21.4 reactions approaching equilibrium -...
TRANSCRIPT
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21.4 Reactions approaching equilibrium(a) First-order reactions close to equilibrium
(2015) Chemical Kinetics by M Lim 1
1
1 k
kA B
−
→←[ ] [ ] [ ]
[ ] [ ]( )1 1
1 1
1 10
1 1
+
−
−
−−
−
+
= −
+=
+
k k t
d Ak A k B
dtk k eA A
k k
[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ]( )
( )[ ] [ ]
[ ][ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ]
0 0
1 1 1 1 0
1 1 1 0
11 10
1 1
10
1 1
11 10 0
1 1
10
1 1
1
1 1
If 0, at all times
+ +
+
1+
+
1 + ln+
+
+
+
B A B A
d Ak A k B k A k A A
dtk k A k A
d Adtkk k A A
k k
kA Ak k tkk k A Ak k
kA Ak kk A
k k
− −
− −
−−
−
−
−
−−
−
−
−
−
= + =
= − = − −
= − +
− = − −
− = −
−[ ][ ] ( )
[ ][ ]
( )( )
1 1
1 11 1
1
1 10
10
1 1
1 1 1 1
1 1 1 1 1 10
+
+
+ + +
k k t
k k tk k t
A kA k k
ekk k
A k k k e keA k k k k k k
−
−−
−
− +−
−
− +− + − −
− − −
−= =
+= + =
[ ] [ ]
[ ] [ ] [ ]
[ ]
[ ][ ]
10
1 1
0
1
1
1
01
1
−∞
−
∞
−
∞
−
∞
∞
=+
= −
=+
≡ =B kKA
kA Ak k
B A A
k A
k
k k
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21.4 Reactions approaching equilibrium(a) First-order reactions close to equilibrium
(2015) Chemical Kinetics by M Lim 2
[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ]( )
( )[ ] [ ]
[ ][ ] [ ]
[ ] [ ]
[ ] [ ]( )
[ ] [ ]
[ ] [ ]
[ ][ ]
0 0
1 1 1 1 0
1 1 1 0
10
1 1
10
1 11 1
10 0
1 1
10
01 1
10 0
1 1
If 0, at all times
+ +
+
+
+ ln +
+
+
+
B A B A
d Ak A k B k A k A A
dtk k A k A
d AdtkA A
k k
kA Ak k k k tkA Ak k
AkA A Ak kkA A
k k
− −
− −
−
−
−
−−
−
−
−
−
−
−
= + =
= − = − −
= − +
= − −
= − −
−−=
−
( )
[ ][ ]
( )( )
1 1
1 11 1
1
1 1
1
1 1
1 1 1 1
1 1 1 1 1 10
+
+
+ + +
k k t
k k tk k t
kk k
ekk k
A k k k e keA k k k k k k
−
−−
−
− +−
−
− +− + − −
− − −
=
+= + =
( ) ( )
( )( )
( )
[ ]
0
0
0
0
0
1 1
10
1 1
ln ln
ln
+
at
at
at
dx adtx b
x b x b adtx b atx b
x b ex bx b x b e
x b x b e
a k kkb A
k k
−
−
−
−
−
= −−
− − − = −
−= −
−−
=−
− = −
= + −
= +
=
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21.4 Reactions approaching equilibrium(a) First-order reactions close to equilibrium
(2015) Chemical Kinetics by M Lim 3
1
1 k
kA B
−
→←[ ] [ ] [ ]
[ ] [ ] [ ]
[ ][ ]
1 1
1 1
1
1
+
At equilibrium
+ 0
eq eq
eq
eq
d Ak A k B
dt
d Ak A k B
dtB k KA k
−
−
−
= −
= − =
∴ = ≡
[ ] [ ]
[ ] [ ] [ ]
[ ]
[ ][ ]
10
1 1
0
10
1 1
1
1
kA Ak k
B A Ak A
k kB kKA k
−∞
−
∞ ∞
−
∞
−∞
=+
= −
=+
≡ =
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(2015) Chemical Kinetics by M Lim 4
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ]
,
a
a
b
b
k ak
a
k bk
b
a b
a b
C D kex A B C DA B k
E F kC E FC k
D E F k kA B D E FA B k k
−
−
−
−
− −
→+ + =←
→ + =←
+ + + =
A reaction taking place in a sequence of stepsKoverall = a product of Ks for each step
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(b) Relaxation methods for fast reaction
(2015) Chemical Kinetics by M Lim 5
Return of a system to equilibrium
1
1 k
kA B
−
→←Suddenly change K: T jump
p jump2
ln ord K HdT RT
∆=
( )1 1
10
1
, relaxation time: 1 k k tk
x x ek
τ−+−−
=+
=[ ] [ ] [ ] [ ]
[ ] [ ] [ ]( )
[ ]( ) [ ]( )( ) [ ] [ ][ ] [ ]{ }
[ ][ ]
( )( )1 1
1 1
1 1
1 1 1 1
1 1
1
1
1 1
0
+
=
+
0
+
e e
e e
e e
e e
e
e
k k t
x A A B B
