mechanism

# Constructing the differential equations from the reaction mechanism

### Mechanism

A reaction mechanism is a collection of reactions showing how molecules react in steps to lead to the overall stoichiometric reaction. The set of reactions specifies the path (or paths) that reactant molecules take to finally arrive at the product molecules. Starting from reactant molecules intermediates may be formed and the intermediate molecules subsequently react to form products. All species in the reaction appear in at least one step and the sum of the steps gives the overall reaction.

### Differential rate equations

The mechanism leads directly to differential equations that govern the rate of the reaction.

Each species in the mechanism is consumed or formed at a rate with contributions from various steps in the mechanism. Steps that form a species contribute positive terms and those that consume the species contribute negative terms to the rate. Coefficients of species in the mechanism steps appear as powers in the terms of the rate equations. We say a step is bimolecular when two molecules come together before a reaction, unimolecular when one molecule reacts without colliding with another, termolecular if three must collide, etc.

The easiest way to describe how the mechanism gives rise to rate equations is by examples. See the illustrative examples below.

## Caveat - Disclaimer

It is impossible to assign a mechanism with complete certainty to any real chemical reaction. The best that can be done is to find a mechanism that is consistent with the known data about the reaction. Sometimes more than one mechanism is consistent with the data; in such cases new experiments are needed to choose between mechanisms.

### Illustrative Examples

#### Example 1: Bimolecular decomposition

• Mechanism:
• Step one: 2A --> A + A* ; with rate constant k1.

Two molecules of A collide and one acquires energy leaving it in an excited state A*. This is a bimolecular step requiring that two molecules of A come together.

Step two: A* + A --> 2A ; with rate constant k_1.

Step one is reversible. This is also bimolecular since A* must collide with A for deactivation to occur.

Step three: A* --> B ; with rate constant k2.

Excited molecule A* can rearrange to form the product B. This is a unimolecular process since no molecular collisions are required. The overall reaction is found by adding steps one, two and three with multipliers so that the intermediate A* cancels: 2(2A --> A + A*) + (A* + A --> 2A) + (A* --> B) = A --> B.

• Differential rate equations:
• d[A]/dt = -k1 [A]2 + k_1 [A*][A] ;

Species A is consumed in step one and re-generated in step two. The products of concentrations reflects the increased frequency of bimolecular collisions as concentrations of colliders increases.

d[A*]/dt = k1 [A]2 - k_1 [A*][A] - k2 [A*] ;

Species A* is formed in the bimolecular collisions of step one. It is removed in the bimolecular collisions of step two and in the unimolecular step three.

d[B]/dt = k2 [A*] ;

The product species B is created in step three.

The mechanism has yielded three coupled differential equations that might be solved with DynaSys (click here to see the dyanasys solution) or other software tool.

#### Example 2: catalysis

• Mechanism
• Step one: A + B --> C ; with rate constant k1

Bimolecular step in which the reactant A combines with the catalyst B to create the intermediate complex C.

Step two: C --> P ; with rate constant k2

Unimolecular step in which the complex decomposes to product and re-generates the catalyst.

The overall reaction, A --> P, does not contain the catalyst C. Nevertheless, the rate depends on the concentration of C.

• Differential Rate equations
• Reactant:
d[A]/dt = -k1 [A][B]

Reactant forms a complex with catalyst.

Catalyst:
d[B]/dt = -k1 [A][B] + k2 [C]

Catalyst forms complex with reactant or complex decomposes to re-generate catalyst plus product.

Complex:
d[C]/dt = k1 [A][B] - k2 [C]

Complex is generated in bimolecular step one or consumed in step two.

Product:
d[P]/dt = k2 [C]

Product is formed from decomposition of the complex between catalyst and reactant.

Each of these three coupled differential equations expresses the rate of change of one species as a sum of terms that contain a rate constant and powers of one or more species. A MathCad solution can be seen or a DynaSys solution can be seen by clicking here.

### Exercises

Problem 1: The following mechanism consists of consecutive steps with feedback.

step one: A --> X, with rate constant k1;

the reactant rearranges to intermediate X

step two: X --> Y, rate constant k2;

subsequently X decomposes to the new intermediate Y

step three: A + Y --> Y + X, rate constant k3;

Y catalyzes the rearrangement of A to intermediate X (feedback)

step four: Y --> Z, rate constant k4;

Y decomposes to final product Z.

