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Chemical kinetics: accounting for the rate laws. 자연과학대학 화학과 박영 동 교수. The approach to equilibrium. Forward: A → B Rate of formation of B = k r [A] Reverse: B → A Rate of decomposition of B = k r [ B ]. Net r ate of formation of B = k r [A] − k r [ B ]. - PowerPoint PPT Presentation
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Chemical kinetics:accounting for the rate
laws자연과학대학 화학과박영동 교수
The approach to equilibriumForward: A → B Rate of formation of B = kr[A]Reverse: B → A Rate of decomposition of B = kr[B]
Net rate of formation of B = kr[A] − kr[B]
At equilibrium, rate = 0 = kr[A]eq − kr[B] eq
eq
eq
[B][A] '
r
r
kK
k
rate and temperature
/Ea RTrk Ae
( )/' ' Ea G RTrk A e
( )/eq ( )/
( )/eq
[B]"
[A] ' '
Ea RTG RTr
Ea G RTr
k AeK A ek A e
The net rateForward: A → B Rate of formation of B = kr[A]Reverse: B → A Rate of decomposition of B = kr[B]Net rate d[A]/dt = − kr[A] + kr[B] = − kr[A] + kr([A]0 − [A])
How to solve the following differential equation?'
Let's guess y as
' ( )0
or
ax
ax ax
ax
y ay b
y ce d
y ace a ce d bad b
b bd y cea a
d[A]/dt = − (kr + kr) [A] + kr[A]0
( ') 0'[A] [A]
( ')r rk k t r
r r
kce
k k
0 00
'[A] [A] [A] or
( ') ( ')r r
r r r r
k kc c
k k k k
( ')0( ' )[A]
[A]( ')
r rk k tr r
r r
k k ek k
at t = 0,
( ')0(1 )[A]
[B]( ')
r rk k tr
r r
k ek k
The approach to equilibrium( ')
0( ' )[A] [A]
( ')
r rk k tr r
r r
k k ek k
( ')0(1 )[A]
[B]( ')
r rk k tr
r r
k ek k
0eq
'[A] [A]
( ')r
r r
kk k
0eq
[A] [B]
( ')r
r r
kk k
eq
eq
[B][A] '
r
r
kK
k
The approach to equilibrium of a re-action that is first-order in both direc-tions.
Relaxation methods
0 / , where = 'tr rx x e k k
Consecutive reactions
0I / dt A I [A] e Iak ta b a bd k k k k
0
0 0 0
0
0 0
Let I [A] e e
I / [A] e e [A] e ( [A] e e )
( )[A] e e[A] ( )[A] , 0
( ), / ( ), 0from at = 0, [
a b
a b a a b
a b
k t k t
k t k t k t k t k ta b a b
k t k ta b b b
a a b b
a a b a b a
a b c
d dt ak bk k k a b c
k k a k b k cak k k a k cak k k a a k k k c
t
I]=0, 0, -a b b a
0 I (e e )[A]a bk t k ta
b a
kk k
0
e e P 1 [A]
b ak t k ta b
b a
k kk k
Reaction Mechanism and elemen-tary reactions
a unimolecular elementary reaction a bimolecular elementary reaction
A → P v = kr[A] A + B → P v = kr[A] [B]
The formulation of rate laws2 NO(g) + O2(g) → 2 NO2(g) ν = kr[NO]2[O2]
Step 1. Two NO molecules combine to form a dimer:(a) NO +NO →N2O2
Rate of formation of N2O2 = ka[NO]2
Step 2. The N2O2 dimer decomposes into NO molecules:N2O2 → NO + NO,
Rate of decomposition of N2O2 = ka΄[N2O2]
Step 3. Alternatively, an O2 molecule collides with the dimer and results in the formation of NO2:
N2O2 + O2 → NO2 + NO2,Rate of consumption of N2O2 = kb[N2O2][O2]
The steady-state approxi-mation
Rate of formation of NO2 = 2kb[N2O2][O2]
2 NO(g) + O2(g) → 2 NO2(g)
Net rate of formation of N2O2 =ka[NO]2 − ka΄[N2O2] − kb[N2O2][O2] = 0
The steady-state approximation: ka[NO]2 − ka΄[N2O2] − kb[N2O2][O2] = 0
[N2O2] = ka[NO]2 /( ka΄+ kb[O2] )
Rate of formation of NO2 = 2kb[N2O2][O2]= 2kakb [NO]2 [O2]/( ka΄+ kb[O2] )
if ka΄[N2O2]>>kb[N2O2][O2]
Rate = 2kakb [NO]2 [O2]/( ka΄+ kb[O2] ) = (2kakb/ka΄)[NO]2 [O2] kr= (2kakb/ka΄)
The rate-determining step
The rate-determining step is the slowest step of a reaction and acts as a bottleneck.
