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Consider the elementary reaction; A → B
nA0 nB0
Defining the extent of the reaction as ξ .
Elementary Reaction Kinetics
Chapter 35
ni = ni0 ±ν iξ
ksi
ni = number of moles of i
0
For A, nA = nA − ξ
Equilibrium reached
0
For B, nB = nB + ξ
ν i stoichimetric coefficients - associated with a sign
ν i negative for reactants
ν i positive for products
For a general reaction such as;
Example:
V = reaction volume
The unique reaction rate R, can be expressed with respect to
any component i.
R=
1 d [i ]
ν i dt
For the generalized reaction;
R=−
1 d [ NO2 ]
1 d [O2 ] 1 d [ N 2O5 ]
=−
=
4 dt
2 dt
2
dt
V = reaction volume, constant
The exponents in the rate law of reactions and the
stoichiometric coefficients are unrelated in general;
the rate law is an experimentally determined relationship.
Reaction rate law is given by;
where,
α = order w.r.t. A
β = order w.r.t. B
.
.
(α + β + ...) = overall order of the reaction
k = intrinsic reaction rate.
For non elementary rxn. the order w.r.t. reactant ≠ stoichiometric coefficient.
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Elementary reaction: One step reaction
Slope (reactant)
Rate for the
elementary reaction:
M s-1
R=
-ve
d [ B]
dt
Rate R, changes with time.
+
Overall reaction order and rate-constant units
M s-1
The sum of the individual orders gives the overall reaction
order.
v
For the unit of rate to come out to be M/s, the units of the rate
constant for third-order reactions must be M−2⋅s−1 since
M/s = (M−2⋅s−1) (M3)
v
For a second-order reaction, the rate constant has units of
M−1⋅s−1 because M/s = (M−1⋅s−1) (M2).
In a first-order reaction, the rate constant has the units s−1
because M/s = (s−1) (M1).
Elementary (single step)
reactions
For elementary reactions, the order w.r.t. reactant = stoichiometric coefficient.
Each exponent (α, β,...) of the overall reaction rate law is
determined and is followed by substitution to the rate equation
⇒ rate constant.
α = order w.r.t. A
β = order w.r.t. B
(α + β + ...) = overall reaction order
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Determining orders –Isolation Method:
For a one step reaction:
Rate Law for the step reaction is of
the form:
If [A]>>0, [A] variation negligible during reaction,
[A] ≅ constant
Rate Law reduces to:
Make runs varying one [reactant] but keeping the others in
excess.
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Plot the appropriate plot. What is it?
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Determining orders –Initial rates Method:
Determining orders –Initial rates Method:
Make runs varying one [reactant] but keeping the others
constant.
Make two runs varying one [reactant] keeping the others in
excess. Generate the appropriate plot;
Find initial reaction rate R for each run;
[B]0 constant
[B]0 constant
Reaction Mechanisms:
A collection of elementary (one step) reactions that would lead
reactants to products. The number of molecules (particles)
involved in the elementary reaction is termed as the
molecularity of that elementary (single) step.
For a given reaction one could propose more than
one mechanism.
Usually the simplest mechanism which is consistent with the
experimental observations (rate law) is accepted correct,
until it is disproven by experimentation.
Examples of elementary (single step) reactions;
k1
A 
→ I1
1 unimolecular
k2
I1 
→ I2
k3
I 2 
→P
2 bimolecular
Elementary Reactions and their Rate Laws.
Elementary reactions are one step reactions
involving one, two or three molecular species
where all species collide (molecularity) to generate
the products of that ‘step’.
The initial step of determining the rate law is to follow the
reactant(s) concentration over time.
That would be followed by, assuming a rate law (i.e. assuming
the orders w.r.t. the reactants) and attempting to fit the
experimental data to the integrated rate law.
First order elementary reaction:
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Upon integration with limiting conditions of [A]0 at t=0 and
[A] at t=t.
First order elementary reaction:
Molecularity = 1
Differential
expression for rate
⇒
Upon integration with limiting conditions of [A]0 at t=0 and
[A] at t=t.
