Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (13.81 MB, 563 trang )
Ch013.qxd
12/22/2006
322
13.1.1
10:35 AM
Page 322
13. Acid–Base Catalysis and Proton-Transfer Reactions
Fast pre-equilibrium: Arrhenius intermediates
When the first step of the mechanism, step (13.II), is a fast equilibrium, the intermediate
X is in equilibrium with the reactants, and corresponds to Arrhenius’s concept of an equilibrium intermediate. This step leads to
[X][Y] = k1 = K
[C][S] k−1
(13.1)
In this expression, the concentrations of C and S, [C] and [S], are their equilibrium concentrations and not their initial concentrations. For initial concentrations, [C]0 and [S]0, we
can write
[C]0 = [C] + [X]
[S]0 = [S] + [X]
(13.2)
(13.3)
Introducing these expressions in eq. (13.1), we obtain
[ X ][ Y ]
=K
([C]0 − [X ]) ([S]0 − [X ])
(13.4)
which is a quadratic equation in X. The reaction rate, expressed in terms of the formation
of the product P in step (13.III), is given by
v = k2 [ X ][ W ]
(13.5)
The intermediate X can be eliminated from this expression using eq. (13.4).
It is often more useful to consider the relative concentrations of the catalyst and substrate and obtain limiting cases for the rate law, rather than carry out all the algebra with
the quadratic equation. Because [X] cannot be larger than [C]0, we can obtain a first limiting case when the initial concentration of the substrate is relatively high, [S]0 [C]0, and,
consequently, [S]0᎐[X] [S]0. Under this approximation eq. (13.4) becomes
[X][Y]
([C]0 − [X])[S]0
=K
(13.6)
or
K [C]0 [S]0
[X] = K
[S]0 + [Y]
(13.7)
Ch013.qxd
12/22/2006
13.1
10:35 AM
Page 323
General Catalytic Mechanisms
323
Figure 13.1 Rate of a catalysed reaction as a function of the initial concentration of substrate, [S]0,
observed when [S]0 [C]0.
and the rate law can be written as
v=
k2 K [C]0 [S]0 [ W ]
K [S]0 + [ Y ]
(13.8)
This expression shows that the reaction rate changes with [S]0, as illustrated in Figure 13.1.
For low substrate concentration, the rate resembles a first-order process with respect to
the substrate. However, for higher substrate concentrations, the rate may attain a plateau
and become independent of the substrate concentration, just like a zero-order reaction
with respect to S. As long as [S]0 [C]0, the reaction rate is first order with respect to the
catalyst.
We have seen this behaviour before in Chapter 10 in single-substrate reactions on the
surface of solids, and will discuss it again in Chapter 14 in the context of enzyme catalysis. For both cases of surface reactions and enzyme catalysis, the species Y and W do not
exist and eq. (13.8) becomes
v=
k2 K [C]0 [S]0
K [S]0 + 1
(13.9)
this equation is equivalent to eq. (10.19) of surface reactions, which is defined per unit
area, and to the Michaelis–Menten equation of enzyme catalysis, eq. (4.133).
Another limiting case is obtained when the catalyst is in excess relative to the substrate,
[C]0 [S]0, and eq. (13.4) becomes
[ X ][ Y ] = K
[C]0 ([S]0 − [ X ])
(13.10)
Ch013.qxd
12/22/2006
10:35 AM
Page 324
324
13. Acid–Base Catalysis and Proton-Transfer Reactions
leading to
v=
k2 K [C]0 [S]0 [ W ]
K [C]0 + [ Y ]
(13.11)
Now the reaction is always first order with respect to the substrate, independent of [S],
provided that [S]0 [C]0, but the partial order with respect to the catalyst may vary
between zero and unity.
Additionally, when K[C]0 is much greater than [Y], the rate of reaction is simply given by
v = k2 [S]0 [ W ]
(13.12)
Ea = E 2
(13.13)
and the activation energy is
where Ea is the activation energy of reaction (13.III).
The activation energies associated with the rates expressed by eqs. (13.8) and (13.11)
can also be obtained when only a very small quantity of the substrate S is present. In such
cases, the rate of reaction is
v = k2
k1
[C]0 [S]0 [W ][Y]−1
k−1
(13.14)
and the activation energy becomes
Ea = E1 + E2 − E −1
(13.15)
where E1 and EϪ1 represent the activation energy of reaction (13.II) and its reverse, respectively.
