gregory yablonsky1, denis constales , daniel branco pinto ... · experimental verification of...
TRANSCRIPT
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Introduction1
Gregory Yablonsky1, Denis Constales2, Daniel Branco Pinto2,
Vladimir Galvita3, Eugene Redekop4 and Guy B. Marin3
1 Parks College of Engineering, Aviation and Technology, Saint Louis University, 3450 Lindell Blvd.,
Saint Louis, MO 63103, USA; E-Mail: [email protected] Department of Mathematical Analysis, Gent University, Galglaan 2, Gent B-9000, Belgium3 Laboratory for Chemical Technology, Ghent University, Technologiepark 914, Gent B-9052, Belgium
4 Department of Chemistry, University of Oslo, Postbox 1033, Blindern,
N - 0315 Oslo, Norway
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Introduction2
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Introduction3
• Joint Kinetics is a New Kinetic
Strategy for Chemical Kinetics and
Heterogeneous Catalysis
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Introduction4
• TRENDS AND EVENTS
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Introduction5
• TRENDS
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Introduction6
Types of Temporal Evolution Relaxation
c
t
c
t
c
t
slowintermediate
fast
Simple exponential relaxation Relaxation with induction period
Relaxation of different components at different time scales
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Introduction7
Types of Temporal Evolution Relaxation
c
t
3
2
1
Relaxation with “overshoots” (1) & (3) and start in
“wrong” direction (2)
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Introduction8
8
Types of Temporal Evolution Relaxation
c
t
I
II
c
t
Relaxation with different steady states
Damped oscillations
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Introduction9
Types of Temporal Evolution Relaxation
c
t
c
t
Regular oscillations
around a steady state
Chaotic oscillations
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Introduction10
Coherency in Trends
• Stoichiometry
• Symmetrical Invariants etc
G.B. Marin & G.S. Yablonsky (2011). Kinetics of Chemical Reactions. Decoding Complexity
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Steady-State Chemical Kinetics: A Primer
EVENTS
• Extremum (Maximum, Minimum)
• Turning Point
• Extinction
• Ignition
• Intersection of kinetic dependences
• Coincidence of events
etc….
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Steady-State Chemical Kinetics: A Primer
CATEGORIZATION of
EVENTS
• “ALWAYS” (Unavoidable)
• “NEVER”
• “SOMETIMES” (Avoidable)
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Steady-State Chemical Kinetics: A Primer
• SIMPLE is COMPLEX
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Steady-State Chemical Kinetics: A Primer
What is Joint Kinetics?
It is defined as the “trends-events’
analysis , i.e. the analysis of special
combinations of kinetic dependences and
events (intersections, coincidences etc).
A map of kinetic events is constructed
and analyzed.
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Steady-State Chemical Kinetics: A Primer
MAIN TOPICS
I. Symmetrical Relations
II. Intersections and Coincidences
III. Kinetic and Thermodynamic Control
IV. Momentary Equilibrium
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I.
Joint Kinetics:
Symmetric Relations
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General Contemporary Dogma
It is impossible to predict the temporal
evolution of a reacting chemical system
based on its description under
equilibrium conditions
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New Result: Reversible Linear Reactions
Batch reactor, 𝐴
𝑘1
𝑘−1
𝐵
• 𝐴𝐴 𝑡 , 𝐵𝐴 𝑡 from (1,0)
• 𝐴𝐵 𝑡 , 𝐵𝐵 𝑡 from 0,1
Remarkably, 𝐵𝐴(𝑡)
𝐴𝐵(𝑡)=
𝑘1
𝑘−1= 𝐾𝑒𝑞 is constant!
ℒ𝐵𝐴 𝑠 =𝑘1
𝑠2 + 𝑘1 + 𝑘−1 𝑠ℒ𝐴𝐵 𝑠 =
𝑘−1𝑠2 + 𝑘1 + 𝑘−1 𝑠
Yablonsky, G.S., Constales, D., Marin, G.B. Equilibrium relationships for non-
equilibrium chemical dependencies. Chem. Eng. Sci. 66 (1) 111-114 (2011).
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Laplace domain solution
The ratio of B from A to A from B is a constant,
in fact the equilibrium constant of the first
reaction step:
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Reversible consecutive reactions
Next case: batch reactor, 𝐴
𝑘1
𝑘−1
𝐵՜𝑘2𝐶
• 𝐴𝐴 𝑡 , 𝐵𝐴 𝑡 , 𝐶𝐴 𝑡 from (1,0,0)
• 𝐴𝐵 𝑡 , 𝐵𝐵 𝑡 , 𝐶𝐵 𝑡 from (0,1,0)
𝑡𝐵𝐴max ≡ 𝑡𝐴𝐵max?
