chemical kinetics: accounting for the rate laws

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


[S]1

[E]02

MKkV


[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


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



[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


[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




[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

chemical kinetics: accounting for the rate laws

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