Derive the corresponding rate equations.

Optionally, solve these equations for concentrations of A, X, Y, Z versus time when k1=1, k2=0.5, k3=0.1, k4=0.1, [A]0 = 1, and [X]0=[Y]0=[Z]0=0.

Problem 2: Write the rate equations for the following proposed mechanism for reaction between nitrogen pentoxide and nitric oxide in the gas phase.

step one: N2O5 --> NO2 + NO3, rate constant k1;
reactant N2O5 dissociates to for two intermediates: NO2 and NO3.
step two: NO2 + NO3 --> N2O5, rate constant k2;
step one is reversible.
step three: NO + NO3 --> 2 NO2, rate constant k3;
reactant NO combines with one of the dissociation intermediates of step 1.

Problem 3: Certain enzymes catalyze the reaction between two substrates: A + B + E --> Y + Z + E. Write the rate equations that arise from the following proposed mechanism for such reactions.

step one: E + A --> EA, k1
step two: EA --> E + A, k_1
step three: EA + B --> EZ + Y, k2
step four: EZ --> E + Z, k3.

{Hint: d[EA]/dt = k1[E][A] - k_1[EA] -k2[EA][B]}

### Glossary of Terms

• Stoichiometry determines the molar ratios of reactants and products in an overall chemical reaction. We express the stoichiometry as a balanced chemical equation. For kinetics it is convient to write this as products minus reactants: npP + nqQ - naA - nbB (instead of the conventional equation naA + nbB ---> npP + nqQ). This indicates that na and nb moles of reactants A and B, resp., produce np and nq moles of products P and Q.
• The rate of a chemical reaction is defined in such a way that it is independent of which reactant or product is monitored. We define the rate, v, of a reaction to be v = (1/ng) d[G]/dt where ng is the signed (positive for products, negative for reactants) stoichiometric coefficient of species G in the reaction. Namely, v = (-1/na) d[A]/dt = (1/np) d[P]/dt, etc.
• It is convenient to refer to the extent of reaction. As the reactants are sonsumed and the products are produced, their concentrations change. If the initial concentrations of A, B, P and Q are [A], [B], [P] and [Q], resp., then the extent of reaction is defined: x = -([A]-[A]0)/na = -([B] - [B]0)/nb = ([P]-[P]0)/np = ([Q]-[Q]0)/nq. Alternately, each species concentration is a function of the extent of reaction: [A] = [A]0 - nax, etc.
• Many reactions follow elementary differential rate laws such as v = k f([A], [B], ...) where f([A], [B], ...) is a function of the concentrations of reactants and products. That is, the rate varies as the concentrations change. A proportionality constant, k, is called the rate constant of the reaction.
• When the rate law has the special form of a product (or quotient) of powers, f([A], [B], ...) = [A]a [B]b [P]p [Q]q then a is the order of the reaction with respect to A, b is the order w.r.t. B, etc. Note that order may be positive, negative, integer, or non-integer. Further, the sum a + b + p + q is the overall order of the reaction rate law.
• NOTE: there is no necessary relation between orders and stoichiometric coefficients. That is, a might differ from na.
• Reaction rate constants are usually temperature dependent; the rate of a reaction usually increases as the temperature rises. The temperature dependence often follows Arrhenius' equation: k(T) = A exp(-Ea/RT) where T is the absolute temperature, R the universal gas constant, Ea is the activation energy (specific to each reaction), and A is the "pre-exponential" or "frequency" or "entropy" factor.
• One objective of chemical kinetics is to solve the differential rate law d[G]/dt = k f([A], [B], ...), and thereby express each species concentration as a function of time: [G](t). Since solution requires integration, we call it the integrated rate law.
• A reaction mechanism is a set of steps at the molecular level. Each step involves combinations or re-arrangements of individual molecular species. The steps in combination describe the path or route that reactant molecules follow to reach the product molecules. The result of all steps is to produce the overall balanced stoichiometric chemical equation for reactants producing products.

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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.