Chapter 13. Acid–Base Catalysis and Proton-Transfer Reactions

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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)



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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)



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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)



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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)



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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



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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



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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)



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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,



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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)



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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.



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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)