13.1.2 Steady-state conditions: van’t Hoff intermediates
If k2[W] kϪ1[Y] in mechanism (13.IIϪ13.III), then [X] is small and the steady-state
approximation can be applied to this mechanism. Laidler called X under these conditions
a van’t Hoff intermediate. The energy profiles corresponding to Arrhenius and van’t Hoff
intermediates are illustrated in Figure 13.2.
Under steady-state conditions, [X] does not change appreciably with time
d [X]
dt
= k1 [C][S] − k−1 [ X ][ Y ] − k2 [ X ][ W ] = 0
(13.16)
Replacing eqs. (13.2) and (13.3) in this expression gives
(
) ([S] − [X]) − k [X][Y] − k [X][W] = 0
k1 [C]0 − [ X ]
0
−1
2
(13.17)
Ch013.qxd
12/22/2006
13.1
10:35 AM
Page 325
General Catalytic Mechanisms
325
Figure 13.2 Potential-energy profiles for catalysed reactions. (a) The rate-determining step is the
second step, occurring after the formation of an Arrhenius intermediate. (b) The rate-determining
step is the first step, which leads to a van’t Hoff intermediate.
The low concentration of X justifies ignoring the quadratic term in [X], and leads to
[X] =
k1 [C]0 [S]0
(
)
k1 [C]0 + [S]0 + k−1 [ Y ] + k2 [ W ]
(13.18)
The rate is therefore
v=
(
k1 k2 [C]0 [S]0 [ W ]
)
k1 [C]0 + [S]0 + k−1 [ Y ] + k2 [ W ]
(13.19)
As mentioned before, Y and W do not exist in surface and enzyme catalysis, and
eq. (13.19) simplifies to
v=
(
k1 k2 [C]0 [S]0
)
k1 [C]0 + [S]0 + k−1 + k2
(13.20)
At high catalyst concentrations, eq. (13.19) reduces to eq. (13.12), and the activation
energy is given by eq. (13.13). For low substrate and catalyst concentrations, eq. (13.20)
becomes
v=
k1 k2
[C]0 [S]0
k−1 + k2
(13.21)
Ch013.qxd
12/22/2006
10:35 AM
Page 326
326
13. Acid–Base Catalysis and Proton-Transfer Reactions
The Arrhenius equation, and consequently the definition of activation energy, does not
apply to this system, except if k2 kϪ1 or kϪ1 k2. In the first case,
v = k1 [C]0 [S]0
(13.22)
Ea = E1
(13.23)
k1 k2
[C]0 [S]0
k−1
(13.24)
and the activation energy is
In the second case,
v=
and the activation energy is that of eq. (13.15).
13.2
GENERAL AND SPECIFIC ACID–BASE CATALYSIS
Arrhenius and Ostwald played very important roles in the early studies on acid–base catalysis, one century ago. Arrhenius contributed to the definition of acids and bases, and established the dependence between the rate constants and the temperature. Additionally, he
also formulated an electrolytic theory of dissociation that ultimately led to him receiving
the 1903 Nobel Prize in Chemistry. Ostwald proposed useful definitions of catalysis and
classifications of catalysts, but he was unable to develop a satisfactory theory of these
effects. This is not surprising, in view of the very limited knowledge of the mechanisms
of catalysis at his time, and of the lack of understanding of how molecular properties can
influence the rates of reactions. Nevertheless, his seminal work on catalysis was rewarded
by him receiving the 1909 Nobel Prize in Chemistry.