Actually, 𝐵𝐴(𝑡)
𝐴𝐵(𝑡)=
𝑘1
𝑘−1= 𝐾𝑒𝑞 is constant!
ℒ𝐵𝐴 𝑠 =𝑘1
𝑠2 + 𝑘1 + 𝑘−1 + 𝑘2 𝑠 + 𝑘1𝑘2ℒ𝐴𝐵 𝑠 =
𝑘−1𝑠2 + 𝑘1 + 𝑘−1 + 𝑘2 𝑠 + 𝑘1𝑘2
Yablonsky, G.S., Constales, D., Marin, G.B. Equilibrium relationships for non-equilibrium chemical dependencies. Chem. Eng. Sci. 66 (1) 111-114 (2011).
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A B C : dual experiments
We explore the behavior of CA(t), CB(t), and
CC(t) relative to each other from two
symmetrical initial conditions:
1) CA,0 = 1, CB,0 = 0; and
2) CA,0 = 0, CB,0 = 1.
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Rigorous proof
• For linear or linearized kinetics with
microreversibility, dx/dt = Kx, the kinetic opretor
K is symmetric in the entropic product. This form
of Onsager reciprocal relations implies that that
the shift in time, exp (Kt), is also a symmetric
operator. This generates the reciprocity relations
between the kinetic curves.
• Yablonsky, Gorban, Constales, Galvita, Marin,”
Reciprocal relations between kinetic curves”,
EPL, 93(2011), 20004
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• This time invariance is a special case of a
general result (Onsager reciprocity) for linear
systems and some non-linear systems
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General time-invariances
• We have also extended it to the TAP
(Temporal Analysis of Products) reactor.
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Thin-Zone (TZ) Temporal Analysis of Products (TAP)
25
timeQMStime
Inlet pulse Exit flow
pulse valve
catalyst
inert
thermocouple
Fexit(t)Finlet(t)
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Thin-zone and Single Particle Reactor
Configurations
Thin-zone
Single-particle
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Inert zone Catalyst zone
Thin-Zone TAP -Reactor (TZTR) Idea
Dimensionless Axial Coordinate
Dim
ensi
on
less
Gas
Con
cen
trati
on
0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.00.00
.25
.50
.75
1.00
1.25
1.50
1.75
2.00
Vacuum
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• It was experimentally proven in TAP-
studies of the Water-Gas- Shift reaction
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General time-invariances
TAP- experiments. Water-Gas-Shift
Two experiments:
1. CO2 is pulsed. CO(t) is measured
2. CO is pulsed. CO2 (t) is measured
Ratio CO2(t) / CO (t) = const
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General time-invariances
• We have also extended it to the TAP
(Temporal Analysis of Products) reactor.
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Experimental Verification of Onsager Reciprocal Relations in
Chemistry
The reciprocal relations were tested experimentally for many
systems. In 1960, D. G. Miller wrote a remarkable review on
experimental verification of the Onsager reciprocal relations
which is often referred to even now. Analyzing many different
cases of irreversible phenomena (thermoelectricity,
electrokinetics, isothermal diffusion, etc), Miller found that these
reciprocal relations are valid. However, regarding the chemical
reactions, Miller’s point was : “The experimental studies of this
phenomenon ...have been inconclusive, and the question is still
open from an experimental point”
( Miller D. G., Chem. Rev., 60 (1960) 15.)
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• A New Understanding:
Knowing the equilibrium composition (or
equilibrium constants) and kinetic
dependences experimentally measured from
some initial conditions, one can determine
kinetic dependences from other initial
conditions.
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Questions
• (1) How many kinetic dependences do we
have to determine in addition to knowing
the equilibrium constants for this purpose?
• (3) What is the procedure for determining
the unknown kinetic dependences
• based on the known ones
• (3) Are all these procedures successful or
not?
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Answer:
For n-linear chemical system, in addition to
the equilibrium composition we have to
know N x (N-1) /2 kinetic dependences.
E.g., for the classical 3-step Wei-Prater
isomerization mechanism, we have to know
(3 x 2) / 2 = 3 kinetic dependences
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General Dogma
A reminder on the general dogma
It is impossible to predict the temporal
evolution of a reacting chemical system
based on its description under
equilibrium conditions
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Revisiting the dogma
NEW UNDERSTANDING
In some cases, knowing the thermodynamic
characteristics and kinetic dependencies
which start from some initial conditions, it is
possible to predict kinetic behavior from
other initial conditions.