The reaction profile for a mecha-nism in which the first step is rate determining.
Unimolecular Reaction and The Lindemann Mechanism
Rate of formation of A* = ka[A]2A + A → A* + A
Net rate of formation of A* =ka[A]2 − ka΄[A*] [A]− kb[A*]= 0
[A*] = ka[A]2 /( ka΄[A]+ kb)
Rate of formation of P = kb[A*]=kakb [A]2 /( ka΄[A]+ kb )
if ka΄[A]>>kb
Rate = kakb [A]2 /( ka΄[A]+ kb) = (kakb/ka΄)[A]
kr= (kakb/ka΄)
Rate of deactivation of A* = ka΄[A*][A]A + A* → A + A
Rate of formation of P = kb[A*]A* → PRate of consumption of A* = kb[A*]
Activation control and diffusion control
Rate of formation of AB = kr,d[A][B]A + B → AB
Rate of formation of P = kr[A][B]
kr= kr,a kr,d/(kr,a + kr,d΄)
Rate of loss of AB = kr,d΄ [AB]AB → A + B
Rate of reactive loss of AB = kr,a [AB]AB → P
kr= kr,di) kr,a >> kr,d΄
ii) kr,a << kr,d΄ kr= kr,a kr,d/ kr,d΄
diffusion-controlled limit
activation-controlled limit
kr,d = For a diffusion-controlled reaction in water, for which η = 8.9 × 10−4 kg m−1 s−1 at 25°C.
kr,d = 7.4 × 109 dm3 mol−1 s−1
kr,d =
¿8 × 8.3145 ×298 J mol−1
3 × 8.9× 10− 4 kg m− 1 s− 1
Diffusion
Table 11.1 Diffusion coefficients at 25°C, D/(10−9 m2 s−1)
The flux of solute particles is propor-tional to the concentration gradient.
J=− D dcdx
Fick’s first law of diffusion
Diffusion
𝜕c𝜕 t
=(J x − J x+dx )
d x=D 𝜕2 c
𝜕 x2
Fick’s second law of diffusion
Diffusion
D=λ2
2𝜏
2d Dt
D=D0 e− Ea /RT
D=kT
6 𝜋𝜂 a𝜂=𝜂0 eE a /RT
Einstein–Smoluchowski equation:
Diffusion
D=λ2
2𝜏
Einstein–Smoluchowski equation:
Suppose an H2O molecule moves through one molecular diameter (about 200 pm) each time it takes a step in a random walk. What is the time for each step at 25°C?
=8.85
Catalysis
A catalyst acts by providing a new reaction pathway between reactants and products, with a lower activation energy than the original pathway.