Ostwald recognised that a catalysed reaction proceeds by an alternative reaction pathway, made possible by the addition of a new species. For a process that in the absence of
catalysts proceeds with a rate constant k0, the general acid–base catalysed reaction follows
the rate expression:
(
)
v = k0 + kH+ ⎡H + ⎤ + kOH − ⎡OH − ⎤ + kHA [HA ] + kA − ⎡ A − ⎤ [S]
⎣ ⎦
⎣
⎦
⎣ ⎦
(13.25)
involving the sum of the rates for all the alternative pathways. In this expression, kHϩ and
kOHϪ are the rate constants for catalysis by proton and hydroxide ion, and kHA and kAϪ are
the catalytic rate constants for the acid HA and the base AϪ. In aqueous solutions the proton is strongly hydrated and, as will be discussed below, it is not present in the form of the
free Hϩ ion. However, for simplicity, except where required to illustrate PT to water, we
will continue to use the representation Hϩfor the proton in the equations, and will also continue to represent the hydroxide ion, OHϪ.
Eq. (13.25) indicates that the reaction is catalysed by an acid, HA, or a base, AϪ, present
in the solution, in addition to catalysis by Hϩ and OHϪ. We refer to catalysis in the former
Ch013.qxd
12/22/2006
13.2
10:35 AM
Page 327
General and Specific Acid–Base Catalysis
327
case (by HA and AϪ) as general acid–base catalysis. However, in many catalytic mechanisms, particularly in aqueous solutions, only Hϩ and OHϪsignificantly influence the rate
of the reaction. In this case we are in the presence of specific acid–base catalysis.
In the general catalytic mechanism represented by (13.II) and (13.III), an acid catalysis
corresponds to replacing X by SHϩ, the catalyst C by the acid AH and Y by that catalyst
without a proton. In the second step, SHϩ transfers a proton to the species, W, for example a water molecule, and gives the product P
⎯⎯
→
HA + S ←⎯⎯ AϪ + SH +
k1
(13.IV)
k−1
+
SH + H 2 O ⎯⎯ P + H 3 O
→
k2
+
(13.V)
This example, where W is a solvent molecule, corresponds to a protolytic transfer. When W
is the conjugate base of the catalyst, WϭAϪ, the mechanism is called prototropic. Table 13.1
presents the four possible combinations for acid and base catalysis that correspond to the
possible identities of species involved in the general catalytic mechanism.
The mechanism of each of the four possible combinations may involve an Arrhenius or
a van’t Hoff intermediate. This leads to eight possible mechanisms, schematically presented in Table 13.2. Each of these mechanisms can be developed using either the preequilibrium or the steady-state approximation to arrive at the corresponding rate law. The
lessons that can be learnt from the treatment of these mechanisms are also indicated in
Table 13.2; some mechanisms lead exclusively to specific acid or base catalysis, while others lead to general acid or base catalysis. Furthermore, the specific catalysis is associated
with the existence of a limiting rate, that is the rate that will not increase indefinitely with
the Hϩ, or OHϪ, concentration, but attain a limiting value equal to k2[S]0.
For specific acid–base catalysis at low pH, eq. (13.25) simplifies to
(
)
v = k0 + kH+ ⎡H + ⎤ [S]
⎣ ⎦
(13.26)
As the pH increases, the participation of the hydroxide ion in the catalysis becomes
increasingly important, and the reaction rate for specific acid–base catalysis becomes
(
)
v = k0 + kH+ ⎡H + ⎤ + kOH− ⎡OH − ⎤ [S]
⎣ ⎦
⎣
⎦
(13.27)
Table 13.1
Nature of the reacting species in general acid–base catalysis in aqueous solutions
Catalysis
C
Y
W
Z
Acid
Acid
Base
Base
HA
H2O
B
H2O
AϪ
OHϪ
BHϩ
H3Oϩ
H2O
B
H2O
HA
H3Oϩ
BHϩ
OHϪ
AϪ
Ch013.qxd
12/22/2006
10:35 AM
Page 328
328
13. Acid–Base Catalysis and Proton-Transfer Reactions
Table 13.2
Summary of mechanisms for acid–base catalysis
Acid catalysis
Arrhenius intermediates
H transferred to solvent
⎯⎯
→
