Collision theory of bimolecular reactions • Arrhenius formula...

物理化學(三) 應用化學系朱超原老師 1

• Collision theory of bimolecular reactions• Arrhenius formula (again)• Fraction number in bimolecular reactions • Unimolecular reactions

VIII. The chemical reaction mechanismsVIII. The chemical reaction mechanisms


VIIIVIII--1. Collision theory of bimolecular reactions1. Collision theory of bimolecular reactions

bBaA products

dt

tBdbdt

tAda

tr 11

Two reactants

For simplicity (a=b=1)

products

products

A A


Reaction mechanisms

Kinetic energy

Internal energy Rotational energyVibrational energyElectronic energy

No kinetic energy change Elastic collision

Internal energy change inelastic collision

Bond forming or/and breaking Reactive collision

collision scattering


Total collision rate

22

VN5.0

21

dv

VNzZ rel

mTk

vv Brel 16

2

Number of collision per unit volume +per unit time

scollisonsofnumber3m

Fraction number of reactions = f fZReaction rate =

fZN Av

1Reaction rate in mole =


AAA 2 products

22

AvV

N/NA

22VN5.01

21

dv

Nf

dtAd

relAv

225.021 AdvfN

dtAd

relAv

Rate constant

AvB Nm

Tkdfk

42

(second-order)

Only dependent on temperature

(Ideal case = elementary)


BA products

dtBd

dtAdfZ

N Av12

1Reaction rate in mole =

212

121212 dV

NVN

vZ 12

128

Tkv B

BA

V/NN

V/NN Av2Av1

BAdvfNdt

AdAv

21212

AvB N

Tkdfk

12

212

8

Rate constant


Example 2HIIH 22

1115 smolL1074.8 fk

At T = 373.15K

Calculate fraction number f

Rate constant

AvB N

Tkdfk

12

212

8

Table A15

6.35397I2

2.72294H2

d×1010/mT/K

mmd 101012 1054.41035.672.221

kgkgmm

mm 272721

2112 103.31066.12542

2542


smolL

smolm

NTk

d AvB

11

38

12

212

1077.7

1077.7

8

-123 JK10380.1 Bk

molN Av /10022.623

At T = 373.15K

md 1012 1054.4

kg2712 103.3

1115 smolL1074.8 fk2611

15101.1

1077.71074.8

f

If T = 473.15K 1110 smolL1053.9 fk

smolL

smolL

NTkd AvB

11

11

12

212

1075.8

15.37315.4731077.7

8

2111

10101.1

1075.81053.9

f


VIIIVIII--2. Arrhenius formula (again)2. Arrhenius formula (again)

Kinetic energy

212122

1 vK

Classical picture

aEK

No reaction aEK

Reactions occur

K

aE


Arrhenius formula

RTE

Ak aexp

Activation energy

Arrhenius preexponential factor

A and Ea are very slow change with respect to temperature

dTkdRTEa

ln2

Activation energy is measured through experiment reaction rate cActivation energy is measured through experiment reaction rate constant onstant

Reactants aE

Products

Activated Complex

aE

Path of a reaction

ΔH


Example 2HIIH 22

1115 smolL1074.8 fkAt T = 373.15K

Rate constant

At T = 473.15K 1110 smolL1053.9 fk

Calculate activation energy and pre-exponential factor

11 exp RT

EAk a

22 exp RT

EAk a

15

2

1

21

21 molJ10703.1ln

kk

RTT

TTEa

11mol8.314JK R

3.432

RTEa 1192 smolL1008.63.43exp kA


PCIIIweek16-1作業

Problems 12.712.8

Page 539


VIIIVIII--3. Fraction number in bimolecular reactions3. Fraction number in bimolecular reactions

Total collision rate

VN

VNdvZ 122121212 12

128

Tkv B

vgMaxwell–Boltzmann distribution 1311 vdvgN number of A particles in 111 vdvv

1311 vdvgN number of B particles in 222 vdvv

Differential collision rate

V

vdvgNV

vdvgNddZ 1

3112

3222

121212 vv

13 vdvg


Maxwell–Boltzmann distribution

Tkmv

Tkmvg

BB 2exp

2

23

222

211 2

121 vmvmH

Center-of –mass coordinate

12 vvv

21

1122M

vvV

mmmm

2221

21 vMVH M

vdvgVdVgM

mmV

NNddZ MM

33212

1221212 v

1


Integral d3VM vdvg

VNN

ddZ 32122

1212 v

If we integral d3v 2122

121212 vV

NNdZ (no new)

Total reaction rate

cv

Classical pict ure

v ( relative speed)

