35.10 Reversible Reactions and Equilibrium
** Reversible reaction:
Assumption:
[B]0 = 0
[B] = [A]0 – [A]
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.10 Reversible Reactions and Equilibrium
加了等於沒加
1
d ([A ](k A + k B ) − k B [ A]0 )
∫[ A ]0 (k A + k B ) ([A](k A + k B ) − k B [A]0 ) =
[A]
 [A](k A + k B ) − k B [A]0 
 = −(k A + k B )t
ln
k A [ A]0


Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.10 Reversible Reactions and Equilibrium
-- Apparent decay rate constant of A,
k f = k A + k B.
-- Apparent formation rate constant of B, kr = kA + kB.
= -kA[A]eq + kB[B]eq
-- Thermodynamics
ΔG = 0
-- Stat. Mech.
-- Kinetics: the ratio of forward and back rate
constants of a reaction.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.10 Reversible Reactions
and Equilibrium
Kc = 104
A= 1012 s-1
Ea = 42 kJ/mol
Example 35.9 P. 915
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
To simulate the
overall decay
35.11 Perturbation-Relaxation Methods
?? What if the initial concentration of the reactant
at a specified time can not be controlled?
Perturbation – relaxation method.
-- The perturbation time scale is rapid compared to the system relaxation.
-- Perturbing the system by changing temperature, pressure, or concentration.
-- System evolution to a new equilibrium state.
** Perturbation by temperature jump (T-jump):
T1
-- Evolution of [A] to [B] as a function of time until the
new equilibrium conditions are reached.
-- Under the new equilibrium conditions:
T2
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.11 Perturbation-Relaxation Methods
[A]eq , pre− jump − ξ 0 = [A]eq
[B]eq , pre− jump + ξ 0 = [B]eq
** At t = 0, immediately after the
T-jump:
[A]0 = [A]eq + ξ
[B]0 = [B]eq - ξ
ξ = ξ0
** At any given time t, after the
T-jump:
[A] = [A]eq + ξ
[B] = [B]eq - ξ
d [ A ] dξ
+
+
=
= − k A [A ] + k B [ B]
dt
dt
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.11 Perturbation-Relaxation Methods
d [ A ] dξ
+
+
=
= − k A [A ] + k B [ B]
dt
dt
=0
** Relaxation time:
**
+
A
+
B
K, (k + k )
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
+
A
k ,k
+
B
35.12 The Autoionization of Water: An T-Jump Example
** Autoionization of water:
1st order forward rxn
2nd order reverse rxn
-- Differential rate equations (1):
-- At 298 K, new equilibrium:
k f+ [H 2 O] = k r+ [H + ][OH − ]
-- Differential rate equations (2):
+
r
k ξ
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
2
A good approximation
under the initial rate
conditions
35.12 The Autoionization of Water: An T-Jump Example
** Autoionization of water:
-- Differential rate equations (3):
** Relaxation time = 37 μs :
(
+
f
+
r
(
+
−
37 µs = τ = k + k [H ]eq + [OH ]eq
** pH = 7:
** Solutions of the forward and reverse reactions:
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
))
−1
35.12 The Autoionization of Water: An T-Jump Example
交大-98甲組
交大 甲組,台大某年
甲組 台大某年
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
36.2 The Pre-equilibrium Approximation
** Pre-equilibrium conditions: Equilibrium among a subset of species is established
before product formation occurs.
37.2.1 General Solution
Approximation:
(1) Equilibrium between reactants and the intermediate is maintained during the
course of the reaction.
(2) The intermediate undergoes decay to form products.
-- The reaction is second-order overall and first-order wrt both reactants.
--
keff = kpkf/kr
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
36.2.2 A Pre-equilibrium Example
**
----
Balanced equation:
2nd-order of NO determined experimentally.
First-order in O2 determined experimentally.
Rate constant decreases as the temperature increases.
Not a simple
termolecular
reaction in one step.
** Reaction mechanism:
-- Pre-equilibrium:
-- Rate of product formation:
-- The reaction mechanisms predicts 2nd-order in [NO] and 1st-order in [O2].