Michaelis-Menten kinetics
ES complex is a reaction intermediate
E + S ES → E + P k2→
→ k1
k-1
Mechanism
0[S])[ES]([S][E]
)[ES]([ES])[S][E]([ES][ES][E][E] [ES][E][E]
0)[ES]([E][S][ES]
[ES][P]rate initial
12101
2101
00
211
2
kkkk
kkkdt
d
kkkdt
d
kdt
d
Steady state approximation
Michaelis-Menten Kinetics
[S]1
[E][ES][P][S]
1
[E]
[S]1
[E][S])(
[S][E][ES]
[ES][P]
022
0
1
21
0
121
01
2
M
M
Kkk
dtdV
Kk
kkkkkk
kdt
drateinitial
Michaelis-Menten Kinetics
[S]1
[E]02
MKkV
Michaelis-Menten Kinetics
[S][S][E]
[S]
[E]~
[S]1
[E] 020202 kK
kK
kK
kVMMM
max0202
1[E]
[S]1
[E] VkK
kVM
i) When [S] → 0
ii) When [S] → ∞
0th Order Reaction!
1st Order Reaction
[S]1
max
MKVV
Michaelis-Menten Kinetics
Vmax approachedasymptotically
V is initial rate (near time zero)
Michaelis-Menten Equation
V varies with [S]
[S]1
max
MKVV
Determining initial rate (when [P] is low)Ignore the reverse reaction
slope=
Range of KM values
KM provides approximation of [S] in vivo for many enzymes
Lineweaver-Burk Plot
[S]111
maxmax VK
VVM
[S]1
max
MKVV
Allosteric enzyme kinetics
Sigmoidal dependence of V0 on [S], not Michaelis-Menten
Enzymes have multiple subunitsand multiple active sites
Substrate binding may be cooperative
Enzyme inhibition
Kinetics of competitive inhibitor
Increase [S] toovercomeinhibition
Vmax attainable,KM is increased
Ki =dissociationconstant forinhibitor
0)[ES]/[I]1
[S](/[I]1[S][E]
)[ES]()[S]/[I]1[ES][E]([ES]
/[I]1[ES][E][E] [EI] [ES][E][E]
[E][I][E][I][EI] 0[EI][E][I][EI]
0)[ES]([E][S][ES]
[ES][P]
I
121
I
01
21I
01
I
00
I'1
'1'
1'1
211
2
Kkkk
Kk
kkK
kdt
dK
Kkkkk
dtd
kkkdt
d
kdt
drate
Steady state approximation
Kinetics of competitive inhibitor
[S])/[I]1(1
[S])/[I]1(1
[E][ES]
[S])/[I]1(1
[E][S])/[I]1)((
[S][E][ES]
[ES][P]
I
max022
I
0
1I21
01
2
KKV
KKkkVrate
KKkKkkk
kdt
drate
MIM
M
Kinetics of competitive inhibitor
[S]1)/[I]1(11
max
I
max VKK
VVM
교차축과에서 y 11
maxVV
Competitive inhibitor
Slope: increasedmax
I )/[I]1(V
KKM
max
1V
maxVKM
Vmax unaltered
Kinetics of non-competitive inhibitor
Increasing [S] cannotovercome inhibition
Less E available,Vmax is lower,KM remains the samefor available E
)/[I] 1)(1[S]
()/[I] 1)(1[S]
(
[E]
)/[I] 1)(1[S]
(
[E][ES]
)}/[I](1)/[I] 1([S]
{ [ES]
)/[I](1 [ES] )/[I] 1[E](/[ES][I] /[E][I] [ES][E][E] 0)[ES]( [E][S])[I])[ES]/(( [E][S]
[I])[ES]([EIS][E][S][ES] /[ES][I][EIS]
/[E][I][EI]
[ES][P]
I
max
I
02
I
0
II
IIII0
211I211
211
I
I
2
KKV
KKkV
KK
KKKKKKK
kkkKkkkkk
kkkkkdt
dK
K
kdt
drate
MM
M
M
II
II
Steady state approximation
Kinetics of non-competitive inhibitor
Kinetics of non-competitive inhibitor
