Chemical Kinetics Lecture Notes 332x5d
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Introduction of Chemical Kinetics
Properties of Umass Boston
Scope of Chemical Kinetics Chemical Kinetics deals with the rates of chemical reactions and with how the rates depend on factors such as concentration and temperature. Such studies are important in providing essential evidence as to the mechanisms of chemical processes
Properties of Umass Boston
Macroscopic and Microscopic Kinetics • Macroscopic kinetics describes the branch of kinetics, which results relate to the behavior of a very large group of molecules in thermal equilibrium. • Microscopic kinetics is to investigate the molecules in well-defined states, which will provide information about the dynamic of both reactive and unreactive collisions. (crossed molecular beams). Properties of Umass Boston
Kinetics covers many areas
Properties of Umass Boston
Definition of rate For reaction
A + 3B → 2 Z
d [ A] d [ B] νB = dt dt d[Z ] rate of formation ν Z = dt considering Volume change during the reaction 1 d [ B] 1 dnB 1 d [ B] [ B] dV ν nB = [ B] × V ; dnB = Vd [ B] + [ B]dV ν = B = = = + 3V dt 3 dt 3V dt − 3 3 dt rate of
consumption ν A =
rate of
reaction ν =
νA
=
νB
−1 − 3
=
νZ 2
Δn A ΔnB ΔnZ Δn = = = 2 −1 −3 γ 1 dξ rate of reaction ν = rate of conversion V dt Empirical rate equation Extent of
ν = k[ A]n [ B]m
reaction ξ =
dξ dt
reaction order
Properties of Umass Boston
Analysis of results-differential • Order with respect to concentration, true order. • Order with respect to time For reaction A → Z dC ν = − A = kC A n ; lnν = ln k + n ln C A dt However , the reaction may be reversible
ν=
k1
A⇔ Z k −1
dC Z = k1 (C A0 − C Z ) − k −1C Z dt
Properties of Umass Boston
Analysis of results-integration First − order reactions dx = k (Ca0 − x); dt int egration
A→Z
dx = kdt 0 Ca − x
dx ∫ Ca0 − x = ∫ kdt
Ca0 ln( 0 ) = kt Ca − x
with int itial condition t = 0 x = 0
or ln(Ca0 − x) = ln Ca0 + kt
Properties of Umass Boston
Half Life • Half-life (t1/2): the time required for the concentration of a particular reactant to reach a value that is half-way between its initial and final values.
For
first order reaction C ln( ) = kt1/ 2 C −C /2 ln 2 t1/ 2 = kA For sec ond order reaction 1 t1/ 2 = kC 0 For third order reaction 3 t1/ 2 = 2k (C 0 ) 2
Properties of Umass Boston
Second-order reaction Second − order reactions 2 A → Z dx dx = k (Ca0 − x) 2 ; = kdt 0 2 (Ca − x) dt int egration
dx ∫ (Ca0 − x) 2 = ∫ kdt
1 1 − = kt 0 0 Ca − x Ca
or
with int itial condition t = 0 x = 0
x = kt x x Ca (Ca − x)
Properties of Umass Boston
Properties of Umass Boston
Measurement methods-Flow system • Static system Vs flow system • Flow system
Properties of Umass Boston
Stop flow for fast reaction
Properties of Umass Boston
Relaxation method •
The method differs from conventional kinetic methods, the system is initially at equilibrium, when the condition suddenly changes the system will become non-equilibrium and relaxes to new equilibrium.
