Department of Mathematics

IDEA: Internet Differential Equations Activities


Constructing the differential equations from the reaction mechanism

Table of contents:

  1. About Reaction Mechanism
  2. Differential Rate Equations
  3. Illustrative Examples
  4. Exercises
  5. Glossary of terms


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

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

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

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.


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

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.

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.

{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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