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.
Stoichiometric coefficients 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: \[\frac{\dd[\text{NO}]}{\dd t}=-2r,\quad \frac{\dd[{\text{O}_2}]}{\dd t}=-r,\quad \frac{\dd[{\text{NO}_2}]}{\dd t}=2r\]
