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Catalysis

Field: Chemical Kinetics

The process of increasing the rate of a chemical reaction by adding a substance known as a catalyst.

Sequence of Expressions

Consider a general reaction Reactants→kuncatProducts\text{Reactants} \xrightarrow{k_{uncat}} \text{Products} and a catalyzed reaction Reactants→kcatProducts\text{Reactants} \xrightarrow{k_{cat}} \text{Products}. A substance Cat\text{Cat} acts as a catalyst if it provides an alternative reaction pathway such that the overall rate constant kcatk_{cat} satisfies the condition:\nkcat>kuncatk_{cat} > k_{uncat} \n\nFurthermore, the catalyst concentration [Cat][\text{Cat}] must satisfy the conservation law: Δ[Cat]overall=0\Delta[\text{Cat}]_{\text{overall}} = 0 over the reaction duration.
k=Aexp⁡(−EaRT)k = A \exp\left(-\frac{E_a}{R T}\right)\n\nWhere kk is the rate constant (Time−1\text{Time}^{-1}), AA is the pre-exponential factor (or frequency factor, Time−1\text{Time}^{-1}), EaE_a is the activation energy (Energy\text{Energy}), RR is the ideal gas constant (Energy/(Temperature⋅Mole)\text{Energy}/(\text{Temperature} \cdot \text{Mole})), and TT is the absolute temperature (Temperature\text{Temperature}). This relationship governs the temperature dependence of the rate constant.
Define the reaction rate vv for an enzyme-catalyzed reaction involving substrate [S][S] as:\nd[P]dt=v=Vmax[S]Km+[S]\frac{d[P]}{dt} = v = \frac{V_{max}[S]}{K_m + [S]} \n\nHere, VmaxV_{max} is the maximum reaction velocity achieved when [S]≫Km[S] \gg K_m, and KmK_m is the Michaelis constant, defined by the equilibrium condition Km=k−1+k2[S]k1K_m = \frac{k_{-1} + k_{2}[S]}{k_{1}} (or simply Km=k−1k1K_m = \frac{k_{-1}}{k_1} under simplified assumptions).
For a general reaction aA+bB→cC+dDa\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D}, the rate of reaction RR is defined by the differential rate law:\nR=1rd[Products]dt=k[A]a[B]bR = \frac{1}{r} \frac{d[\text{Products}]}{dt} = k [\text{A}]^a [\text{B}]^b \n\nHere, kk is the rate constant, which has units of Concentration−(a+b−1)⋅Time−1\text{Concentration}^{-(a+b-1)} \cdot \text{Time}^{-1}. The rate constant kk is the proportionality factor relating the reaction rate to the concentrations of the reactants.
For a surface reaction A→ProductsA \rightarrow Products catalyzed by a surface SS, the rate rr is modeled based on the fractional coverage θ\theta of active sites: r=kmaxθ[A]r = k_{max} \theta [A] where θ\theta is the surface coverage, defined by the Langmuir isotherm for adsorption: θ=KA[A]1+KA[A]\theta = \frac{K_A [A]}{1 + K_A [A]} and KAK_A is the adsorption equilibrium constant.
For a general reversible reaction aA+bB⇌cC+dDaA + bB \rightleftharpoons cC + dD, the equilibrium constant KK is defined by the ratio of the activities (approximated by concentrations) of products to reactants at equilibrium: K=[C]c[D]d[A]a[B]bK = \frac{[C]^c [D]^d}{[A]^a [B]^b} where the activities are evaluated when the system reaches thermodynamic equilibrium.
Let EreactantsE_{reactants} be the potential energy of the reactants and EtransitionE_{transition} be the potential energy of the transition state along the reaction coordinate ξ\xi. The activation energy EaE_a is formally defined as the minimum energy barrier height: \nEa=Etransition−EreactantsE_a = E_{transition} - E_{reactants} \n\nThis value represents the maximum potential energy difference required for the system to transition from the reactant valley to the product valley.
Let ZZ be the collision frequency between species AA and BB. The rate constant kk for the reaction A+B→ProductsA + B \rightarrow Products is given by the product of the collision frequency ZZ and the steric factor PP and the Boltzmann factor f(E)f(E): k=PZNe−Ea/RTk = P \frac{Z}{N} e^{-E_a/RT} where NN is the number density of reactants, EaE_a is the activation energy, and RR is the universal gas constant.
Define the rate constant kk using the Eyring equation, which relates the rate to the free energy of activation ΔG‡\Delta G^{\ddagger}: k=kBThe−ΔG‡/RTk = \frac{k_B T}{h} e^{-\Delta G^{\ddagger}/RT} where kBk_B is the Boltzmann constant, hh is Planck's constant, TT is the absolute temperature, and ΔG‡\Delta G^{\ddagger} is the standard free energy change for forming the transition state.
Consider a reaction sequence involving a catalyst CC: A+C→k1B+CA + C \xrightarrow{k_1} B + C and B+C→k2Products+CB + C \xrightarrow{k_2} Products + C. The overall rate RR is determined by the rate-determining step (RDS), rRDSr_{RDS}, and the catalyst concentration [C][C] must be conserved throughout the cycle. The net rate of product formation is: R=min⁡(k1[A][C],k2[B][C])R = \min \left( k_1 [A] [C], k_2 [B] [C] \right) subject to the constraint that the net change in [C][C] over the cycle is zero.