BH + + S ←⎯ SH + + B
⎯
SH + + H 2 O ⎯⎯ P + H 3 O+
→
Hϩ transferred to solute
⎯⎯
→
BH + + S ←⎯ SH + + B
⎯
SH + + B ⎯⎯ P + BH +
→
Hϩ transferred from solvent
⎯⎯
→
B + SH ←⎯ S− + BH +
⎯
S− + H 2 O ⎯⎯ P + OH −
→
Hϩ transferred from solute
⎯⎯
→
B+SH ←⎯ S− + BH +
⎯
S− + BH + ⎯⎯ P + B
→
van’t Hoff intermediates
k2 kϪ1 [B]
Specific Hϩcatalysis
With limiting rate
k2 kϪ1
General catalysis
No limiting rate
k2 kϪ1 [BHϩ]
Specific OHϪcatalysis
With limiting rate
k2 kϪ1
General catalysis
No limiting rate
ϩ
k2 kϪ1 [B], k2 k1 [BHϩ]
General catalysis
No limiting rate
k2 kϪ1
General catalysis
No limiting rate
k2 kϪ1 [BHϩ], k2 k1 [B]
General catalysis
No limiting rate
k2 kϪ1
General catalysis
No limiting rate
It is convenient to define a pseudo-first-order rate constant
k =
v
S]T
[
(13.28)
where [S]T is the total concentration of the substrate, and express the reaction rate constant
in terms of
k = k0 + kH+ ⎡H + ⎤ + kOH− ⎡OH − ⎤
⎣ ⎦
⎣
⎦
(13.29)
In aqueous solutions, the equilibrium
H + + OH −
H2 O
(13.VI)
where [Hϩ][OHϪ]ϭKw, leads to
k − Kw
k = k0 + kH+ ⎡H + ⎤ + OH +
⎣ ⎦ ⎡H ⎤
⎣ ⎦
(13.30)
Usually, it is possible to consider a region of sufficiently low pH where the catalysis is
dominated by the Hϩ ion, and eq. (13.30) simplifies to
k = kH + ⎡ H + ⎤
⎣ ⎦
(13.31)
Ch013.qxd
12/22/2006
13.3
10:35 AM
Page 329
Mechanistic Interpretation of the pH Dependence of the Rates
329
or
log k = log kH+ − pH
(13.32)
It may also be possible to find a region of sufficiently high pH where the catalysis is dominated by the OHϪ ion, and eq. (13.30) becomes
k =
kOH− K w
⎡H + ⎤
⎣ ⎦
(13.33)
or
log k = log kOH− − pK w + pH
(13.34)
Another special case is that of a small spontaneous reaction, in the absence of catalysis,
that is, k0 (Kw kHϩ kOHϪ)1/2, and the rate of reaction as a function of the pH goes through
a minimum. The value of that minimum can be obtained differentiating eq. (13.30):
k − Kw
dk
= kH+ − OH 2
+
d ⎡H ⎤
⎡H + ⎤
⎣ ⎦
⎣ ⎦
(13.35)
equating the derivative to zero and expressing the Hϩ concentration as
⎡H + ⎤ =
⎣ ⎦ min
kOH− K w
kH+
(13.36)
This expression allowed Wijs [2,3], at the end of the 19th century, to measure the catalytic
rate constants for the hydrolysis of ethyl acetate at low and high pHs and obtain the ionic
product of water, Kw.
13.3 MECHANISTIC INTERPRETATION OF THE pH DEPENDENCE
OF THE RATES
The above discussion shows that the dependence of the reaction rate upon the pH contains
very important information on the reaction mechanism. Each rate must be measured at
constant pH, which usually involves measuring it in a buffer solution. In addition, usually
an inert salt is added to maintain ionic strength constant to avoid the salt effects discussed
in Chapter 9. In fact, experimentally, the rates are measured at different buffer concentrations, keeping the pH and the ionic strength constant. Under these conditions, and for a
constant substrate concentration, there is a linear dependence between the rate and the
buffer concentration, as illustrated in Figure 13.3. Extrapolating to zero buffer concentration, one obtains the rate for a constant pH. When general acid–base catalysis is present,
Ch013.qxd
12/22/2006
330
10:35 AM
Page 330
13. Acid–Base Catalysis and Proton-Transfer Reactions
Figure 13.3 Representation of the method employed to obtain the pH dependence of the pseudofirst-order rates (kψ) as a function of the pH. (a) Rates measured for different buffer concentrations
at pH 4.5. (b) Pseudo-first-order rates at different pHs.
the rate depends on the nature of the buffer, and the extrapolation gives different rates for
different buffers. Such cases of general acid–base catalysis will be discussed in the next
section. Here, we focus on the mechanistic interpretation of the pseudo-first-order rate
constants measured as a function of the pH in aqueous solutions.