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• Goal: develop dual kinetic experiments for
invariant quantities in electrochemical
reactions:
DUAL KINETIC
CHRONOAMPEROMETRY
(Kiss, 2014-2015)
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Dual Kinetic Chronoamperometry
Consider first order reversible electrochemical reaction on a rotating disk electrode [7]
A+ + e- B
dcA
dt=
2
a-k
f(V )c
A+ k
r(V )c
B( ) +2D
A
a2c
A
0 - cA( )
dcB
dt=
2
ak
f(V )c
A- k
r(V )c
B( ) +2D
B
a2c
B
0 - cB( )
a: Nernst diffusion layer thickness
cA and cB: near surface concentrations of A and
B
cA0 and cB
0: bulk concentration of A and B
kf(V) and kr(V): potential (V) dependent
forward and reverse first-order rate constant
DA, DB: diffusion constants of A and B.
kf0 and kr
0: rate constants at V=0
β: transfer coefficient
R: gas constant, T: temperature
Q: reaction quotient
kf(V ) = k
f
0 exp -(1- b )zF
RTV
é
ëê
ù
ûú
kr(V ) = k
r
0 exp bzF
RTV
é
ëê
ù
ûú
Thermodynamic equilibrium (Nernst potential):
E = E0-
RT
zFlnQ = E
0-
RT
zFln
cB
0
cA
0
Current
i(t) = zF -k
f(V)c
A(t) + k
r(V)c
B(t)é
ëùû
(2)
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Methodology• Thermodynamic equilibrium: The electrode potential is set to V=E
and
limt®¥
i(t) = 0, limt®¥
cA(t) = c
A
0 , limt®¥
cB(t) = c
B
0
• We shall seek invariant quantities in the ‘relaxation’ process to this equilibrium state defined by reaction quotient at the equilibrium state Q = cB
0/cA0
• The experiments shall involve initial conditions with only one species present. To achieve this, we use pre-polarization experiment:
– ‘Scenario 1’: the electrode potential is set to large negative value in the mass-transfer limited region where all A is reduced to B, therefore, cA(t=0)=0, cB(t=0)=cB
0+cA0
– ‘Scenario 2’: the electrode potential is set to large positive value in the mass-transfer limited region where all B is oxidized to A, therefore, cB(t=0)=0, cA(t=0)=cB
0+cA0
• We shall measure the concentration of A in ‘Scenario 1’ and the concentration of B in ‘Scenario 2’.
• Hypothesis: for simple electrochemical reaction studied here, the ratio of near surface concentrations cB/cA at any time t will be an invariant quantity determined by the bulk concentrations cB
0/cA0
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Dual Kinetic ChronoamperometryModel simulations
• Equations 2 were numerically integrated.
• cA0 =0.1 M, cB
0=0.2 M, Q=2 a=0.01 cm, z=1, T=298 K, DA =DB=10-5 cm2/s, β=0.5kf
0 = 0.01 cm/s, and kr0=0.001 cm/s
• With these parameters the equilibrium: V=E=0.041 V, cA =0.1 M, cB=0.2 M
‘Scenario 1’
V=-0.5 V for t< 0 s : cathodic limit
V=0.041 V for t≥ 0 s : Nernst potential
‘Scenario 2’
V=0.5 V for t< 0 s : anodic limit
V=0.041 V for t≥ 0 s : Nernst potential
Time (s) Time (s)
c(t)
c(t)
Dual kinetic analysis
cA(t)cB(t)
cB(t)/cA(
t)
The ratio of concentrations
from dual kinetic experiments
with symmetrical initial
conditions is invariant quantity
= Q: hypothesis is confirmed
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Dual Kinetic ChronoamperometryExperiments
• Using ferrocyanide/ferricyanide system (A: ferricyanide, B: ferrocyanide)
• Applied potential is used to control the state of the system– Potential for the equilibrium state is open circuit potential (current
=0) given by the Nernst equation based on the initial concentrations of the species
– The symmetrical initial conditions were obtained with pre-polarization at the mass transfer limited regions in the anodic and cathodic direction.
– The concentrations of ferrocyanide and ferricyanide were determined by shielding experiments with a ring electrode: ring-disk experimentsDisk: reaction takes placeRing: measures concentrations
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Equipment
Top
view
Pt
disk
Au
ring
Water
bathWater
bath
Nitrogen gas
feed
Hg/Hg2SO4/sat. K2SO4
reference electrode
Pt counter
electrode
Rotating disk
instrument
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Equilibrium and limiting potentials
• Equilibrium potential (OCP) at current = 0
• Limiting potentials were found by linear sweep voltametry (LSV)– Start from equilibrium
potential (open circuit potential, OCP) and scan potential until current no longer increases
– Scan in both anodic and cathodic directions
Mass transfer limiting
regions
Cathodic limiting
region
Anodic limiting
region
OC
P
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Calculating concentration from ring current
• Shielding experiments: the change in ring current is proportional to concentration [7]
• For ferrocyanide: the ring is set to anodic potential of V=0.5 V. – When disk is at OCP, there is no shielding, therefore, the ring
current ir,1 corresponds to a near surface concentration that equals the bulk concentration:[cB
0, ir,1]
– When disk is at anodic overpotential at mass transfer limit: all ferrocyanide is oxidized, the ring current ir,2 is fully shielded, therefore, the near surface concentration is zero [0, ir,2]
– We used linear interpolation from above data for calculation of concentrations from the ring currents during the chronoamperometric experiment on the disk
• For ferrycianide concentration analogous technique was used but with cathodic potential of V=-0.8 V.
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Chronoamperometry
• During experiment disk and ring currents are measured at a data acquisition rate of 1 kHz.
• Dual kinetic measurement:
• ‘Scenario 1’: The disk current was cathodically pre-polarized at V=-0.8 V. At t=0 the disk potential was quickly switched to OCP. For ferro/ferricyanideconcentration the ring is polarized anodically(V=0.5 V)/cathodically (V=-0.8 V) for the entire experiment.
• ‘Scenario 2’:The disk current was anodically pre-polarized at V=0.5 V. At t=0 the disk potential was quickly switched to OCP. For ferro/ferricyanideconcentration the ring is polarized anodically(V=0.5 V)/cathodically (V=-0.8 V) for the entire experiment.
• c(ferrocyanide)0= 0.02 Mc(ferricyanide)0= 0.01 M
Initial
ring
current
value
Final
ring
current
value
____ring
current
____disk
current
‘Scenario 2’, with anodic
polarization of the ring current for
measurement of ferrocyanide
concenration
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Concentrations:Experimental Results
___Fe(II)
___Fe(II)
c (M)
Time
(s)
Time
(s)
c(Fe(II))
/c(Fe(III))
Time
(s)
Scenario 1 Scenario 2
c (M)
Dual kinetic analysis
c(Fe(II))c(Fe(III))
The ratio of concentrations
from dual kinetic experiments
with symmetrical initial
conditions is approximately
invariant quantity ≈ Q=2:
hypothesis is experimentally
confirmed
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Conclusions• Reciprocal relations between the kinetic curves provide a unique possibility to extract
the non-steady state trajectory starting from one initial condition based only on the
equilibrium constant and the trajectory which starts from the symmetrical initial
condition.
• Dual kinetic chronoamperometry is proposed as a novel technique for exploration
of kinetic features of electrochemical reactions
• Kinetic information is extracted from two experiments: each experiment consisted of
setting the the disk electrode to an equivalent far-from-equilibrium potential, such as
the anodic or cathodic limit, and allowing each to relax to equilibrium defined by the
Nernst potential.
• Numerical simulations indicate that the proper ratio of the transient kinetic curves
obtained from cathodic and anodic mass transfer limited regions give thermodynamic
time invariances related to the reaction quotient of the bulk concentrations.
• Experimental tests with the ferrocyanide/ferricyanide system further confirm the
principle: the concentrations of the oxidized and reduced species followed reciprocal
paths as they relaxed toward equilibrium as long as both started from an equivalent
state.
• Simplifying principles can exists in far-from-equilibrium chemical systems
• The results could impact (bio)fuel cell, sensor, and battery technology by predicting
the concentrations and currents of the underlying non-steady state processes in a
wide domain from thermodynamic principles and limited kinetic information.
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References
[1] N. G. van Kampen, Nonlinear irreversible processes, Physica
67 (1), 1–22 (1973).
[2] H. Stockel, Linear and Nonlinear Generalizations of Onsager’s
Reciprocity Relations. Treatment of an Example of Chemical
Reaction Kinetics, Fortschritte der
Physik/Progress of Physics 31, 165–184 (1983).
[3] M. Ozera, I. Provaznik, J. Theor. Biol. 233, 237–243 (2005).
[4] G. S. Yablonsky et al. EPL 93, 20004 (2011).
[5] G. S. Yablonsky, D. Constales, G. B. Marin, Chem. Eng. Sci.
66, 111–114 (2011).
[6] D. Constales et al., Chem. Eng. Sci. 66, 4683 (2011).
[7] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New
York, 1980.
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II. EVENTS
INTERSECTIONS
COINCIDENCES
49
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• Concepts of Events
What are Events?
Extrema, Intersections, Coincidences,
Turning Points etc…
Map of Events.
Concepts of Ensemble of Experiments:
Experiments with different Initial
conditions
50
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A. Coincidences
• Surprising properties of the simple kinetic
models; in particular, A->B->C.
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Solutions (known before)
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Intersections
Depending on the parameter values and initial
conditions, transient concentration curves of
species A and B may intersect once or not
intersect at all. The concentration transients of
species C always intersect the concentration
transients of A and B.
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Coincidences (cont’d)
• A simple problem is posed: what do we
know about the points of intersection, the
maximum point of CB(t), and their
ordering?
Example: k1=k2
We call it Euler point.
k1 = k2 = 1 s-1
tB,max = 1 s
CA,intersect=CB,intersect=1/e
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Coincidences (cont’d)
• Nonlinear problem, even for a linear
system.
• Many analytical results can be obtained.
• Of 612 possible arrangements, only six
can actually occur.
• We introduce separation points for
domains
• A(cme), G(olden), E(uler),
L(ambert),O(sculation), T(riad) points.
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Coincidences (cont’d)
• Acme, k2=k1/2
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Coincidences (cont’d)
• Lambert, k2=1.1739… k1
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Consecutive reactions
Linear kinetics in batch reactor, 𝐴՜𝑘1𝐵՜𝑘2𝐶
Yablonsky, G.S., Constales, D., Marin, G.B. Coincidences in chemical kinetics:
Surprising news about simple reactions. Chem. Eng. Sci. 65 (23) 6065-6076
(2010).
58
(𝑡𝐵,max , 𝑣𝐵,max)
(𝑡𝐵=𝐶 , 𝑣𝐵=𝐶)
(𝑡𝐴=𝐶 , 𝑣𝐴=𝐶)
(𝑡𝐴=𝐵,𝑣𝐴=𝐵)
𝑡𝐴=𝐵 < 𝑡𝐴=𝐶 < 𝑡𝐵,max < 𝑡𝐵=𝐶
𝑣𝐴=𝐶 < 𝑣𝐴=𝐵 < 𝑣𝐵=𝐶 < 𝑣𝐵,max
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Coincidences (cont’d)
Inspecting the peculiarities of the
experimental data, we may immediately infer
the domain of the parameters.
Intersections, extrema and their ordering are
an important source of as yet unexploited
information.
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Consecutive reactions with one reversible step
Batch reactor, 𝐴
𝑘1
𝑘−1
𝐵՜𝑘2𝐶
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Coincidences
The intersections between concentration transients can be traced
systematically and represented in parameter space: e.g. we show here
the comparison between the intersection times of A from A with B
from A (t1) and that of A from B with B from B (t2). The rate constants
are used as barycentric coordinates.
The coincidence proper (t1=t2) occurs on the curved
line separating the yellow and blue domains.
() t1< t2, () t1> t2,
() only t1 exists,
() only t2 exists,
() no intersections
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Parametric subdomains
Combining all intersections in time and value an intricate map is obtained…
where each different patch is a qualitatively separate subdomain:
It is similar to abstract compositions
by Felix De Boeck (1898-1995):
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• SIMPLE IS COMPLEX
63
AIChE Annual Meeting, October 2012, Danckwerts Memorial Lecture
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III.
Thermodynamic and Kinetic Control:
Switching Point
64
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
“while the endo isomer is formed more
rapidly, longer reaction times, as well
as relatively elevated temperatures,
result in higher exo/endo ratios. These
facts must be considered in the light of
the remarkable stability of the exo-
compound on the one hand, and the
very facile dissociation of the endo
isomer on the other”
* First mention in 1944, reaction of
fulvene with maleic anhydride:
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
Typical reaction in basic chemistry textbooks:
hydrohalogenation of 1,3-butadiene
66
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
67
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Laboratory for Chemical Technology, Ghent University
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l
68
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Laboratory for Chemical Technology, Ghent University
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application
From: Caravaca M., Sanchez-Andrada P., Soto A., Alajarin M., Phys. Chem.