-- The formation of N2O2 is exothermic.
Kc ↓ as the temperature ↑.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.13 Potential Energy Surfaces
?? How to connect the reaction kinetics to energies of the reaction?
-- the equivalence to the Arrhenius activation energy?
Potential energy surfaces
** A simple mode of a triatomic reaction: AB + C
A + BC
-- Intermediate ABC is formed.
-- ABC and AC are not products.
?? How to describe the course of the reaction?
The potential energy surfaces is a four-dimension function contains information
about RAB, RBC, θ, energy.
** Assume a AB + C reaction, of which the optimal reaction path is that
AB interacts with C co-linearly.
theta is 180°.
(An example reaction, not including all triatomic reactions.)
The potential energy surface (energy) is a function of RAB and RBC.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.13 Potential Energy Surfaces
** Potential energy surface (PES) of a co-linear AB + C
A + BC reaction:
Transition state (TS)
0
0
∞
A+ B +C
RBC = ∞
RAB = ∞
Fixed RBC
Varying RAB
Fixed RAB
Varying RBC
PES: The variation of the potential energy
with a change along the reaction coordinates.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.13 Potential Energy Surfaces
**
Contour plot
RAB = ∞
PES of RBC
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.13 Potential Energy Surfaces
** Reaction pathway (dashed line between points c and d) 反應沿著最低位能表面的反應座標(reaction coordinate)進行.
** Activated complex (an intermediate) – The maximum in energy along the
reaction pathway.
** Transition state – The maximum in energy (with the right mode of motion)
along the reaction pathway.
-- Studied by the molecular-beam technique.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.13 Potential Energy Surfaces
Cross beam
Molecular beam allows studies to study:
(1) Collisions between molecules in pre-selected
energy states.
Translational – rotational sectors and supersonic
expansion.
Vibrational – laser excitation
Orientation – electric field
(2) Determine the states of the products of a reactive
collision. (Spectroscopic means)
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Activated Complex Theory
** The complex is not stable and has a
lifetime on the order of one of a few
vibrational periods. (~10-14 s)
A+ B <==> AB‡
P
** Transition-state theory:
(1) An equilibrium exists between the
reactants and the activated complex.
(2) The reaction coordinate of the activated complex leading to the product can be
mapped onto a single energetic degree of freedom of the activated complex.
= The transition state along the reaction coordinate.
i.e. If product formation involves the breaking of a bond, then the vibrational
degree of freedom corresponding to bond stretching is taken to be the
reactive coordinate.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Activated Complex Theory
** The presence of equilibrium between reactants and the activated complex.
** The rate of reaction:
?? What is k2?
-- k2 is related to the vibrational frequency (ν) associated with the bond breakage.
-- Only a fraction of the activated complex at the transition state will continue along
the reaction coordinate and result in product formation.
k2 = κν
κ: transmisssion coefficient
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Activated Complex Theory
k2 = κν
aA + bB <==> cC + dD
c
** Decompose Kc‡ into two parts:
(1) Partition function corresponding to the vibrational motion along the reactive
coordinate:
qvib, rc = kBT/hν (現在不必知道why)
(2) The remaining energetic degrees of freedom:
k BT ‡
k =κ 0 Kc
hc
R=
κk BT
hc0
[A][B]
-- To evaluate
:The rotational and vibrational partition functions for the
complex requires knowledge of the structure of the activated complex.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
d
q  q 