d Adx k A k Bdt dt
k A x k B x
k k x k A k B
k A k B
BkKk A
dx k k xdt
x x e −
−
−
− −
−
−
−
− +
≡ − = −
= − = − −
− − +
= − + −
− =
= =
= −
∴ =
[ ] [ ]( )
[ ] [ ]( ) [ ] [ ]( )
1
1
f
r
f
r
k
k
f r e e
k
k
f re e e e
A B C
k k B C
A B C D
k A B k C D
τ
τ
→ +←
= + +
→+ +←
= + + +
x
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(b) Relaxation methods for fast reaction
(2015) Chemical Kinetics by M Lim 6
Ex 21.4 When Kw=1.008×10−14 at 298K, τ=37 µs and pH = 7.What is kf and kr for H2O ↔ H+ + OH− ?
( ) ( ) ( )
[ ]
( ) ( ) ( )
( )
1
2
1
2 1 2
1611
1 1
22 2
1 2 2 2 2
112
51 2
1O ,
1.81 10 , O 55.6
1 2
1 1.4 102
2.4 10
k
e ek
we e
e
w w we e
w
H H aq O
M M s
H aq k k H OH
H OHk K K k k Kk H
k k H OH k K k K K k K K
k M sK K
k k K
τ
τ
τ
+ − + −
+ −−
+ −
−
−
− −
→ + = + +←
= = ≡ = × =
= + + = + + = +
= = ×+
= = × 1s−
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(2015) Chemical Kinetics by M Lim 7
21.5 The T dependence of reaction rate• Arrhenius equation (an empirical observation)
aE RTk Ae−= ln ln aEk A RT= −
• Non-Arrhenius: Ea(T) (we can still define Ea)
( )2 ln ln
1ad k d kE RT RdT d T
≡ = −
( )2
ln ln
ln ln or 1
a
a a
Ek ART
d k E d k EdT RT d T R
= −
= = −
High Ea: stronger T-dependence of k
min
: pre-exponential factor (frequency factor): activation energy ( that s must have to form s)a
AE E R P
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(2015) Chemical Kinetics by M Lim 8
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(2015) Chemical Kinetics by M Lim 9
21.5(b) The interpretation of the parameters
aE RTk Ae−=Activation complex
(transition state)
reaction coordinate
Ex 21.5 Determining theArrhenius parameters.
Slope = −Ea/R = −22.7Intercept = ln A = 27.7
( )1 1 31
27.7 12 1 1
22.7 8.314 10
189 1.1 10
aE JK mol K
kJ molA e M s
− −
−
− −
= × ×
=
= = ×
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(2015) Chemical Kinetics by M Lim 10
Review 21-21
1 k
kA B
−
→←
( )1 1
1
1
01 1
1, relaxation time: k k t
kKk
x x ek k
τ−−
− +
−
=
= =+
aE RTk Ae−=
( )2 ln ln
1ad k d kE RT RdT d T
≡ = −
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(2015) Chemical Kinetics by M Lim 11
21.6 Elementary reactions
Molecularity: # of molecules coming together to react in elementary reaction
Unimolecular (1, 1st-order) ex, cyclopropane → propane
Bimolecular (2, 2nd order)ex, CH3I + CH3CH2O- → CH3CH2OCH3 + I-
Termolecular (3, third-order; rare)
(Real reactions at the molecular level)
Elementary bimolecular → 2nd-order kinetics2nd-order kinetics → the reaction might be complex
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(2015) Chemical Kinetics by M Lim 12
21.7 Consecutive elementary reactions a bk kA I P→ →
[ ] [ ]
[ ] [ ] [ ]
[ ] [ ]
a
a b
b
d Ak A
dtd I
k A k Idt
d Pk I
dt
= −
= −
=
[ ] [ ][ ] [ ] [ ] [ ]
[ ] [ ]
[ ] [ ] ( )[ ] [ ] [ ] [ ]
[ ]
0 0
0
0
0
0
0
If 0
1
a
a b
b a
k t
a k t k t
b a
k t k ta b
b a
I P
P I A A
A A e
k AI e e
k kP A I A
k e k eAk k
−
− −
− −
= =
+ + =
=
= −−
= − −
−= + −
10a bk k=
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(2015) Chemical Kinetics by M Lim 13
21.7 Consecutive elementary reactions a bk kA I P→ →
10a bk k=
Ex 21.6 At what time will I be present in the greatest concentration?