[ A]
= e− kt
[ A]0
From the product’s ‘view’;
[ A]
= e− kt ; [ A] = [ A]0 e− kt
[ A]0
Linearization
Exponential rise of [P]
i.e.
[A]
ln
= -kt
[A]0
Linear plots are easier to prove the assumption and extract
parameters such as k.
If [P] is followed and a first order reaction w.r.t. A is
assumed, what would be your plot?.
Second order elementary reaction Type I:
Half life – for first order reactions
Molecularity = 2
Time t1/2 for [A]0 to reduce to [A]0/2.
ln
[ P]
= 1 − e − kt
[ A]0
[A]
= -kt
[A]0
Independent of [A]0
Differential
expression for rate
define keff = 2k
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Upon integration with limiting conditions of [A]0 at t=0 and
[A] at t=t.
Second order elementary reaction Type II:
Molecularity = 2
[A]0
= 1+ [A]0 kt
[A]
and
[A]
1
=
[A]0 1+[A]0 kt
The relationships between concentrations are:
Component concentrations
at time t,
;conservation of matter
Differential
expression for rate
d [ A]
= − k[ A][ B ] = − k[ A]( ∆ + [ A])
dt
[ A]
d [ A]
∫[ A]0 [ A](∆ + [ A]) = −kdt
using the general solution;
Consecutive first order elementary reactions:
Differential rate
equation A, loss only
Differential rate
equation I,
gain and loss
⇒
What would be the integrated equation for this case if [A]0 = [B]0?
For I;
Solves as:
All concentrations are related
as;
⇒
Differential rate
equation P,
gain only
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Concentration profiles of A, I and P for different k values.
Concentration profiles of A, I and P for different k values.
Concentration profiles of A, I and P for different k values.
[I]max and time to reach [I]max
⇓
Regardless of the [A]0 value of tmax is a constant.
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Overall rate determining step:
Depends on the relative values of rate constants. Smaller
k generally controls the overall reaction rate.
Smaller k generally controls the rate.
How small should it be?
~ 1st order decay of I
kA>>kI
rds = step 2
i.
kA>>kI
1
ii.
For kA > 20 kI
rls approximation
valid.
1st order decay of I
rls =rds
rate limiting or
rate determining
step
kI>>kA
1
1st order decay of A
Steady state approximation:
~ 1st order decay of A
kI >> kA
rds = step 1
Differential rate Equations:
For kI > 25 kA
rls approximation
valid.
consecutive.reactions.xls
Steady state approximation (SSA) - visualization:
Applying SSA to the intermediates:
[I] very small and do not change
much. Therefore approximated
as,
and
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For P (i.e. [P]ss = [P] assuming SSA)
Validity of SSA
Using
At SS,
~ 1st order decay of A !!
d [ I ]ss
=0
dt
k 2 [ A] e− kAt
− A 0
=0
k1
i.e. k1 >> k A2 [ A]0 e− k At
small k A makes, e− k At → 1
then k1 >> k A2 [ A]0
Parallel Reactions:
kA (0.02/s) < k1(0.2/s) = k2
SSA predicts a considerably
different concentration
profile!! if kA is not very low,
as is here.
Differential rate equations:
Solves as:
d [ B]
= k B [ A] = k B [ A]0 e− ( kB + kC )t
dt
[B]
∫
0
[ B]
d[ B] =
∫k
B
[ A]0 e −( kB + kC )t dt
k B > kC
0
Also;
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Product (fractional) yield, parallel reactions:
Yield of component i;
where,
assume kB =2kC
For the parallel reaction;
e.g.
Temperature dependence of k:
k
Much faster rise of k than √T.
Reaction progress: Energy diagram
Part of Ea – energy required to
overcome repulsive forces among
electron clouds of reacting
molecules.
High T
Ea
Reaction at Equilibrium
One step elementary reactions in both directions
>Ea
<Ea
Low T
Reaction coordinate
diagram
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;substituting for [B].
Differential rate equations:
⇓
;conservation of matter
and
Once the equilibrium
is
reached at t >>0;
K eq =
[ B ]eq
[ A]eq
=
kA
kB
Division:
KC is dependent only on the ki values (ratio of, in this case).
apparent decay constant = (kA + kB)
Single step reactions
k
Energy hump to overcome as the reaction moves
toward products.