[S])/[I](1/[I]1
)/[I] 1)(1[S]
(1
)/[I] 1)(1[S]
()/[I] 1)(1[S]
(
[E]
max
I
max
I
max
I
I
max
I
02
M
M
MM
KV
KV
KV
KK
V
KKV
KKkV
[S]1
''11
maxmax VK
VVM
교차축과때일 x 1[S]1
MK
IKVV
/[I]1' maxmax
[S]111
maxmax VK
VVM
Cf. Michaelis-Menten Eqn
Noncompetitive inhibitor
K M u
nalte
red
Vmax decreased
Enzyme inhibition by DIPFGroup - specific reagents react with R groups of amino acids
diisopropylphosphofluoridate
DIPF (nerve gas) reacts with Ser in acetylcholinesterase
Enzyme inhibition by iodoacetamide
A group - specific inhibitor
Affinity inhibitor: covalent modification
Example 11.1Determining the catalytic efficiency of an
enzyme
The Lineweaver–Burke plot
CO2(g) + H2O(l) → HCO3-(aq) + H+(aq)
the hydration of CO2 in red blood cells
[CO2]/(mmol dm-
3)1.25 2.5 5 20
v/(mmol dm-3 s-1) 2.78 × 10-2
5.00 × 10-2
8.33 × 10-2
1.67 × 10-1
1/[CO2]/(mmol dm-3) 0.8 0.4 0.2 0.05
1/v/(mmol dm-3 s-1) 36.0 20.0 12.0 5.99
y = 40.0 x + 4.00=0.25 mmol dm-3 s-1
[S]111
maxmax VK
VVM
KM
V max=40.0
=10.0 mmol dm-3 s-1
at [E]0=2.3 nmol dm−3:
competitive inhibition
noncompetitive inhibition
Explosions
The explosion limits of the H2/O2 reaction. In the explosive regions the reaction proceeds explosively when heated homogeneously.
H2(g) + Br2(g) → 2 HBr(g)Step 1. Initiation: Br2 → Br· + Br·
Rate of consumption of Br2 = ka[Br2]
Step 2. Propagation:
Step 3. Retardation:
Step 4. Termination: Br· + ·Br + M → Br2 + MRate of formation of Br2 = ke[Br]2
H = kb[Br][H2] − kc[H][Br2] − kd[H][HBr] = 0Br = 2ka[Br2] − kb[Br][H2] + kc[H][Br2] + kd[H][HBr] − 2ke[Br]2 = 0
HBr = kb[Br][H2] + kc[H][Br2] − kd[H][HBr]Net rate of formation of HBr
Rate of formation of HBr1/2 3/2
b a c 2 2
2 d c
2 ( / ) [H ][Br ][Br ] ( / )[HBr]
k k kk k
H2(g) + Br2(g) → 2 HBr(g)
1. Calculate the mean activity coefficient γ ± of the solution.
2. Calculate solubility of Hg2(IO3)2 at this temperature in unit of mol dm-3.
Debye-Huckel Limiting Law is a valid approximation when strong electrolyte ions are in the solution at low concentration.
Consider the solubility of Hg2(IO3)2(Ks= 1.3×10-18 )in KCl( 0.05 M).
The DHLL for aqueous solution can be written asln γ ± = -1.173 |z+z-|I1/2.
Assume DHLL applies to this solutioon.
Vari-able Value
KCl 0.05I 0.05
γ ± 0.591803Ks 1.3E-18S 1.161762E-6
Hg2(IO3)2(s) Hg⇄ 22+ (aq) + 2 IO3
−(aq)
KCl(s) K⇄ + (aq) + Cl−(aq)
Ks= (aHg22+)(aIO3
-) 2= (γ±sHg22+)(γ±sIO3
-) 2
Ks=4 (γ±s)3= 1.3×10-18
s= 1.16×10-6
sHg22+=s; sIO3
- =2s
4 s3= Ks/(γ±)3
I= ½ (0.05 + 0.05 +22×[Hg22+]+ [IO3
−]) = 0.05 because ×[Hg2
2+],+[IO3−] << 0.05.
ln γ ± = -1.173 |2∙1| 0.051/2