 

NA   NA 

Kp =
a
0
0 b
 q A   qB 

 

 NA   NA 
0
C
0
D
35.14 Activated Complex Theory
** Connecting the results to thermodynamics:
Eyring equation
k BT ‡
k =κ 0 Kc
hc
k BT ∆S /R‡ − ∆H ‡/ RT
k= 0e e
hc
κ = 1
-- Pre-exponential term is temperature dependent.
-- Similar exponential dependence on temperature as that predicted by the
Arrhenius equation.
-- Arrhenius activation energy:
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Activated Complex Theory
** Arrhenius equation
Transition state results into the Arrhenius equation
(1)
(2)
 d ln K c

 dT

d ln K p
dT
‡

=?


[Chap. 6.10, P. 165
Variation of Kp with T]
[
d ln K c (RT )
=
dT
∆ν
] = d ln K
dT
c
 d ln R dT 
T
+ ∆υ 
+
dT 
 dT


=0
d ln K c ∆H
=
dT
RT
0
reaction
2
∆υRT ∆U
−
=
2
RT
RT
Ea = RT + ΔU‡
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
0
reaction
2
=1/T
35.14 Activated Complex Theory
** Ea = RT + ΔU‡
-- Thermodynamic definition: ΔU‡ = ΔH‡ - Δ(PV)‡
-- Approximation for a reaction taking places in a solution:
P is a constant, and ΔV is negligible
Ea = RT + ΔH‡
Eyring equation:
k BT ∆S /R‡ − ∆H ‡/ RT
k= 0e e
hc
** For a bimolecular reaction in solution:
k = Ae
− Ea / RT
k BTe ∆S /R −( RT + ∆H ‡)/RT
=
e
e
0
hc
** For a unimolecular reaction in solution:
ek BT ∆S+ +
A=
e
h
R
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
ek BT ∆S+ +
A=
e
o
hc
R
35.14 Activated Complex Theory
k = Ae-Ea/RT
(1) Arrhenius equation,
(1) From transition theory: Ea = RT + ΔU‡ = RT + ΔH‡ - Δ(PV)‡
k BT ∆S /R‡ − ∆H ‡/ RT
k= 0e e
hc
(2) For a gas-phase reaction, ΔU‡ = ΔH‡ - Δ(PV)‡
-- Unimolecular, A
A*:
-- Bimolecular, A + B
AB*:
-- Trimolecular, A+B+C
ABC*:
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Activated Complex Theory
-- Unimolecular, A
A*:
-- Bimolecular, A + B
AB*:
-- Trimolecular, A+B+C
ABC*:
** Both the Arrhenius activation energy and pre-exponential terms are
temperature dependent.
** When ΔH‡ >> RT, the temperature dependency of Ea will be modest.
** When ΔH‡ < 0, the reaction may become faster as temperature is decreased!
(negative temperature dependency)
** The entropy difference, ΔS, is important in determining the rate.
-- When ΔH‡ ~ 0, the reaction rate is determined by entropic rather than
enthalpic factors.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Activated Complex Theory
Ea = RT + ΔU‡ = RT + ΔH‡ - Δ(PV)‡
The TS has a lower entropy than the reactants.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.15 Diffusion Controlled Reactions
kd
A + B →
AB
kr
AB →
A+ B
kp
AB → P
* Steady-state approximation:
d[AB]/dt = k d [A][B] − k r [AB] − k p [AB] = 0
k pk d
k d [ A][ B ]
, Rate =
[ A][ B ]
kr + k p
kr + kp
** Diffusion controlled limit: kp >> kr
[ AB ] =
-- Diffusion for the reactants limits the rate of
product formation
The solution-phase reaction
undergo diffusion in solution
until they encounter each
other.
Rate = kd[A][B]
** Activation controlled limit: kp << kd
(kinetic controlled)
k pkd
Rate =
[ A][ B ]
kr
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.14 Diffusion Controlled Reactions
kd
A + B →
AB
kr
AB →
A+ B
kp
AB → P
Rate =
k pkd
kr + kp
[ A][ B]
** The rate constant for diffusion: kd = 4πNA(rA + rB)DAB
** Diffusion constant (Stokes-Einstein equation): D = kBT/6πηr and DAB = (DA + DB)
Example 1: CH3COO- protonation (aq) Example 2: Hemoglobin-O2 binding rate
DCH3COO- = 1.1E-5 cm2s-1
DH+ = 9.3E-5 cm2s-1
(rA + rB) = 0.5 nm
kd ~ 3.9x1010 M-1s-1
< Neutralization rate
(kf ~1011 M-1s-1)
`∵ Coulombic attraction
At 25°C, pH = 7.4
DHb = 7.6E-7 cm2s-1
DO2 = 2.2E-5 cm2s-1
(rA + rB) = (3.5 + 0.2) nm
Experimentally, k ~ 4x107 M-1s-1
Calculated kd ~ 6.4x1010 M-1s-1
The reaction is not diffusion controlled.
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung
35.3.2
Determining Reaction Orders
補充
Measurement techniques
** Time scale
Slow - Chemical method
Fast - Physical method
10-3 s
10-6 s
10-12, 10-15 s
Femtosecond chemistry
-- Sensitivity
e. g. Absorption vs fluorescence
Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung