Introduction

Kinetics

aA + bB cC + dDk

General Reaction:

General Rate Equations:

Rate=k [A ]a [B ]b=− 1𝑎𝑑 [ 𝐴 ]𝑑𝑡

=− 1𝑏𝑑 [𝐵 ]𝑑𝑡

=1𝑐𝑑[𝐶 ]𝑑𝑡

= 1𝑑𝑑 [𝐷 ]𝑑𝑡

where k is a the “rate constant”

Upadhyay

1) Separate variables

2) Apply approximations if necessary

3) Integrate over relevant limits ([A]o, ([B]o)

4) Algebreically solve for variable of interest ([A](t))

Wait!! k is contant? NOT TRUE!!!

Obviously k = f(T)

Rates as a Function of Temperature

Nobelprize.org

𝑑 𝑙𝑛𝐾𝑑𝑇

=∆𝐻𝑅𝑇 2

where K is the equilibrium constant

and is the change of enthalpy.

Proposed in 1884:

Jacobus Henricus van’t Hoff

k=A𝑒−

𝐸𝑎

𝑅𝑇

where A is the “pre-exponential factor” (A(T)) and Ea is the energy of activation.

Proposed in 1889:

Svante August Arrhenius

Awarded Nobel Prize in Chemistry in 1901: “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions".

Awarded Nobel Prize in Chemistry in 1903: "in recognition of the extraordinary services he has rendered to the advancement of chemistry by his electrolytic theory of dissociation".

Collision TheoryMax Trautz (1916)William Lewis (1918)

Objective:

To develop a model describing how the rate constant of a reaction varies with changing temperature considering energetic collisions.

General Assumptions:

1) Molecules are hard spheres in temperature dependent translation.2) The molecules undergo collisions, and any collision with sufficient energy

(E*) will result in a reaction.3) Concentration affects reaction rate due to it’s effects on collision rate.

The Impact Parameter

bmax = r1 + r2

r1

r2b

bmax

Impact Parameter:

Distance of closest possible approach of the center of the species involved in a collision.

-There exists some maximum impact parameter above which a collision will not occur (bmax). Ie:

b > bmax no collision no reaction

b < bmax collision occurs

reaction may occur if sufficient energy is transferred

b = “impact parameter”

Reactant A

Reactant B

Area of collision = π b2max =

“hard-sphere cross section”

v

The Cross Section and Collision Frequency

V (bmax, Δt) = (vΔt)π

-Collision frequency (Z) per unit time and volume is given by: Z = π b2max<v>n1n2

where n1 and n2 are proportional to the number of collision partners present (concetration).

-Reactive cross section(σ) is the sum of the hard sphere cross sections of collision over time and depends on the probability of a reaction having sufficient energy for a reaction to occur.

Collisional Energy Transfer

Maxium Energy Transfer in Collision

v

b = 0

Two Extremes:

b > bmax

v No Energy Transfer in Collision

Quantitative Collisional Energy

αb bmax

v

vlc

α

-The energy transfer of a collision (Et) depends on the velocity of approach relative to the line of centers of the collisional partners.

-Relationships allow determination of vlc:

sin (𝛼 )= 𝑏𝜈𝑙𝑐

=𝑏

𝑏𝑚𝑎𝑥cos (𝛼 )=

𝜈 𝑙𝑐

𝑣

The energy along the line of centers is given by (KE=1/2mv2):

where μ is the reduced mass of the system.𝐸𝑙𝑐=12μ𝜈❑

2𝑙𝑐=¿

Quantitative Collisional Energy

αb bmax

v

vlc

α

-The energy transfer of a collision (Et) depends on the velocity of approach relative to the line of centers of the collisional partners.

-Relationships allow determination of vlc:

sin (𝛼 )= 𝑏𝜈𝑙𝑐

=𝑏

𝑏𝑚𝑎𝑥cos (𝛼 )=

𝜈 𝑙𝑐

𝑣

The energy along the line of centers is given by (KE=1/2mv2):

where μ is the reduced mass of the system.𝐸𝑙𝑐=12μ𝜈❑

2𝑙𝑐=¿

Elc > E* Elc < E* REACTION!

Recall Initial Assumption:

Probability of Reaction

db

b

> E*

P(Et,b) = 0

P(Et,b) = 1

>

Reaction Probability as a function of energy:

σ (𝐸𝑟 )=∫0

∞

𝑃 (𝐸𝑟 ,𝑏)2𝜋 𝑏𝑑𝑏

σ (𝐸𝑟 )=∫0

𝑏 ′

𝑃 (𝐸𝑟 ,𝑏)2𝜋 𝑏𝑑𝑏Solve

σ (𝐸𝑟 )=π𝑏❑2 𝑚𝑎𝑥(1−E∗E 𝑡 )

Physical limits provide limits of integration

Putting Collision Theory TogetherThe Maxwell-Boltzmann energy distribution:

𝐺 (𝐸𝑡 )𝑑𝐸𝑡=2 π ( 1𝜋𝑘𝑏𝑇 )

