Catalyst

A catalyst makes a chemical reaction proceed faster by participating in an alternative reaction mechanism and being regenerated. This makes repeated conversion possible: the same catalyst can help many successive reactants become reaction products. An enzyme is a biological catalyst. The overall standard change in Gibbs free energy is unchanged; see the IUPAC Gold Book: catalyst.

Regeneration does not mean that the catalyst sits unchanged throughout every step. For example, a schematic reaction mechanism can temporarily bind A to a catalyst C in a reaction intermediate AC:

A + C AC net: A ⇌ AC, ⇌ B + C, ⇌ B .

(1)

Adding the steps cancels both the reaction intermediate AC and the catalyst C from the net chemical equation. AC is produced and later consumed; C is used and later recovered. These letters describe possible roles, not a measured reaction mechanism. The alternative route is useful only if its chemical kinetics permits faster conversion. See Flowers et al., Chemistry 2e, §12.7: catalysts and alternative mechanisms.

For a worked barrier example, consider an idealized reversible A-to-B conversion at fixed temperature

T = 300 K

. Stipulate standard molar Gibbs free energy levels

G_{A}^{\circ} = 0

and

G_{B}^{\circ} = - 20 kJ mo l^{- 1}

, with effective transition state levels

80

without a catalyst and

50 kJ mo l^{- 1}

with a catalyst. Use the same standard states and the same zero for both routes; the catalytic route includes the regenerated C at both endpoints. Unlike the separate potential-energy profile example, this illustration uses Gibbs free energy at one specified temperature.

Figure 1.
A catalyst lowers the stipulated Gibbs energy of activation in both directions while retaining the same endpoints
. Original smooth schematic Gibbs free energy profiles at $300 K$ , with levels in $kJ mo l^{- 1}$ . The horizontal reaction coordinate describes progress along each schematic route, not time. These are chosen effective barriers, not measured potential energy surfaces, actual molecular paths, or the two-step reaction mechanism above. A real catalytic route can have several transition states and reaction intermediates. **Details**.

The Gibbs energy of activation is measured from the starting endpoint in the chosen direction. Thus the four barriers are

Δ G_{f, uncat}^{‡\circ} Δ G_{r, uncat}^{‡\circ} = 80 - 0 = 80, = 80 - (- 20) = 100, Δ G_{f, cat}^{‡\circ} Δ G_{r, cat}^{‡\circ} = 50 - 0 = 50, = 50 - (- 20) = 70, all in kJ mo l^{- 1} .

(2)

Both directional barriers fall by

30 kJ mo l^{- 1}

. The endpoint difference stays

Δ_{r} G^{\circ} = - 20 kJ mo l^{- 1}

To turn those barriers into rate constants, use the Eyring form of transition state theory for this effective first-order reaction pair, with a common prefactor and unit probability of successful passage:

k = ν exp (- \frac{Δ G ^{‡\circ}}{RT}), ν = \frac{k _{B} T}{h} .

(3)

Here

k_{B}

is the Boltzmann constant,

h

is the Planck constant, and

R

is the molar gas constant. The common prefactor cancels from the comparisons. These Gibbs energies of activation are not the empirical activation energy

E_{a}

of the Arrhenius equation. With the stipulated value

R = 8.314462618 J mo l^{- 1} K^{- 1}

, the speedup is

\frac{k _{f, cat}}{k _{f, uncat}} = \frac{k _{r, cat}}{k _{r, uncat}} = exp (\frac{30000}{8.314462618 \times 300}) \approx 1.67248 \times 1 0^{5} .

(4)

The exponent is a dimensionless quantity: convert

30 kJ mo l^{- 1}

30000 J mo l^{- 1}

before dividing by

RT

. The exponential function explains why a modest barrier reduction can have a dramatic kinetic effect. This number follows from the chosen free-energy barriers and common prefactor; it is not a universal catalytic multiplier or a prediction across temperatures.

