Reaction rate

Chemical reaction kinetics: Basic principles

Reaction rate

We return to the subject of chemical reactions and cover the basic principles that enable us to create kinetic models for various reaction mechanisms.

A chemical reaction is described in its simplest form by a reaction equation $a\,\text{A}+b\,\text{B}+\cdots \longrightarrow p\,\text{P}+q\,\text{Q}+\cdots\text,$ where the symbols $\text{A, B, }\ldots$ represent reactants (in fact, molecules) that react with each other and yield products $\text{P, Q, } \ldots$ . The numbers $a, b, \ldots$ and $p, q, \ldots$ indicate that $a$ molecules A react with $b$ molecules $B$ , etc., and yield $p$ molecules P, $q$ molecules Q $\ldots$ They are called the stoichiometric coefficients of the reaction equation.

We will now introduce the concept of reaction rate. Suppose that $n_\text{A0}$ symbolizes the amount of the reactant A at the beginning of the reaction, at time $t=0$ , and that $n_\text{A}$ symbolizes the amount of A at time $t$ . The extent of conversion $\xi$ is defined by $n_\text{A}=n_\text{A0} -a\cdot \xi$ and then the rate of conversion $v$ in terms of quantities is equal to the derivative of the rate of product formation: $v=\frac{\dd\xi}{\dd t}$ The rate of conversion can also be expressed as the derivative of the quantity of substance A: $\frac{\dd n_\text{A}}{\dd t}=-a\frac{\dd \xi}{\dd t}$ and thus $v=-\frac{1}{a}\frac{dn_\text{A}}{\dd t}$ Usually it is more convenient to use concentrations instead of amounts. The concentration $C_\text{A}$ of substance A in a volume $V$ is given by $C_\text{A}=n_\text{A}/V$ and so we can define the conversion rate $r$ in terms of concentrations as $r=\frac{v}{V}=-\frac{1}{a}\frac{\dd C_\text{A}}{\dd t}$ We could just as have been looking at a reaction product and define the formation rate in terms of quantities and concentrations formed. The conversion rate and the rate of formation of a reaction are equal to each other and, therefore, one uses the term reaction rate and denotes it with the symbol $r$ .

Agreement Chemists usually denote the concentration of a substance A with [A]. Henceforth we will preferably use this format. In mathematical formulas, the square brackets indicate concentrations in this context.

For the general reaction $a\,\text{A}+b\,\text{B}+\cdots \longrightarrow p\,\text{P}+q\,\text{Q}+\cdots$ we have for the rate of reaction $r$ $r=-\frac{1}{a}\frac{\dd[\text{A}]}{\dd t} =-\frac{1}{b}\frac{\dd[\text{B}]}{\dd t}=\frac{1}{p}\frac{\dd[\text{P}]}{\dd t}=\frac{1}{q}\frac{\dd[\text{Q}]}{\dd t}=\cdots$

For the chemical reaction $2\text{NO} + \text{O}_2 \longrightarrow 2\text{NO}_2$ which describes the formation of nitrogen dioxide from nitrogen monoxide and oxygen, the reaction rate $r$ is in terms of gas concentration: $\frac{\dd[\text{NO}]}{\dd t}=-2r,\quad \frac{\dd[{\text{O}_2}]}{\dd t}=-r,\quad \frac{\dd[{\text{NO}_2}]}{\dd t}=2r$