Reaction probability

0

1

cvv All reactions

cvv no reaction


Total reaction rate vdvvgVNN

dreactionZ 32122

1212

cv BBdvv

Tkv

TkVNNdreactionZ 3

212

312

2122

1212 2exp4

2

2121

2exp 23

212 c

c

avc

v Be

av

advv

Tkv

Tka

B212

F ract ion number in bimolecul ar re actionsF ract ion number in bimolecul ar re actions

RTE

cTk

B

cTkv

B

cc

B

c

B

c

eRTE

eTk

eTk

vf

11

21 2

212

212


Example 2HIIH 22 At T = 373.15K

26101.1 fFraction number of reaction

Tk

B

cTkv

B

c Bc

B

c

eTk

eTk

vf

1

21 2

212

212

26101.11 xex 10ln261.1ln)1ln( xxx = 63.9454499236402

Txk Bc

eVmolkJmolJxRTEc 2/198/1098.15

11mol8.314JK R


new4

BA products

Elementary bimolecular reaction (two reactants)

BAkdt

Ad

RTE

cAv

BC

eRTE

NTkdk

18

12

212

RTE

Ak aexp ac EE

Compare with Arrhenius formula

Rate constant


Example 2HIIH 22

molJEc /1098.15

T1 = 293.15K T2 = 303.15K

11mol8.314JK R

KREc 41038.2

14exp

exp

1

2

RTERTE

C

C

983.0

1

1

2

1

11

22

1

2

TT

RTE

T

RTE

T

TATA

c

c

Change fast

Change slow


v (relative speed)

Reaction probability

cc

c

vvv

vvv

p1

0

Total reaction rate

cv

c

BBdv

vv

vTk

vTkV

NNdreactionZ 12

exp42

32

123

122

1221212

RTE

TkTkv c

B

c

B

c

eeef

2

212

Rate constant

fraction number

RTE

AvB

C

eNTkdk

12

212

8

0

1

cv


Example 2HIIH 22 At T = 373.15K

26101.1 fFraction number of reaction

262 101.1

212

TkTkv

B

c

B

c

eef

1.1ln10ln26 x x = 59.77Txk Bc

eVmolkJmolJxRTEc 93.1/186/1086.15

11mol8.314JK R

Better than …

15 molJ10703.1(exp) aE


Example 2HIIH 22 At T = 373.15K1115 smolL1074.8 fkRate constant

15 molJ10703.1(exp) aEActivation energy

Calculate effective collision diameter

RTE

AvB

C

eNTkdk

12

212

8

904.54RTEa

smolmN

TkAv

B

2612

10128

242721218 1043.1102.11074.8 dCross section 221212 10093.5 md

A4.0104 1112 md


Steric factor in bimolecular reactions

If two molecules are not in right orientation in collision, thereis no reaction no matter how large kinetic energy.

Simple way to consider this:

RTEc

ef

Rate constant

fraction number

RTE

AvB

C

eNTkdk

12

212

8

Steric factor


Example 2HIIH 22

At T = 373.15K

11mol8.314JK R

RTE

Ak aexp

1115 smolL1074.8 k 15 molJ10703.1 aE 119 smolL100.6 A

experiment

Calculate steric factor RTE

AvB

C

eNTkdk

12

212

8

smolL

NTkd AvB

11

12

212

1077.7

8

Av

B NTk

dA12

212

8

0077.0


PCIIIweek16-2作業

Problems 12.1512.16

Page 540


BA products BAkdt

Ad

Rate constant

RTE

Ak aexpExperiment

RTEc

ef

Theory

fraction number

RTE

AvB

C

eNTkdk

12

212

8

2

1221

cAvC vNE

If A=B, kk 21

2

21 Ak

dtAd


VIIIVIII--4. 4. UnimolecularUnimolecular reactionsreactions

A CB

Akdt

Ad

2Akdt

Ad

Experiment found this reaction

neither nor

Lindemann mechanism (2 steps)

AA AA *

*A CB

(1)

(2)


(1) *A as a product AAkAk

dtAd

dtAd **

21

12

1

(2) *A as a reactant ** 2 Akdt

BddtAd

forward reverse

Compare two steps *** 2121 AkAAkAkdtAd

*2 AkdtBd

At equilibrium *A 0* dtAd Akk

AkA12

21*

Akk

Akkdt

Bd

12

212


rate =

AkkAkk

dtBd

dtAdtv

12

212

A CB

(a) 21 kAk

Ak

kkdt

Bddt

Adtv1

12

unik(b) 21 kAk

21 AkdtBd

dtAdtv

First-order

Second-order

Collision theory of bimolecular reactions • Arrhenius formula...

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