Figure 2.
Both directional rate constants increase by about $1.67 \times 1 0^{5}$ , while their forward-to-reverse ratio stays about $3036$
. Original bar plots at $300 K$ . Left: base-ten logarithms of the dimensionless normalized rate constants $k / ν$ for the four stipulated barriers. A less negative value means faster conversion. Right: the logarithm of $k_{f} / k_{r}$ is identical for the two routes. Absolute units factor out through the common $ν = k_{B} T / h$ ; no temperature dependence outside this one model point is asserted. **Details**.

Assume ideal thermodynamic activities with a common concentration standard state, and directional rate laws

J_{f} = k_{f} [A]

and

J_{r} = k_{r} [B]

. At chemical equilibrium, equal reaction rates give

K = [B] / [A] = k_{f} / k_{r}

for this model. The barriers give

K_{uncat} = \frac{e ^{- 80000/ (RT)}}{e ^{- 100000/ (RT)}} = e^{20000/ (RT)} = \frac{e ^{- 50000/ (RT)}}{e ^{- 70000/ (RT)}} = K_{cat} \approx 3035.58.

(5)

The catalyst speeds up both directions, so it changes how quickly chemical equilibrium is approached without changing this equilibrium constant. For a general chemical reaction, the thermodynamic relation is

K = exp [- Δ_{r} G^{\circ} / (RT)]

with consistent standard states; an arbitrary multistep reaction mechanism need not permit

K = k_{f} / k_{r}

for two measured rate constants. At the same temperature, pressure, and thermodynamic activities, a catalyst also leaves the actual overall reaction change in Gibbs free energy unchanged. See 13.3 Shifting Equilibria: Le Châtelier’s Principle (§13.3: a catalyst does not shift equilibrium). To drive an unfavourable overall conversion, a process such as energy coupling must change the combined transformation.

Precatalyst

 0  0

A supplied precursor that must be chemically converted into the catalytically active form. Activation stoichiometry need not equal the reagent consumption of each subsequent catalytic turnover.

Transition state theory ( $k = (k_{B} T / h) exp [- Δ G^{‡\circ} / (RT)]$ )

 1  0

Transition state theory relates an elementary reaction's rate constant to a Gibbs energy of activation. The Eyring expression used in the catalyst example assumes every passage counted at the transition state succeeds. More generally, a factor accounting for unsuccessful passage multiplies it. Comparing barriers alone gives the quoted speedup only when the remaining prefactors agree. Free-energy profiles can depend on temperature, so the example's values at

300 K

do not determine a temperature curve. See the IUPAC Gold Book: transition state theory.

Force-tilted activation barrier ( $k (F) = k_{0} e^{F Δ x / (k_{B} T)}$ )

 0  0

A constant tensile force changes a one-coordinate free-energy landscape from

G_{0} (x)

G_{0} (x) - F x

. To first order, an escape barrier decreases by

F Δ x

, where

Δ x

is the saddle-to-initial-minimum displacement at zero force. An approximately fixed attempt prefactor gives the displayed rate law. Its characteristic force is

F_{0} = k_{B} T /Δ x

. Large changes in saddle location or competing pathways invalidate a constant activation distance. The barrier is measured relative to the initial state; an absolute saddle energy is not enough to determine an escape rate.

Activation volume ( $Δ V^{‡\circ}$ )

 0  0

The pressure derivative of the standard Gibbs energy of activation:

Δ V^{‡\circ} = (\partial Δ G^{‡\circ} / \partial p)_{T}

. In the corresponding rate convention,

Δ V^{‡\circ} = - RT (\partial ln k / \partial p)_{T}

. A negative value makes increasing pressure accelerate the reaction. Solvent reorganization can contribute as well as the intrinsic reacting geometry.

Activation entropy ( $Δ S^{‡\circ}$ )

 0  0

The standard entropy difference between the activated-complex reference state and the reactants. Its numerical value depends on standard-state conventions; association, molecular reorientation, internal motions and solvent reorganization can all contribute.

Activation enthalpy ( $Δ H^{‡\circ}$ )

 0  0

The standard enthalpy difference between the activated-complex reference state and the reactants at the same temperature and pressure. Under the usual Eyring assumptions it controls the slope of