Activated complex theory

• Using the concepts of statistical thermodynamics.

• Steric factor appears automatically in the expression of rate constants.

27.4 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‡

K‡ = ( `‡` is represented by `±` in the math style)

• The partial pressure and the molar concentration has the following relationship:pJ = RT[J]

• thus[C‡] = K‡ [A][B]

• The activated complex falls apart by unimolecular decay into products, P,

C‡ → P v = k‡[C‡] • So v = k‡ K‡ [A][B]

Define k2 = k‡ K‡v = k2[A][B]

BA

C

pp

pp

p

RT

p

RT

p

RT

(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 20.54, 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

cq

hv

kT

q

K

hv

kTRTE

BA

CA eqq

qNK /0

K‡ =

(c) The rate constant

combine all the parts together, one gets

k2 = k‡ K‡ = κv

then we get

k2 = κ (Eyring equation)

p

RT

hv

kTp

RT

K

CK

h

kT

(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:

k2 = κ

k2 = κ NA

3J

mJ

Vq

212 /)( kTm

h

JJ

p

RTVm

C

mC

VIkTq

2

2

RTE

mC

BAA eIkT

V

N

p

RT

h

kT /023

33 2

RTEeru

kT //

0221

8

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 02

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 cab 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

27.6 Reaction 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.

27.7 Potential energy surface

• Can be constructed from experimental measurements or from Molecular Orbital calculations, semi-empirical methods,……

Potential energy surfaces, pt. 2.

Various trajectories through the potential energy surface

27.8 Results from experiments and calculations

(a) The direction of the attack and separation

Attractive and repulsive surfaces

Classical trajectories