Involves bringing AB and C closer together (collision);
breaking A-B and making B-C bonds and separation of A
and BC molecular entities.
i.e. energy demand followed by an energy loss.
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Energetics of elementary reactions:
Transition state.
AB + C
→ ←
A +
←
BC
→
The transition state (‡) of a chemical reaction is a precise atomic
configuration along the reaction coordinate, located at the
highest potential energy position along this reaction coordinate.
Transition state has an equal probability of forming either the
reactants or the products of the given reaction.
The activated complex differs from the transition state.
The activated complex refers to all the configurations that the
atoms acquire in the transformation to products. The activated
complex refers to any point along the energy profile along
reaction coordinate of a reaction in progress.
Involves breaking
A-B and making B-C
bonds.
‡
The reaction path follows an energy profile, but it is not
necessarily a simple linear one.
Consider the bond formation of AB and BC.
B +C
→ ←
Potential energy surface
A + B
→ ←
Morse curve.
In reactions - Formation of bonds and break up of bonds occur
at the same time on way to and away from the transition state.
Energy plots therefore cannot be 2 dimensional but are three
dimensional. Morse curve.
Position of the transition state (‡) on the PE surface –
saddle point ( ).
Saddle point
Stationary point on the PE surface.
Minimum on blue curve.
Maximum on the red curve.
Blue and red curves are in
orthogonal planes.
Reaction coordinate
= lowest energy path.
Reaction coordinate is a ‘one-dimensional coordinate’.
It represents the reaction progress along a reaction pathway.
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Contour plot showing equi-potential heights.
Reaction coordinate d → c = lowest energy
path.
Reaction coordinate diagram.
Activated Complex (Transition State ) Theory;
elementary bimolecular reaction.
k1
Assumptions:
k2
k−1
1.
K≠ =
[ AB ≠ ]
[ A][ B ]
;from thermodynamics
2.
Therefore the reaction rate R;
For A;
d [ A]
= − k1[ A][ B] + k−1[ AB ≠ ]
dt
k-1
k2
Assumption 1 ⇒
One of two bonds can break (κ ), one leading to the product,
other back to the reactants κ = transmission coefficient (≈ 1);
probability of the ‘complex’ formed dissociating into products.
For P;
‘K’ dimensionless.
ν vibrational frequency of the complex of breaking bond –
no restoration.
Frequency of vibration = rate of breaking one bond.
R = k[A][B]
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R = k[A][B]
Using thermodynamic functions to replace the partition function
based ν K c ≠, where ∆G‡, ∆H‡ and ∆S‡ are ‘quantities’ of
activation.
Arrhenius’ Ea
Pre-exponent = Arrhenius’ A
we get
i.e.
Besides the energy requirement (T) to overcome the activation
energy (Ea), the proper orientation of reactants (entropy, ∆S‡ ),
the phase of molecular vibrations (transmission factor) are the
factors determining the intrinsic reaction rates (k) of elementary
reactions.
Diffusion Controlled Reactions:
Gas phase – reactants collide with least
hindrance with the molecules present
in the reaction mixture.
Expressions for A and Ea depend on the molecularity
and the physical phase of the reaction.
Condensed phase – ‘reactants’
collide with solvent molecules
and therefore has a lesser
chance to collide with the
reactant. ‘Reactants’ diffuses
thro’ the solvent.
In both phases the average molecular
velocities are the same.
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Intermediate can
Reactants collide
after diffusion forming revert to reactants.
an intermediate AB.
Intermediate can
generate products.
kd
i.e.
Reaction rate:
kr
Applying SSA to the intermediate AB;
Substituting for [AB]
Also kinetic theory gives;
where DAB = DA + DB ; diffusion coefficients of A and B
for limiting
cases
kp>>kr
Diffusion controlled
limit (reaction).
kp<<kr
Activation controlled
limit (reaction).
and
Diffusion controlled reactions are dependent on the viscosity
of the medium.
Ionic reactions are faster in solution because of the additional
attractive impetus due to Coulombic attractions
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