of the reaction order Determination of the reaction

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Physical Chemistry EPM/10
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Determination of the
reaction order
Reaction order is a purely formal quantity, hence,
its may be determined only experimentally.
In the case of elementary reactions their order is
equal to their molecularity (more later).
The subject of the considerations to follow may be
summarized as: „How to find the reaction order on the basis
of experimental data”?
It’s worthy to notice that the methods presented in next slides are of historical
importance rather, because nowadays – at the age of computerized instrumentation
(measuring and analytical devices) – it is rather easy to acquire large data sets c=f(t),
which subsequently may be fitted to kinetic equations of different orders using
computerized regression techniques.
Physical Chemistry EPM/10
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Determination of the
reaction order (2)
Those regression techniques, however, are based strictly on the theory
presented earlier and the methods presented below. These methods have
their origins in times where data collection was not so easy and
computation methods were more tedious. Therefore, planning of
experiments required much more sophisticated approach.
These old methods are also very useful for analysis of kinetic data (small data sets) in
text problem solving exercises. Hence, their application is essential for chemists. It is
always useful to inspect the data carefully if any regularities discussed in next sections
may be observed. Quite frequently, especially if the experiment was designed in a way
favoring such observations, one can notice, for example, independence of half-life time
of concentration. Such observations are still important even at the age of computers,
when they may suggest which model (order) should be tested first.
Physical Chemistry EPM/10
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Determination of the
reaction order (3)
Equation testing method
This is the simplest (most primitive), though most effective approach (esp. if no
regularities might be observed). In this method we use the data in integrated rate laws of
several orders and check, which one yields invariable rate constant (independent of
varying concentrations of reagents).
Graphical method
In this method, we plot the data in linearized systems of coordinates (suitable for
different reaction orders) and look, which system yields a straight line plot. Its slope is
the rate constant.
One can notice that the regression method is a generalization of the both above
mentioned methods.
Physical Chemistry EPM/10
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Determination of the
reaction order (4)
Half-reaction time method
We saw before that half-reaction time id directly proportional to the initial
concentration for 0th order, independent of the initial concentration for 1st order, and is
inversely proportional for the 2nd order. This may be generalized by formula:
1− n After finding the half life-time for several initial
concentrations, proportionality may be tested using this
1/ 2
0
formula for different orders n.
τ
∝c
Integral method by Ostwald-Zawidzki-Noyes
This is a variant of the former method, in which half life-time (or reduced time, i.e. time
of the same fraction of reactants transformed) must be determined for two different
initial concentrations. Subsequently, reaction order n may be found from the formulae:
t1  c02 

=
t 2  c 01 
n −1
n = 1+
or
ln t1 − ln t 2
ln c02 − ln c01
Physical Chemistry EPM/10
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Determination of the
reaction order (5)
Differential method by van’t Hoffa
In this method one utilizes no concentrations at given time but rates of reactions at
given time. It may be employed when reactions are not very fast. Most frequently it is
utilized for initial rates, when the following formula is valid:
ln v01 − ln v02
ln c01 − ln c02
Isolation method (initial rates)
n=
In this method initial rates are measured in several experiments planned in a way
permitting elimination of indluence of one or more reactants on the rate observed. An
example is shown in the next slides. The method is important because it permits to find
the partial orders versus each of the reactants!!!
Physical Chemistry EPM/10
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Determination of the
reaction order (6)
Example:
X2 + Y → XY + X
#
Initial concentration
mmol/dm3
X2
Y
1
450
270
10,8
2
150
270
1,2
3
450
90
3,6
Initial rate
mmol/(dm3·min)
Physical Chemistry EPM/10
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Determination of the
reaction order (7)
When we write rate laws for experiments #1 and #3 – we can eliminate
influence of reactant X2, whose initial concentrations are the same in these
experiments, by dividing both sides of the rate laws:
n
n
v01 = kcXm 01c Y0
v03 = kcXm 03 c Y0
c Xm 01 = cXm
1
3
n
n
v01 cY01  c Y01 
ln v01 − ln v03

