Chemical kinetics Chapter 19

The Danish National Research Foundation’s

Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital - DENMARK

Chemical kinetics

Chapter 19

Sune Nørhøj Jespersen


Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital

Learning objectives

After working with this material, you should be able to:• Define reaction rate constants.

• Write down basic differential equations for chemical kinetics, including law of mass action, and solve simple cases.

• Explain the principle of detailed balance.• State the Arrhenius law



Reaction rates

Equilibrium:

Kinetics:

A BK

A Bkf

kr

Rate constants:

kf: Probalility of one A molecule to convert to B per unit time.

kr: Same but in the opposite direction.



Kinetic rate equations

In a short time interval dt

and like wise

Coupled first order differential equations

r fd A k dt B k dt A

r f

d Ak B k A

dt

f r

d Bk A k B

dt



General solution

d xR x

dt

A

xB

f r

f r

k kk

k k

exp( ) (0)

(0)t T

x kt x

Ve V x

1

2

0

0

Eigenvalues of k (in this example,

one will be 0); V eigenvectors



General solution

For example:

1 2 1 21 2 1 2[ ( )] (0) (0)t t t tA t a e a e A b e b e B

Can be solved analytically only in simple cases for more involved reaction.

Otherwise numerical solutions are valuable.



Example

A Bkf

r fk k

f

d Ak A

dt

( ) (0) fk tA t A e

N A B

( ) (0) fk tB t N A e

if is conserved

?



Detailed balance

From reversibility of microscopic laws

Principle of detailed balance

This implies steady state

Comparing to the equilibrium constant of chapter 13:

f req eqk A k B

0eq eq

d A d B

dt dt

eq f

req

B kK

A k



Example

I

A B

kAI

kIA

kIBkBI

kBA

kAB

kAI kBI

kBAA B

I

Figure 19.1 The principle of detailed balance is satisfied by mechanism (a) between three

states A,I, and B, but violated by mechanism (b). Forward rates must equal reverse rates

for each pair of states.



Mass action

fkaA bB cC p

Here we stipulate a b c

f

d pk A B C

dt

aA

2a 0 0

0 0

0 0 0

0 0

x x

x x

x

x x

2

A AN

N N

is proportional to the propability that a A molecules are close

Number of pairs Note

1A

N



Arrhenius

2kA B r

2

d pk A B

dt

2

lnd K h

dT kT

2

ln,f a

d k E

dT kT

2

'ln ar Ed k

dT kT

/aE kTfk Ae

2k depends strongly on temperature.

In equilibrium:

Kinetics:

, 'a aE E activation energies.



Arrhenius

Reactants Products

aE 'aE

Transition stateE

lowE

highE

activationE

highTlowT

Energy

Population

Chemical kinetics Chapter 19

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