(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

qq

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