3/2

√𝐸𝑡 𝑒(− 𝐸𝑡

𝑘𝑏𝑇 )𝑑𝐸𝑡

Reaction rate = k(Et)n1n2

¿𝑄>¿∫𝑄𝑓 (𝑄 ) 𝑑𝑄k

vt = (2Et/μ)1/2

k(Et) = σ(Et)vt

and

where

k (𝑇 )=π𝑏❑2 𝑚𝑎𝑥√ 8𝑘𝑏𝑇

𝜋𝜇𝑒− 𝐸∗

𝑘𝑏𝑇

Integrate & solve for k

Mathematical relationship:

Shortcomings of Collision Theory-Rates predicted by collision theory trend higher than experimental evidence indicates

At least partially due to the “steric factor”: compensated for by introducing a constant (p):

𝐤 (𝑻 )=𝒑 𝝅𝒃❑𝟐𝒎𝒂𝒙 √𝟖𝒌𝒃𝑻

𝝅𝝁𝒆− 𝑬∗

𝒌𝒃𝑻

Non-reactive approach zone

Reactive approach zone

-Also, what about reaction order?

Reactive Species:

2 3 4 n

Possible Collisions:

1 3 6

Wouldn’t we expect all reactions initiated by collision

to be bimolecular?

Yet unimolecular reactions exist! Rate ≈ Z = π b2

max<v>n1n2

Lindemann’s Mechanism (1922)Three Steps to Reaction:

ActivationRate=𝑑 [A∗]𝑑𝑡

=𝑘1 [A ] 2

A + A A + A*k1

Activation:

A + A* A + Ak -1

Dectivation:

A* Pk2

Decomposition:

Dectivation Rate=−𝑑 [A∗]𝑑𝑡

=𝑘− 1 [ A ]¿

Decomposition Rate=−𝑑[A∗]𝑑𝑡

=𝑘2¿

Apply steady state conditions with respect to [A*] and then solve for rate of reaction:

ReactionRate=𝑘1𝑘2 [ A ]2  𝑘2+𝑘− 1 [ A ]

ReactionRate=𝑘1𝑘2 [ A ]  

𝑘−1

ReactionRate=𝑘1 [ A ]2k -1 [A] >> k 2

k 2 >> k -1 [A]

Reduce based on two pressure limits.

Hinshelwood Theory (1926)Four Steps to Reaction:

A≠ P

A + A A + A*k1

Activation/Deacivation:

k -1

Decomposition 1& 2:

A* A≠k2

Adapted from R. A. Marcus, J. Chem. Phys. 43, 2658 (1965)

Consider multiple internal degrees of freedom (s):

Cyril N. Hinshelwood

𝑓 = 1𝑘𝑏𝑇

𝑒(− 𝐸

𝑘𝑏𝑇 )𝑑𝐸

Fraction of molecules with energy between E and E+dE is given by

𝑓 = 1(𝑘𝑏𝑇 ) 𝑠

𝑒(− 𝐸1

𝑘𝑏𝑇 )𝑑𝐸 1𝑑𝐸2 ·· ·𝑑𝐸𝑛

Awarded Nobel Prize in Chemistry in 1956: "for [his] researches into the mechanism of chemical reactions.”

Hinshelwood Theory

𝑓 = 1(𝑘𝑏𝑇 ) 𝑠

𝑒(− 𝐸

𝑘𝑏𝑇 )𝑑𝐸1𝑑𝐸2 ·· ·𝑑𝐸 s

Integrate over limits

0 < E < s-1𝑑𝐸1𝑑𝐸2 · ··𝑑𝐸 s 𝐸𝑠

𝑠 ! between E and E + dE

𝐸𝑠− 1

(𝑠−1)!𝑑𝐸

molecules with energy between E and E + dE

Replace to get expression of equilibrium proportion of

𝐹= 1(𝑠−1 )! ( 𝐸

𝑘𝑏𝑇 )𝑠− 1 𝐸

𝑘𝑏𝑇𝑒(− 𝐸

𝑘𝑏𝑇 )𝑑𝐸

Differentiate forrange of energies

𝐹=𝑑𝑘1𝑘− 1 𝑑𝑘1

𝑘− 1= 1

(𝑠−1 ) ! ( 𝐸𝑘𝑏𝑇 )

𝑠 −1 1𝑘𝑏𝑇

𝑒(− 𝐸𝑘𝑏𝑇 )𝑑𝐸

𝑘1𝑘− 1

=1

(𝑠−1 )! ( 𝐸∗

𝑘𝑏𝑇 )𝑠−1

𝑒(− 𝐸∗

𝑘𝑏𝑇 )=

[𝐴∗ ][ 𝐴]

Integrate from E* to

RRKM Theory

Comparison of the Theories

The Reaction Coordinate

H-H (Å)

1.5 2.0

2

4

Ener

gy (e

V)

D-H (Å)

Houston

H-H eq1

D-Heq2

1 2

1

2

“Transition State”

Transition State Theory

General Assumptions:

1) An eqillibrium exists between the reactants “activated state (“transition state”) of a chemical reaction.