For let
k1
first − order a0 = sum of
A⇔ Z
reaction
k −1
concentration of
x = concentration of
Z
dx = k1 (a0 − x) − k −1 x dt at final equilibrum
xe = [ Z ]e
A Z
k1 (a0 − xe ) − k −1 xe = 0 dΔx d ( x − xe ) dx = = = k1 (a0 − x) − k −1 x dt dt dt combine the last two equations dΔx = k1 ( xe − x) − k −1 ( xe − x) = −(k1 + k −1 )Δx dt int egration with conditon when t = 0 Δx = (Δx) 0 ln[
(Δx) 0 ] = ( k k + k −1 )t Δx (Δx) 0 = e called Δx 1 τ= k1 + k −1
The time correspondes to ln[
(Δx) 0 ] = 1 = ( k k + k −1 )τ ; Δx
at equilibrium K =
k1 ; both k1 , k 2 k −1
Properties of Umass Boston
relaxation time τ
can be det er min ed
Shock tube method (t1/2=10-3-10-6) Diaphragm Driver gas with high pressure
Mixture of reactants at low pressure (about 0.001 atm)
The rupture of the membrane will create a narrow shock front travels along the tube at supersonic speed. As the shock front es through each element of volume there is considerable temperature rise to 1000-10000 K at 1μs. Properties of Umass Boston
Flash photolysis (t1/2=10-6)
Properties of Umass Boston
Properties of Umass Boston
Temperature Dependence for k k1
For reaction at equilibrium
A+ B⇔ Y + Z k −1
∂ ln K c ΔU 0 ( )p = (?) ∂T RT 2 k1[ A][ B ] = k −1[Y ][ Z ];
(
[Y ][ Z ] k = Kc = 1 [ A][ B] k −1
d ln k1 d ln k −1 ΔU 0 = = dT dT RT 2 d ln k1 E d ln k −1 E−1 Then ; = 12 ; = dT RT dT RT 2 so k = Ae − E / RT van' t
Kc
equilibrum cons tan t
while E1 − E−1 = ΔU 0
Hoff ' s equation
Y+Z U ΔU A+B
Properties of Umass Boston
Arrhenius Equation •
Arrhenius pointed out that the magnitudes of the temperature effects on rates are usually much too large to be explicable on the basis of how temperature affects the molecular translational energies.
Transition species or
A + B → A ⋅ ⋅ ⋅B * → Y + Z
A + B is at equilibrium with
[ A ⋅ ⋅B * ] ∝ [ A][ B]e E1 / RT ;
and
the reaction rate ν ∝ [ A ⋅ ⋅ ⋅ B * ] A····B*
U
A ⋅ ⋅ ⋅ ⋅B *
E-1
Y+Z
E1 ΔU A+B
Properties of Umass Boston
Arrhenius Equation • •
Arrhenius concluded that an equilibrium is established between normal and active reactant molecules and that this equilibrium shifts in the manner predicted by van’t Hoff’s equation. The Arrhenius equation was accepted because it provides an insight into how reactions proceed. k = Ae − Ea / RT or ln K = ln A −
Ea RT
E ∂ ln K =− a R ∂( 1 ) T where
or
A : pre − exp onential factor Ea : activation energy
Properties of Umass Boston
Arrhenius parameters k = Ae − Ea / RT Ea RT A : pre − exp onential factor Ea : activation energy
or ln K = ln A −
Properties of Umass Boston
ing for the rate laws reaction mechanism • Elementary reaction: reactant molecules collide and exchange energy, atoms, or groups of atoms, or undergo other kind of change, and forming product molecules. H+Br2→HBr+Br, d[Br2]/dt = k[H][Br2] • Molecularity: the number of molecules coming together to react in an elementary reaction. For the above reaction molecularity is 2. • Reaction order and molecularity: reaction order is an empirical quantity, and obrained from the experimental rate law; molecularity refers to an elementary reactoin proposed as individual step in a mechanism. One reaction may involve several elementary reactions.