There are 5 fundamental types of pH dependence, and they are illustrated in Figure 13.4 by
the dependence of the logarithm of the rate as a function of pH (to a first approximation, the
negative of the logarithm of the hydrogen ion concentration) [4]. The experimentally observed
pH dependences can be divided in sections that correspond to these fundamental types. Types
(a), (b) and (c) correspond to reactions that are catalysed by Hϩ, not catalysed, and catalysed
by OHϪ, respectively. The catalysis by Hϩ is described by eq. (13.32), which explains the origin of the slope Ϫ1, corresponding to a linear dependence of rate upon hydrogen ion concentration. The absence of significant acid–base catalysis occurs when k0 dominates all the other
terms in the right-hand side of eq. (13.30), and leads to a rate that is independent of the pH of
the solution. The catalysis by OHϪ is described by eq. (13.34), which shows the origin of the
slope ϩ1. The other two types, (d) and (e), correspond to cases where the rate constant
decreases, case (d), or stops increasing, case (e), after a certain pH. These cases cannot correspond to a change in mechanism because such a change is always associated with the predominance of a competitive reaction, after a certain pH, and that necessarily leads to an
increase in the rate. This is not the case, because the bends followed by the downward trend
of the curves represented in (d) and (e) reflect a decrease in the rates after a given pH. Thus,
each of these curves with a downward trend must be associated with a single mechanism.
There are two alternative mechanisms that may lead to such breaks followed by downward
trends; the presence of a fast pre-equilibrium or a change in the rate-determining step.
In the case of a mechanism of acid catalysis involving a fast pre-equilibrium
S + H+
k1
k−1
SH +
2
SH + ⎯k⎯ P
→
(13.VII)
Ch013.qxd
12/22/2006
13.3
10:35 AM
Page 331
Mechanistic Interpretation of the pH Dependence of the Rates
331
Figure 13.4 Fundamental curves representing the dependence of the logarithm of the pseudo-firstorder rate of acid–base catalysis on the pH of the solution, without involving a change in the reaction mechanism.
Ch013.qxd
12/22/2006
10:35 AM
Page 332
332
13. Acid–Base Catalysis and Proton-Transfer Reactions
where the acidity constant is KaϭkϪ1/k1, it is convenient to express the rate constant in
terms of the fraction of protonated substrate
fSH =
+
⎡SH + ⎤
⎣
⎦ = ⎡H ⎤
⎣ ⎦
⎡SH + ⎤ + [S] K a + ⎡H + ⎤
⎣
⎦
⎣ ⎦
(13.37)
because [SHϩ] ϭ fSH[S]T and the rate constant is v ϭ k2[SHϩ]ϭk2 fSH[S]T. Using these relations and eq. (13.28), gives
k = k2 fSH =
k2 ⎡H + ⎤
⎣ ⎦
Ka + ⎡H + ⎤
⎣ ⎦
(13.38)
This expression is equivalent to
k =
c4 ⎡ H + ⎤
⎣ ⎦
c5 + ⎡H + ⎤
⎣ ⎦
(13.39)
where c4 and c5 are constants, which is the equation of the curve represented in Figure 13.4d.
In contrast, the rate constant of a mechanism of base catalysis involving a fast preequilibrium
k1
S + H+
SH +
k−1
2
SH + ⎯k⎯ P
→
(13.VII)
where the acidity constant is Kaϭk1/kϪ1, is conveniently expressed in terms of the fraction
of un-protonated substrate
fS =
[S]
⎡SH + ⎤ + ⎡S ⎤
⎣
⎦ ⎣ ⎦
=
Ka
Ka + ⎡H + ⎤
⎣ ⎦
(13.40)
because
k = k2 fS =
k2 K a
Ka + ⎡H + ⎤
⎣ ⎦
(13.41)
and has a clear analogy with the mathematical expression of the curve represented in
Figure 13.4e
k =
where c6 and c7 are constants.
c6
c 7 + ⎡H + ⎤
⎣ ⎦
(13.42)