3.2.2 Nonelementary Rate Laws

A large number of both homogeneous and heterogeneous reactions do not follow simple rate laws. Examples of reactions that don’t follow simple elementary rate laws are discussed below.

Homogeneous Reactions

The overall order of a reaction does not have to be an integer, nor does the order have to be an integer with respect to any individual component. As an example, consider the gas-phase synthesis of phosgene,

CO + CL2 → COCL2

in which the kinetic rate law is

image

This reaction is first order with respect to carbon monoxide, three-halves order with respect to chlorine, and five-halves order overall.

Sometimes reactions have complex rate expressions that cannot be separated into solely temperature-dependent and concentration-dependent portions. In the decomposition of nitrous oxide,

2N2O →2N2 + O2

the kinetic rate law is

image

Both kN2O and k’ are strongly temperature-dependent. When a rate expression such as the one given above occurs, we cannot state an overall reaction order. Here we can only speak of reaction orders under certain limiting conditions. For example, at very low concentrations of oxygen, the second term in the denominator would be negligible with respect to 1 (1 >> k′CO2), and the reaction would be “apparent” first order with respect to nitrous oxide and first order overall. However, if the concentration of oxygen were large enough so that the number 1 in the denominator were insignificant in comparison with the second term, k′CO2 (k′CO2 >> 1), the apparent reaction order would be –1 with respect to oxygen and first order with respect to nitrous oxide giving an overall apparent zero order. Rate expressions of this type are very common for liquid and gaseous reactions promoted by solid catalysts (see Chapter 10). They also occur in homogeneous reaction systems with reactive intermediates (see Chapter 9).

Important resources for rate laws

It is interesting to note that although the reaction orders often correspond to the stoichiometric coefficients, as evidenced for the reaction between hydrogen and iodine, just discussed to form HI, the rate expression for the reaction between hydrogen and another halogen, bromine, is quite complex. This nonelementary reaction

H2 + Br2 → 2HBr

proceeds by a free-radical mechanism, and its reaction rate law is

3-8

image

Rate laws of this form usually involve a number of elementary reactions and at least one active intermediate. An active intermediate is a high-energy molecule that reacts virtually as fast as it is formed. As a result, it is present in very small concentrations. Active intermediates (e.g., A*) can be formed by collision or interaction with other molecules.

A + M → A* + M

Here the activation occurs when translational kinetic energy is transferred into energy stored in internal degrees of freedom, particularly vibrational degrees of freedom.3 An unstable molecule (i.e., active intermediate) is not formed solely as a consequence of the molecule moving at a high velocity (high translational kinetic energy). The energy must be absorbed into the chemical bonds where high-amplitude oscillations will lead to bond ruptures, molecular rearrangement, and decomposition. In the absence of photochemical effects or similar phenomena, the transfer of translational energy to vibrational energy to produce an active intermediate can occur only as a consequence of molecular collision or interaction. Collision theory is discussed in the Professional Reference Shelf in Chapter 3.

In Chapter 9, we will discuss reaction mechanisms and pathways that lead to nonelementary rate laws, such as the rate of formation of HBr shown in Equation (3-8).

Heterogeneous Reactions

Historically, it has been the practice in many gas-solid catalyzed reactions to write the rate law in terms of partial pressures rather than concentrations. In heterogeneous catalysis it is the weight of catalyst that is important, rather than the reactor volume. Consequently, we use image in order to write the rate law in terms of mol per kg of catalyst per time in order to design PBRs. An example of a heterogeneous reaction and corresponding rate law is the hydrodemethylation of toluene (T) to form benzene (B) and methane (M) carried out over a solid catalyst.

image

The rate of disappearance of toluene per mass of catalyst, image, i.e., (mol/mass/time) follows Langmuir-Hinshelwood kinetics (discussed in Chapter 10), and the rate law was found experimentally to be

image

where the prime in image notes typical units are in per gram of catalyst (mol/kg cat/s), PT, PH2, and PB are partial pressures of toluene, hydrogen, and benzene in (kPa or atm) and KB and KT are the adsorption constants for benzene and toluene respectively, with units of kPa–1 (or atm–1). The specific reaction rate k has units of

image

You will find that almost all heterogeneous catalytic reactions will have a term such as (1 + KAPA + ...) or (1 + KAPA + ...)2 in the denominator of the rate law (cf. Chapter 10).

To express the rate of reaction in terms of concentration rather than partial pressure, we simply substitute for Pi using the ideal gas law

3-9

image

The rate of reaction per unit weight (i.e., mass) catalyst, image (e.g., image), and the rate of reaction per unit volume, –rA, are related through the bulk density ρb (mass of solid/volume) of the catalyst particles in the fluid media:

image

In fluidized catalytic beds, the bulk density, ρb, is normally a function of the volumetric flow rate through the bed.

In summary on reaction orders, they cannot be deduced from reaction stoichiometry. Even though a number of reactions follow elementary rate laws, at least as many reactions do not. One must determine the reaction order from the literature or from experiments.

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