= n = 
n=
v03 cY0
c
ln
c Y01 − ln c Y03
3
 Y03 
2
2
2
2 03
Even without taking the logarithms, one can see that concentration of Y is 3
times smaller in #3 than in #1 and the initial rate is 3 times smaller, too. Thus,
the reaction order vs. Y is 1. Treating experiments #1 and #2 in an analogous
manner, eliminating influence of reactant Y, we see that concentration of X2 is 3
times smaller in #2 than in #1, whereas the initial rate is 9 times smaller,
meaning 2nd order vs. X2.
Physical Chemistry EPM/10
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Determination of the
reaction order (8)
Experimental measurements of concentrations in time may be
difficult. If we measure during the course of reaction any additive
quantity X (e.g. pressure, volume, electric conductivity, density,
absorption of light), we can calculate concentration after time t
using the formula:
ct
X − Xt
= ∞
c0 X ∞ − X 0
where subscript ∞ means measurement after time,
when the additive quantity in question does not change
anymore (within the uncertainty limits of the used
measuring or analytical technique).
Physical Chemistry EPM/10
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Elementary reactions &
complex reactions
Chemical reactions usually do not occur as they are written in chemical
equations, which represent their summarical stoichiometry only.
Majority of reactions are – from the point of view of their kinetics –
complex reactions. It means that their occur is several steps (stages),
each of the latter being an elementary reaction. All elementary reactions
constituting given complex reaction show its mechanism.
Elementary (simple) reactions do occur as indicated by their
equations. Therefore, their order may be inferred on the basis of
their molecularity, i.e., the number of molecules which must meet
(collide) to result in the chemical reaction. For elementary
reactions order = number of molecules of the reactants.
Physical Chemistry EPM/10
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Reaction rates dependence
on temperature
Rate of majority of chemical reactions (and all elementary
reactions) increases with temperature.
A useful approximate rule (van’t Hoff’s) is assumtion that reaction
rate increases twice when temperature is raised by 10oC (K).
kT +10
=2
kT
Dependence of reaction rate on temperature
means actually dependence of its rate constant
on temperature.
More exact relation between temperature and reaction rate constant is
given by Arrhenius equation:
k= A ⋅ e
−
Ea
RT
Physical Chemistry EPM/10
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Arrhenius equation
EA – activation energy
A – preexponential factor (frequency factor)
k= A ⋅ e
ln k= ln A −
−
Ea
RT
Ea
RT
Results of measurements of k as a
function of T, plotted in a linearized
system of coordinates lnk=f(1/T),
permit determination of the
parameters of Arrhenius equation.
Slope:
Ea
tan α= −
Physical Chemistry EPM/10
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12
Arrhenius equation (2)
EA – activation energy is the lowest energy
that the reactants must have to get
transformed to products. It may also be
interpreted as the fraction of collisions
between molecules of sufficient kinetic
energy to exceed Ea. This is given by
Boltzmann distribution.
A – preexponential factor is requently
known as the frequency factor reflects the
frequency of collisions regardless their
energy.
The product of both represents the number
of successful collisions in time.
k= A ⋅ e
e
Physical Chemistry EPM/10
−
−
Ea
RT
Ea
RT
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Kinetics of reversible
reactions
So far we have treated all reaction as running to completion (in stoichiometric sense),
though sometimes „completion” was achieved after time t=∞. In reality, reactions run to
the point of equilibrium rather and we must find a suitable description of such
situations.
Equilibrium is not a static state (altough observation may suggest that the reaction does
not occur at this state – „nothing happens”). Actually, a dynamic equilibrium exists,
when two reactions occur: „left to right” and „right to left”, but their rates are the same.
k1
→
A←
B
k -1

v1 = k1c∞A v−1 = k −1c∞B
k1c∞A = k −1c∞B
k1
c
= ∞B = K
k −1 c∞A
v1 = v−1
K is reaction
(concentration)equilibrium constant.
Physical Chemistry EPM/10
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Kinetics of reversible
reactions (2)
dcA
= k1c A − k −1c B
dt
 k c − k −1cB0 
 = (k1 + k −1 ) t
ln  1 A0
 k1cA − k −1c B 
Rate law (differential) may be expressed as:
which, after integration, assumes
the following form:
If the equilibrium composition (concentrations) is known, one can write:
−
c −c
ln  A0 ∞A
 c A − c∞A

 = (k1 + k −1 ) t

The case with two second order reactions is more complex,
becoming very complex when reactions back and forth are of
different orders.
Physical Chemistry EPM/10
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Parallel reactions
If, at given conditions, one reactant yields more than one product,
which may be represented by the scheme:
then, assuming that all three reactions
are 1st order reactions, one can write:
dcA
= k1cA + k 2 cA + k 3 cA
dt
c
ln 0A = (k1 + k 2 + k 3 ) t
obtaining after integration:
cA
−
Such situations can be met in organic chemistry, when three isomers may
be obtained (e.g. o-, m-, and p-).
Chem. Fiz. TCH II/18
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Catalysis
Catalysis is a process of changing the reaction rate by certain
substances known as catalysts. Usually, the „change” means increase
in the reaction rate (acceleration). Sometimes, however, the desired
effect is slowing the reaction down. Such a process is known as
inhibition and the „negative catalysts” – as inhibitors.
Inhibitors are frequently used in fighting corrosion (if unavoidable then at
least slow it down) and in cosmetics industry.
Catalyst is a substance that changes the rate of reaction without being
consumed in the reaction, hence it is not shown in the stoichiometric
equation. Catalysts actually react (take part in reactions), chging their
mechanisms (othwerwise, how can they influence the reaction rate?).
Physical Chemistry EPM/10
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Catalysis (2)
There are two basic types of catalysis:
• homogenous – when catalyst is present in the same phase as the
reactants (practically it means catalysis in solutions)
• heterogenous – when catalyst is present in the different phase from
the reactants (practically a solid catalyst acting on reactants in solution
or in a gas phase).
In reaction mechanisms of complex reaction, there is usually one step,
which is slow and determines the rate of the whole complex reaction. It
acts very much like a part of a three-lane highway, where repairs are
being made and only one lane is open, thus limiting the traffic speed
along the highway. This step is known as the rate limiting step.
Catalysts usually target this step.
Physical Chemistry EPM/10
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