Properties of Umass Boston
Consecutive elementary reactions ka kb A ⎯⎯→ I ⎯⎯→ P d [ A] = − k a [ A] ⇒ dt
t d [ A] ∫[ A]0 [ A] = −∫0 ka dt [ A]
[ A] = [ A]0 e − kat d[I ] = k a [ A] − kb [ I ] ⇒ dt solving the first − order [I ] =
d[I ] + kb [ I ] = k a [ A]0 e − kat dt differential equation [ I ]0 = 0
ka (e − kat − e − kbt )[ A]0 kb − k a
and [ A] + [ I ] + [ P] = [ A]0
k a e − kb t − kb e − k a t [ P] = {1 + }[ A]0 kb − k a d [ P] = kb [ I ] dt Properties of Umass Boston
Properties of Umass Boston
Rate determine step
Properties of Umass Boston
Steady-state approximation ka kb A ⎯⎯→ I ⎯⎯→ P k a << kb
d[ I ] so k a [ A] − kb [ I ] ≈ 0 ≈0 dt k [ I ] ≈ ( a )[ A] kb d [ P] k = kb [ I ] ≈ kb ( a )[ A] = k a [ A] kb dt [ A] = [ A]0 e − k at d [ P] = k a [ A]0 e − k at dt [P]
∫
0
t
d [ P] = k a [ A]0 ∫ e − ka t dt 0
[ P] = (1 − e − k at )[ A]0
Properties of Umass Boston
Pre-equilibria ka
kb A + B ⇔ I ⎯⎯→ P k a'
when k a' >> kb
but
not
when kb >> k a'
ka [I ] = ' K= [ A][ B] k a d [ P] = kb [ I ] = kb K [ A][ B] = k[ A][ B] dt k a kb where k = Kkb = ' ka Properties of Umass Boston
Example 7 overall reaction 2 N 2O5 ( g ) → 4 NO2 ( g ) + O2 ( g ) mechanism ka
N 2O5 ⇔ NO2 + NO5 k a'
NO2 + NO3 → NO2 + O2 + NO
kb
NO + N 2O5 → NO2 + NO2 + NO2 NO and
NO3
kc
are int ermediates
d [ NO ] = kb [ NO2 ][ NO3 ] − kc [ NO ][ N 2O5 ] ≈ 0 dt d [ NO3 ] = k a [ N 2O5 ] − k a' [ NO2 ][ NO3 ] − kb [ NO2 ][ NO3 ] ≈ 0 dt d [ N 2O5 ] = − k a [ N 2O5 ] + k a' [ NO2 ][ NO3 ] − kc [ NO ][ N 2O5 ] dt 2k k [ N O ] = − a 'b 2 5 k a + kb
Properties of Umass Boston
Properties of Umass Boston
Scope of Chemical Kinetics Chemical Kinetics deals with the rates of chemical reactions and with how the rates depend on factors such as concentration and temperature. Such studies are important in providing essential evidence as to the mechanisms of chemical processes
Properties of Umass Boston
Macroscopic and Microscopic Kinetics • Macroscopic kinetics describes the branch of kinetics, which results relate to the behavior of a very large group of molecules in thermal equilibrium. • Microscopic kinetics is to investigate the molecules in well-defined states, which will provide information about the dynamic of both reactive and unreactive collisions. (crossed molecular beams). Properties of Umass Boston
Kinetics covers many areas
Properties of Umass Boston
Definition of rate For reaction
A + 3B → 2 Z
d [ A] d [ B] νB = dt dt d[Z ] rate of formation ν Z = dt considering Volume change during the reaction 1 d [ B] 1 dnB 1 d [ B] [ B] dV ν nB = [ B] × V ; dnB = Vd [ B] + [ B]dV ν = B = = = + 3V dt 3 dt 3V dt − 3 3 dt rate of
consumption ν A =
rate of
reaction ν =
νA
=
νB
−1 − 3
=
νZ 2
Δn A ΔnB ΔnZ Δn = = = 2 −1 −3 γ 1 dξ rate of reaction ν = rate of conversion V dt Empirical rate equation Extent of
ν = k[ A]n [ B]m
reaction ξ =
dξ dt
reaction order
Properties of Umass Boston
Analysis of results-differential • Order with respect to concentration, true order. • Order with respect to time For reaction A → Z dC ν = − A = kC A n ; lnν = ln k + n ln C A dt However , the reaction may be reversible
ν=
k1
A⇔ Z k −1
dC Z = k1 (C A0 − C Z ) − k −1C Z dt
Properties of Umass Boston
Analysis of results-integration First − order reactions dx = k (Ca0 − x); dt int egration
A→Z
dx = kdt 0 Ca − x
dx ∫ Ca0 − x = ∫ kdt
Ca0 ln( 0 ) = kt Ca − x
with int itial condition t = 0 x = 0
or ln(Ca0 − x) = ln Ca0 + kt
Properties of Umass Boston
Half Life • Half-life (t1/2): the time required for the concentration of a particular reactant to reach a value that is half-way between its initial and final values.