[ ] [ ] ( )[ ] [ ] ( )
( )
0
0
max
0
1 ln
− −
− −
− −
−
= −−
= − − =−
=
=
∴ =−
a b
a b
a b
a b
a k t k t
b a
a k t k ta b
b ak t k t
a b
k k ta
b
a
a b b
k AI e e
k kk Ad I
k e k edt k k
k e k ek ek
ktk k k
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(2015) Chemical Kinetics by M Lim 14
21.7(b) The steady-state approximation
[ ] [ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ]
[ ][ ] ( )[ ]
0
0
0
0
1
a
a
a
a b
a
b
k ta
b
b a
k ta
k t
d Ik A k I
dtkI Ak
kI A ek
d Pk I k A
dtk A e
P e A
−
−
−
= − ≈
∴ =
≈
= =
=
≈ −
[ ] 0 , a bd I
eg k kdt
≈
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(2015) Chemical Kinetics by M Lim 15
[ ] [ ] ( )[ ] ( )
[ ] [ ]
[ ]
[ ] ( )
0
0
0
0
0
If ,
,
0
1
1
1
b a
a b
a
b a
a
a
b ak t k t
b a b
a k t k t
b a
a k t
b
k t k ta b
b a
k tb
b
k t
k k
e e k k kk A
I e ek kk A
ek
k e k eP Ak k
k eAk
A e
− −
− −
−
− −
−
−
>>
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(2015) Chemical Kinetics by M Lim 16
21.7(d) Pre-equilibria a b
a
k k
kA B I P
−
→+ →←
[ ][ ][ ]
[ ] [ ] [ ][ ]
[ ][ ]
When ,
aa b
a
b b
a b
a
I kk k KA B k
d Pk I k K A B
dtk kk A B kk
−−
−
>> = =
= =
= ≡
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(2015) Chemical Kinetics by M Lim 17
a ba
k k
kA B I P
−
→+ →←
[ ] [ ][ ] [ ] [ ]
[ ] [ ][ ]
[ ] [ ] [ ][ ] [ ][ ]
0
When ,
−
−
−
−
−−
= − − ≈
≈+
= = ≡+
=+
>> ≡
a a b
a
a b
b ab
a b
b ar
a b
a ba b
a
d Ik A B k I k I
dtk A B
Ik k
d P k k A Bk I k A B
dt k kk kk
k kk kk k kk
Ex 21.8 Without ignoring the kb.
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(2015) Chemical Kinetics by M Lim 18
21.7(e) Kinetic and thermodynamic control of reaction
[ ][ ][ ][ ]
[ ][ ]
1
2
1
2
1 1
2 2
1 1
2 2
If
Rate of formation of
Rate of formation of
k
k
A B PA B P
P k A B
P k A B
P kP k
+ →
+ →
=
=
∴ =
Kinetic control: (before it reaches equilibrium)
Thermodynamic control: (when it is allowed to reach equilibrium)the ratio is determined by the standard Gibbs energies
21.4 Reactions approaching equilibrium�(a) First-order reactions close to equilibrium21.4 Reactions approaching equilibrium�(a) First-order reactions close to equilibrium21.4 Reactions approaching equilibrium�(a) First-order reactions close to equilibrium슬라이드 번호 4(b) Relaxation methods for fast reaction(b) Relaxation methods for fast reaction슬라이드 번호 7슬라이드 번호 8슬라이드 번호 9슬라이드 번호 10슬라이드 번호 11슬라이드 번호 12슬라이드 번호 13슬라이드 번호 14슬라이드 번호 15슬라이드 번호 16슬라이드 번호 17슬라이드 번호 18