(b) Analytical solutions of the material balance equation Using numerical method: Euler method or 4...
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Transcript of (b) Analytical solutions of the material balance equation Using numerical method: Euler method or 4...
(b) Analytical solutions of the material balance equation
][][][
Jkx
JD
t
J
2
2
ktt kt eJdteJkJ ][][][ *0
21
40
2
/
/
)(][
DtA
enJ
Dtx
Using numerical method: Euler method or 4th order Runge-Kutta method to Integrate the differential equation set.
Fig. 1 Snapshots of stationary 2D spots (A) and stripes (B) in a thin layer and 2D images of the corresponding 3D structures (A→B, C; B→E to G) in a capillary.
T Bánsági et al. Science 2011;331:1309-1312
Published by AAAS
Fig. 3 Stationary structures in numerical simulations.
T Bánsági et al. Science 2011;331:1309-1312
Published by AAAS
Stationary structures in numerical simulations. Spots (A), hexagonal close-packing (B), labyrinthine (C), tube (D), half-pipe (E), and lamellar (F) emerging from asymmetric [(A), (B), and (E)], symmetric [(D) and (F)], and random (C) initial conditions in a cylindrical domain. Numerical results are obtained from the model:
dx/dτ = (1/ε)[fz(q – x)/(q + x) + x(1 – mz)/(ε1 + 1 – mz) – x2] + ∇2x; dz/dτ = x(1 – mz)/(ε1 + 1 – mz) – z + dz∇2z,
where x and z denote the activator, HBrO2 and the oxidized form of the catalyst, respectively; dz is the ratio of diffusion coefficients Dz/Dx; and τ is the dimensionless time. Parameters (dimensionless units): q = 0.0002; m = 0.0007; ε1 = 0.02; ε = 2.2; f = (A) 1.1, (B) 0.93, [(C) to (F)] 0.88; and dz = 10. Size of domains (dimensionless): diameter = 20 [(A) to (C) and (F)]; 14 [(D) and (E)]; height = 40.
Transition State Theory(Activated complex theory)
• Using the concepts of statistical thermodynamics.
• Steric factor appears automatically in the expression of rate constants.
24.4 The Eyring equation • The transition state theory pictures a reaction between A and B as proceeding
through the formation of an activated complex in a pre-equilibrium: A + B ↔ C‡
( `‡` is represented by `±` in the math style)
• The partial pressure and the molar concentration have the following relationship:pJ = RT[J]
• thus
• The activated complex falls apart by unimolecular decay into products, P,
C‡ → P v = k‡[C‡] • So
Define
v = k2[A][B]
BA
C
pp
ppK
]][[][ BAp
RTKC
]][[ BAKp
RTkv
Kp
RTkk 2
(a) The rate of decay of the activated complex
k‡ = κv
where κ is the transmission coefficient. κ is assumed to be about 1 in the absence of information to the contrary. v is the frequency of the vibration-like motion along the reaction-coordinate.
(b) The concentration of the activated complexBased on Equation 17.54 (or 20.54 in 7th edition), we have
with ∆E0 = E0(C‡) - E0(A) - E0(B)
are the standard molar partition functions.
provided hv/kT << 1, the above partition function can be simplified to
Therefore we can write qC‡ ≈
where denotes the partition function for all the other modes of the complex.
RTE
BA
CAe
qNK /0
Jq
kThveq
/
1
1
hv
kT
kThv
q
)( 11
1
cqhv
kT
Cq
K
hv
kTRTE
BA
CA eqq
qNK /0
K‡ =
(c) The rate constant
combine all the parts together, one gets
then we get
(Eyring equation)
To calculate the equilibrium constant in the Eyring equation, one needs to know the partition function of reactants and the activated complexes.
Obtaining info about the activated complex is a challeging task.
C
Kh
kTk 2
Kp
RT
hv
kTvk
Kp
RTkk
2
2
(d) The collisions of structureless particles
A + B → AB
Because A and B are structureless atoms, the only contribution to their partition functions are the translational terms:
3J
mJ
Vq
2/1)2( kTm
h
JJ
p
RTVm
C
mC
VIkTq
2
2
RTE
mC
BAA eIkT
V
N
p
RT
h
kTk /
23
33
20
2
RTEA er
u
kTNk /2
2/1
20
8
24.5 Thermodynamic aspects
Kinetics Salt EffectIonic reaction A + B ↔ C‡ C‡ → P d[P]/dt = k‡[C‡]
the thermodynamic equilibrium constant
Then
d[P]/dt = k2[A][B]
Assuming is the rate constant when the activity coefficients are 1 ( )
Debye-Huckle limiting law with A = 0.509
log(k2) = log( ) + 2AZAZBI1/2 (Analyze this equation)
]][[
][
]][[
][
BA
CK
BA
C
aa
aK
BA
C
BA
C
K
Kkk
2
02k Kkk 0
2
K
kk
02
2
212 /)log( IAzJJ
02k
Experimental tests of the kinetic salt effect
• Example: The rate constant for the base hydrolysis of [CoBr(NH3)5]2+ varies with ionic strength as tabulated below. What can be deduced about the charge of the activated complex in the rate-determining stage?
I 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300
k/ko 0.718 0.631 0.562 0.515 0.475 0.447
Solution:
I1/2 0.071 0.100 0.122 0.141 0.158 0.173
Log(k/ko) -0.14 -0.20 -0.25 -0.29 -0.32 -0.35
24.6 Reactive Collisions
• Properties of incoming molecules
can be controlled:
1. Translational energy.
2. Vibration energy.
3. Different orientations.
• The detection of product molecules:
1. Angular distribution of products.
2. Energy distribution in the product.
infrared chemiluminescence, a process in which vibrationally excited molecules emit infrared radiation as they return to their ground states.
laser-induced fluorescence, a technique in which a laser is used to excite a product molecule from a specific vibration–rotation level and then the intensity of fluorescence is monitored.
multiphoton ionization (MPI), a process in which the absorption of several photons by a molecule results in ionization.
Detection techniques
reaction product imaging, a technique for the determination of the angular distribution of products.
resonant multiphoton ionization (REMPI), a technique in which one or more photons promote a molecule to an electronically excited state and then additional photons are used to generate ions from the excited state.
state-to-state cross-section, σnn', the cross section for reaction in which a specified initial state changes into a specified final state.
state-to-state rate constant, the rate constant for a specified state-to-state reaction; knn = σnnvrelNA.
24.6 Reactive collisions (cont..)
24.7 Potential energy surface
• Can be constructed from experimental measurements or from Molecular Orbital calculations, semi-empirical methods,……
Contour plot
Potential energy is a function of the relative positions of all the atoms takingpart in the reaction.
Potential energy surfaces, pt. 2.
Various trajectories through the potential energy surface
24.8 Results from experiments and calculations
(a) The direction of the attack and separation
Attractive and repulsive surfaces
Crossing crowded dance floors.
S Bradforth Science 2011;331:1398-1399
Published by AAAS
Classical trajectories