For
first order reaction C ln( ) = kt1/ 2 C −C /2 ln 2 t1/ 2 = kA For sec ond order reaction 1 t1/ 2 = kC 0 For third order reaction 3 t1/ 2 = 2k (C 0 ) 2
Properties of Umass Boston
Second-order reaction Second − order reactions 2 A → Z dx dx = k (Ca0 − x) 2 ; = kdt 0 2 (Ca − x) dt int egration
dx ∫ (Ca0 − x) 2 = ∫ kdt
1 1 − = kt 0 0 Ca − x Ca
or
with int itial condition t = 0 x = 0
x = kt x x Ca (Ca − x)
Properties of Umass Boston
Properties of Umass Boston
Measurement methods-Flow system • Static system Vs flow system • Flow system
Properties of Umass Boston
Stop flow for fast reaction
Properties of Umass Boston
Relaxation method •
The method differs from conventional kinetic methods, the system is initially at equilibrium, when the condition suddenly changes the system will become non-equilibrium and relaxes to new equilibrium.
For let
k1
first − order a0 = sum of
A⇔ Z
reaction
k −1
concentration of
x = concentration of
Z
dx = k1 (a0 − x) − k −1 x dt at final equilibrum
xe = [ Z ]e
A Z
k1 (a0 − xe ) − k −1 xe = 0 dΔx d ( x − xe ) dx = = = k1 (a0 − x) − k −1 x dt dt dt combine the last two equations dΔx = k1 ( xe − x) − k −1 ( xe − x) = −(k1 + k −1 )Δx dt int egration with conditon when t = 0 Δx = (Δx) 0 ln[
(Δx) 0 ] = ( k k + k −1 )t Δx (Δx) 0 = e called Δx 1 τ= k1 + k −1
The time correspondes to ln[
(Δx) 0 ] = 1 = ( k k + k −1 )τ ; Δx
at equilibrium K =
k1 ; both k1 , k 2 k −1
Properties of Umass Boston
relaxation time τ
can be det er min ed
Shock tube method (t1/2=10-3-10-6) Diaphragm Driver gas with high pressure
Mixture of reactants at low pressure (about 0.001 atm)
The rupture of the membrane will create a narrow shock front travels along the tube at supersonic speed. As the shock front es through each element of volume there is considerable temperature rise to 1000-10000 K at 1μs. Properties of Umass Boston
Flash photolysis (t1/2=10-6)
Properties of Umass Boston
Properties of Umass Boston
Temperature Dependence for k k1
For reaction at equilibrium
A+ B⇔ Y + Z k −1
∂ ln K c ΔU 0 ( )p = (?) ∂T RT 2 k1[ A][ B ] = k −1[Y ][ Z ];
(
[Y ][ Z ] k = Kc = 1 [ A][ B] k −1
d ln k1 d ln k −1 ΔU 0 = = dT dT RT 2 d ln k1 E d ln k −1 E−1 Then ; = 12 ; = dT RT dT RT 2 so k = Ae − E / RT van' t
Kc
equilibrum cons tan t
while E1 − E−1 = ΔU 0
Hoff ' s equation
Y+Z U ΔU A+B
Properties of Umass Boston
Arrhenius Equation •
Arrhenius pointed out that the magnitudes of the temperature effects on rates are usually much too large to be explicable on the basis of how temperature affects the molecular translational energies.