Chem. Phys., 16 (2014) 25409. 69
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
crossing point
and
switching point
T.HERMODYN,PROdUCT.
70
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
KIN. PRODUCT
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
dA(t)
dt= − k1 + k2 A t + k3K t + k4T(t)
)dK(t
dt= k1A t − k3K t
)dT(t
dt= k2A t − k4T t
mathematical resolution
71
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
mathematical resolution
72
A(t) = Aeq + Ao − Aeq − Ax e−λp t + Axe−λm t
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
dA(t)
dt= − k1 + k2 A t + k3K t + k4T(t)
)dK(t
dt= k1A t − k3K t
)dT(t
dt= k2A t − k4T t
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
kinetic vs thermodynamic control
mathematical resolution
73
A(t) = Aeq + Ao − Aeq − Ax e−λp t + Axe−λm t
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
dA(t)
dt= − k1 + k2 A t + k3K t + k4T(t)
)dK(t
dt= k1A t − k3K t
)dT(t
dt= k2A t − k4T t
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Laboratory for Chemical Technology, Ghent University
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the switching point ts
equality of rates of formation
74
ቤ)dK(t
dtt=ts
= ቤ)dT(t
dtt=ts
K(t) ---
T(t) ---
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
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Laboratory for Chemical Technology, Ghent University
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the switching point ts
equality of rates of formation
75
K(t) ---
T(t) ---
K’(t) ―
T’(t) ―
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇ቤ)dK(t
dtt=ts
= ቤ)dT(t
dtt=ts
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
the switching point ts
equality of rates of formation
76
ts =
logλp − Δ
λm − Δ
λp − λm
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇ቤ)dK(t
dtt=ts
= ቤ)dT(t
dtt=ts
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the switching point ts
equality of rates of formation
77
where ∆= λpλmKeq − Ko − Teq − To
Aok1 − Kok3 − Aok2 − Tok4ts =
logλp − Δ
λm − Δ
λp − λm
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇ቤ)dK(t
dtt=ts
= ቤ)dT(t
dtt=ts
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Laboratory for Chemical Technology, Ghent University
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the switching point ts
78
wherets =
logλp − Δ
λm − Δ
λp − λm
if Ko = To = 0 ∆=λpλm
Ao
Keq − Teq
k1 − k2
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
∆= λpλmKeq − Ko − Teq − To
Aok1 − Kok3 − Aok2 − Tok4
ቤ)dK(t
dtt=ts
= ቤ)dT(t
dtt=ts
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
79
initial concentration of the products
𝐴𝑘1⇄𝑘3
𝐾 𝐴𝑘2⇄𝑘4
𝑇
ts =
logλp − Δ
λm − Δ
λp − λm ∆= λpλm𝑝𝑟𝑜𝑑𝐾 − 𝑝𝑟𝑜𝑑𝑇
𝑟𝑎𝑡𝑒𝐾(𝑡=0) − 𝑟𝑎𝑡𝑒𝑇(𝑡=0)
𝑝𝑟𝑜𝑑𝐾 < 𝑝𝑟𝑜𝑑𝑇
𝑟𝑎𝑡𝑒𝐾(𝑡=0) > 𝑟𝑎𝑡𝑒𝑇(𝑡=0)
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
conclusions
80
* The distinction between the kinetic and the thermodynamic control regimes should be made
according to the rate of formation of the products. The kinetic control regime extends from the
beginning of the reaction until the switching point; under this regime, the product with the largest
rate of formation is the kinetic product. After the switching point, the thermodynamic control
regime is settled, and now the thermodynamic product has the largest rate of formation.
* The value of the switching time ranges between zero and a maximum value
logλpλm
λp − λm
the switching time, occurs the maximum of concentration of the kinetic product, and eventually, the
;
after
crossing of the concentration profiles of the competing products.
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• III. Momentary Equlibrium
81
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Thin-Zone (TZ) Temporal Analysis of Products (TAP)
82
timeQMStime
Inlet pulse Exit flow
pulse valve
catalyst
inert
thermocouple
Fexit(t)Finlet(t)
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Thin-zone and Single Particle Reactor
Configurations
Thin-zone
Single-particle
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Inert zone Catalyst zone
Thin-Zone TAP -Reactor (TZTR) Idea
Dimensionless Axial Coordinate
Dim
ensi
on
less
Gas
Con
cen
trati
on
0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.00.00
.25
.50
.75
1.00
1.25
1.50
1.75
2.00
Vacuum
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Introduction
85
• Adsorption equilibrium is an essential step of catalytic processes,
which is often used to measure the total concentration of catalytic sites.