2) The difference in energy (Eact)between these two states must be supplied for the to the reaction for the it to form.

3) A certain vibrational degree of freedom exists that is necessarily active for the transition state to dissociate

Henry Eyring and John Polanyi (1931)

Objective:

To develop a model describing how the rate constant of a reaction varies with changing temperature considering unstable transition states.

John Polanyi

Awarded Nobel Prize in Chemistry in 1986: for his contributions concerning the dynamics of chemical elementary processes“.

General Reaction:

Transition State Theory

A + B X≠ C + DK≠

[X≠] = K≠ [A][B]

X≠ C + D through

vibrational mode v withEvib=hv=kbT.Rate of reaction = [X≠]v = K≠ [A][B] = k[A][B]

3

k = K≠ [A][B]

RT lnK = -∆G∘

Standard Enthalpy Change:

∆G∘ = ∆H∘ - T∆S∘

Gibb’s-Helmholtz Relation:

1

2

∆H∘ = Eact

k (𝑇 )=𝑘𝑏𝑇𝒉

𝑒∆𝑺 ≠

𝑹  𝑒−𝑬𝒂𝒄𝒕

𝑹𝑻

General Scheme for Vibrational Control of Reactions

Simple Infrared Excitation

Stimulated Raman Excitation

Infrared Multiphoton Excitation

Vibrational Overtone Excitation

Stimulated Emission Pumping

Preparation of select vibrational mode(s)

1

Photoacoustic Spectroscopy

Resonance Enhanced Multiphoton Imaging (REMPI)

Detection of Mode Specific Products

3

Run the Reaction2

Rotovibrational Spectrum

Wiki

The different energies of the allowed transitions lead to elaborate vibrational spectrum:

The selection rule for rotational state transitions is given by:

∆J = ±1

Photoacoustic Effect

Laser pulses shoot sample with infrared or near-ir frequency radiation.

1Sample heats up from the radiation, which activates modes of

2

vibration and rotation. This in turn creates pressure wavesin the air.

3

The pressure waves are detected as sound by a microphone.

Fourier

Transform

Photoacoustic Spectrum

http://www.shimadzu.com/an/ftir/support/ftirtalk/talk7/intro.html

Photoacoustic Spectroscopy Technique

Braz. J. Phys. vol.32 no.2b São Paulo June 2002

Gas Flow

SemipermiableMirrors

Microphone

CellWindow

ResonatingChamber

Buffer Gas Volumes

Wiki PAS

Ultrafast Lasers

Path APath B

Length = L

PULSEWiki

Bond Selected Reaction of CH3D + Cl

CH3D + Cl

H

H

H

D

CH3 + DCl

CH2D + HCl

CalculatedEndothermicity

2200 cm-1

1800 cm-1

1

2

Cl

Pathway 2Pathway 1

ClD

Two

Possible

Abstraction

Pathways

Expected Product Disbribution CH3D + Cl

G. D. Boone, F. Agyin, D. J. Robichaud, F.-M. Tao, and S. A. Hewitt, J.Phys. Chem. A 105, 1456 ~2001

“Primary Kinetic Isotope Effect”

Calculated Potentials: In the thermal reaction, what would we expect the product distribution to be?

On these considerations, we expect the major product to be CH2D in the absence of

state specific vibrational excitation.

Energy

Frequency

CH3D + Cl Experimental

1) A molecular beam was created using a 1:1:4 mixture of CH3D, Cl2, and He with a 660 torr backing pressure.

2) Deuteromethane was vibrationally excited with 2.3 μm (4300 cm-1) laser pulses (excitation laser).

3) Molecular chlorine was dissociated with a 355 nm laser pulse (dissociation laser).

4) After a 200-250 ns delay, a 2+1 REMPI produces ions from either the CH2D or CH3 product (probe laser).

5) Time-of-flight mass spectrometry detects products.

Simultaneously, the laser beam is directed through a cell containing 15 torr CH3D for collection of absorbtion spectrum to computationally verify vibrational assignments.

CH3D + Cl Experimental

Monitors the generation of a product of a specific product over a range of vibrational excitation laser wavelengths.

Provides information similar to photoacoustic spectroscopy. Allows determination of degree of generation of a specific

product with respect to individual vibrational modes.

Action Spectra

REMPI Excitation Spectra Monitors the generation of multiple products through mass

resolved spectrometric detection after REMPI. Allow determination of product distributions from vibrational

mode specific reactants.

J. Chem. Phys., Vol. 119, No. 9, 1 September 2003

Less StableProduct

More StableProduct

Thermal Reaction of CH3D + Cl6x

Stronger Signal

6x Weaker Signal

REMPI spectrum:

Reactive Vibrational Energies

J. Chem. Phys., Vol. 119, No. 9, 1 September 2003

Parallel Transition

Perpendicular Transition

A1 Symmetry

ESymmetry

2v2 2v2

Houston

1

2

Conclusions