Transition species or
A + B → A ⋅ ⋅ ⋅B * → Y + Z
A + B is at equilibrium with
[ A ⋅ ⋅B * ] ∝ [ A][ B]e E1 / RT ;
and
the reaction rate ν ∝ [ A ⋅ ⋅ ⋅ B * ] A····B*
U
A ⋅ ⋅ ⋅ ⋅B *
E-1
Y+Z
E1 ΔU A+B
Properties of Umass Boston
Arrhenius Equation • •
Arrhenius concluded that an equilibrium is established between normal and active reactant molecules and that this equilibrium shifts in the manner predicted by van’t Hoff’s equation. The Arrhenius equation was accepted because it provides an insight into how reactions proceed. k = Ae − Ea / RT or ln K = ln A −
Ea RT
E ∂ ln K =− a R ∂( 1 ) T where
or
A : pre − exp onential factor Ea : activation energy
Properties of Umass Boston
Arrhenius parameters k = Ae − Ea / RT Ea RT A : pre − exp onential factor Ea : activation energy
or ln K = ln A −
Properties of Umass Boston
ing for the rate laws reaction mechanism • Elementary reaction: reactant molecules collide and exchange energy, atoms, or groups of atoms, or undergo other kind of change, and forming product molecules. H+Br2→HBr+Br, d[Br2]/dt = k[H][Br2] • Molecularity: the number of molecules coming together to react in an elementary reaction. For the above reaction molecularity is 2. • Reaction order and molecularity: reaction order is an empirical quantity, and obrained from the experimental rate law; molecularity refers to an elementary reactoin proposed as individual step in a mechanism. One reaction may involve several elementary reactions.
Properties of Umass Boston
Consecutive elementary reactions ka kb A ⎯⎯→ I ⎯⎯→ P d [ A] = − k a [ A] ⇒ dt
t d [ A] ∫[ A]0 [ A] = −∫0 ka dt [ A]
[ A] = [ A]0 e − kat d[I ] = k a [ A] − kb [ I ] ⇒ dt solving the first − order [I ] =
d[I ] + kb [ I ] = k a [ A]0 e − kat dt differential equation [ I ]0 = 0
ka (e − kat − e − kbt )[ A]0 kb − k a
and [ A] + [ I ] + [ P] = [ A]0
k a e − kb t − kb e − k a t [ P] = {1 + }[ A]0 kb − k a d [ P] = kb [ I ] dt Properties of Umass Boston
Properties of Umass Boston
Rate determine step
Properties of Umass Boston
Steady-state approximation ka kb A ⎯⎯→ I ⎯⎯→ P k a << kb
d[ I ] so k a [ A] − kb [ I ] ≈ 0 ≈0 dt k [ I ] ≈ ( a )[ A] kb d [ P] k = kb [ I ] ≈ kb ( a )[ A] = k a [ A] kb dt [ A] = [ A]0 e − k at d [ P] = k a [ A]0 e − k at dt [P]
∫
0
t
d [ P] = k a [ A]0 ∫ e − ka t dt 0
[ P] = (1 − e − k at )[ A]0
Properties of Umass Boston
Pre-equilibria ka
kb A + B ⇔ I ⎯⎯→ P k a'
when k a' >> kb
but
not
when kb >> k a'
ka [I ] = ' K= [ A][ B] k a d [ P] = kb [ I ] = kb K [ A][ B] = k[ A][ B] dt k a kb where k = Kkb = ' ka Properties of Umass Boston
Example 7 overall reaction 2 N 2O5 ( g ) → 4 NO2 ( g ) + O2 ( g ) mechanism ka
N 2O5 ⇔ NO2 + NO5 k a'
NO2 + NO3 → NO2 + O2 + NO
kb
NO + N 2O5 → NO2 + NO2 + NO2 NO and
NO3
kc
are int ermediates
d [ NO ] = kb [ NO2 ][ NO3 ] − kc [ NO ][ N 2O5 ] ≈ 0 dt d [ NO3 ] = k a [ N 2O5 ] − k a' [ NO2 ][ NO3 ] − kb [ NO2 ][ NO3 ] ≈ 0 dt d [ N 2O5 ] = − k a [ N 2O5 ] + k a' [ NO2 ][ NO3 ] − kc [ NO ][ N 2O5 ] dt 2k k [ N O ] = − a 'b 2 5 k a + kb
Properties of Umass Boston
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