• The total concentration of sites
is typically determined by:
- Very sensitive pressure and microbalance
measurements in equilibrated closed system.
- Chemisorption of irreversibly adsorbing molecules.
- Chemisorption of reversibly adsorbing molecules
at low temperatures (much less than operating).
pi
Cs
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Introduction
86
• Adsorption equilibrium is an essential step of catalytic processes,
which is often used to measure the total concentration of catalytic sites.
• The total concentration of sites
is typically determined by:
- Very sensitive pressure and microbalance
measurements in equilibrated closed system.
- Chemisorption of irreversibly adsorbing molecules.
- Chemisorption of reversibly adsorbing molecules
at low temperatures (much less than operating).
• Measuring the concentration of adsorption sites by
reversible chemisorption under realistic operating temperatures and
non-steady-state conditions presents a significant challenge.
pi
Cs
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Outline
87
•
• TAP experiments and data analysis
• Momentary Equilibrium
• Pulse-Intensity Modulation
• Experimental example
• Conclusions
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Outline
88
• Introduction
• TAP experiments and data analysis
• Momentary Equilibrium
• Pulse-Intensity Modulation
• Experimental example
• Conclusions
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Case study
89
CO + Z ZCOk+
k-
𝐾𝑒𝑞 =𝑘+
𝑘−
Single-site CO adsorption:
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Case study
90
CO + Z ZCOk+
k-
𝐾𝑒𝑞 =𝑘+
𝑘−
𝑅𝐶𝑂 = −𝑑𝐶𝐶𝑂𝑑𝑡
= 𝑘+𝐶𝑍𝐶𝐶𝑂 − 𝑘−𝐶𝑍𝐶𝑂 =
= 𝑘+(𝐶𝑍,𝑡𝑜𝑡−𝐶𝑍𝐶𝑂)𝐶𝐶𝑂 − 𝑘−𝐶𝑍𝐶𝑂
Single-site CO adsorption:
• Kinetics is governed by
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Case study
91
CO + Z ZCOk+
k-
𝐾𝑒𝑞 =𝑘+
𝑘−
We used numerical TZ TAP experiments with realistic noise model
to elucidate kinetic behavior during CO adsorption in TAP.
𝑅𝐶𝑂 = −𝑑𝐶𝐶𝑂𝑑𝑡
= 𝑘+𝐶𝑍𝐶𝐶𝑂 − 𝑘−𝐶𝑍𝐶𝑂 =
= 𝑘+(𝐶𝑍,𝑡𝑜𝑡−𝐶𝑍𝐶𝑂)𝐶𝐶𝑂 − 𝑘−𝐶𝑍𝐶𝑂
Single-site CO adsorption:
• Kinetics is governed by
• Surface CO uptake can be obtained as
𝐶𝑍𝐶𝑂(𝑡) = න0
𝑡
𝑅𝐶𝑂 𝑡′ 𝑑𝑡′
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Intra-pulse kinetic characteristics
92
Pulse-response experiment
in TZ TAP microreactor
RC
O, (m
mol/
kg
ca
t/s)
CC
O, (m
mol/
m3)
CZ
CO
, (m
mol/
kg
ca
t)
t, (s)
RCO
CZCO…
CCO
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Momentary Equilibrium (ME)
93
Pulse-response experiment
in TZ TAP microreactor
RC
O, (m
mol/
kg
ca
t/s)
CC
O, (m
mol/
m3)
CZ
CO
, (m
mol/
kg
ca
t)
ME
t, (s)
RCO
CZCO…
CCO
𝑅𝐶𝑂 𝑡 = 0 ՜ 𝑟+ = 𝑟−
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Momentary Equilibrium (ME)
94
How do we use ME to obtain isotherms and
estimate the total concentration of sites?
Are the surface and gas concentrations in ME related
via the equilibrium constant?
