Post on 20-Jan-2016
description
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital - DENMARK
Chemical kinetics
Chapter 19
Sune Nørhøj Jespersen
The Danish National Research Foundation’s
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
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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.
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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.
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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
?
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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.
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
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.
The Danish National Research Foundation’s
Center of Functionally Integrative NeuroscienceAarhus University / Aarhus University Hospital
Arrhenius
Reactants Products
aE 'aE
Transition stateE
lowE
highE
activationE
highTlowT
Energy
Population