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Outline
95
• Introduction
• TAP experiments and data analysis
• Momentary Equilibrium
• Pulse-Intensity Modulation
• Experimental example
• Conclusions
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Pulse-Intensity Modulation
96
𝐶𝑍𝐶𝑂,𝑀𝐸 =𝐾𝑒𝑞𝐶𝑍,𝑡𝑜𝑡𝐶𝐶𝑂,𝑀𝐸
1 + 𝐾𝑒𝑞𝐶𝐶𝑂,𝑀𝐸
Langmuir isotherm:
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Pulse-Intensity Modulation
97
𝐶𝑍𝐶𝑂,𝑀𝐸 =𝐾𝑒𝑞𝐶𝑍,𝑡𝑜𝑡𝐶𝐶𝑂,𝑀𝐸
1 + 𝐾𝑒𝑞𝐶𝐶𝑂,𝑀𝐸
Langmuir isotherm:
Fexit(t)
t, (s)
Np,CO
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Outline
98
• Introduction
• TAP experiments and data analysis
• Momentary Equilibrium
• Pulse-Intensity Modulation
• Experimental example
• Conclusions
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Experimental example
99
CO adsorption on Pt/Mg(Al)Ox (1 wt. % Pt)
Pulse-intensity range: Np,CO = 1 – 14.5nmolCO
Temperature range: T = 50 – 140 °C
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Experimental example
100
Estimated concentration of adsorption sites: CZ,tot = 0.3-0.55 (mmol
Estimated heat of CO adsorption: ΔHads= - 17.9 (kJ/molCO)
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Rate-concentration trajectories
101
t, (s)
RCO
CZCO
CCO
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Experimental example
102
CO adsorption on Pt/Mg(Al)Ox (1 wt. % Pt)
Pulse-intensity range: Np,CO = 1 – 14.5nmolCO
Temperature range: T = 50 – 140 °C
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Experimental example
103
Estimated concentration of adsorption sites: CZ,tot = 0.3-0.55 (mmol
Estimated heat of CO adsorption: ΔHads= - 17.9 (kJ/molCO)
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CONCLUSIONS
– SIMPLE IS COMPLEX !
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Acknowledgements
John Gleaves
Alexander Gorban
Vladimir Galvita
Rebecca Fushimi
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References
• References (I)
• 1. G. S. Yablonsky, D. Constales, G. Marin, “Coincidences in Chemical Kinetics:
• Surprising News about Simple Reactions”, Chem. Eng. Sci. 65(2010)2325-2332
•
• 2. G. S. Yablonsky, D. Constales, G. Marin, “Equilibrium relationships for non-
• equilibrium chemical dependences” , Chem. Eng.Sci.;66, 1,111-114(2011)
• 3. G. S. Yablonsky, A. N. Gorban, D. Constales, V. Galvita and G.B. Marin
• “Reciprocal Relations Between Kinetic Curves”, EuroPhysics Letters, EPL, 93
• (2011) 20004-20007
• 4. D. Constales, G. S. Yablonsky, V. Galvita, and G.B. Marin, “Thermodynamic time-
• invariances: theory of TAP pulse-response experiments”, Chem. Eng. Sci., 66 (2011) 4683-
• 4689
•
• 5. D. Constales, G.S. Yablonsky, and G.B. Marin, “Thermodynamic time invariances for
• dual kinetic experiments: nonlinear single reactions and more”, Chem. Eng.Sci., 73(2012)20-29
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• References (II)
6. D. Constales, G.S. Yablonsky, G.B. Marin, “Intersections and coincidences in chemical
• kinetics: Linear two-step reversible–irreversible reaction mechanism”, “Computers and
• Mathematics with their Applications“, 65(2013)10, 1614-162
• 7. E. Redekop, G. Yablonsky, V. Galvita, D. Constales, R. Fushimi, J. T. Gleaves, and G. B. Marin,
• “Momentary Equilibrium (ME) in Transient Kinetics and Its Application for Estimating the
• Concentration of Active Sites”, Ind. Eng. Chem. Res., 52 (44), 15417–15427 (2013)
• 8. G. S. Yablonsky, D. Constales and G. B. Marin, “New Types of Complexity in Chemical Kinetics :
• Intersections, Coincidences and Special Symmetric Relationships”, “Advances in Chemical
• Physics”, v. 157 (2014)69-73
• 9. D. Constales, G. S. Yablonsky, G.B. Marin, “Predicting kinetic dependences and closing the
• balance: Wei and Prater revisited”, Chemical Engineering Science, 123(2015)328-333
•
• 10. D. Branco Pinto, G.S. Yablonsky, G.B. Marin, and D. Constales, “New Patterns in Steady-State
• Chemical Kinetics: Intersections, Coincidences, Map of Events (Two-Step Mechanism)”, Entropy,
• 17(10), 2015, 6783-6800
• 11. D. Branco Pinto, G.Yablonsky, G.B. Marin, and D. Constales, “The Switching Point between the
• Kinetic and Thermodynamic Control”, Comp.Chem. Eng., 2015 (submitted)
107