#### Learning Objective

- Produce rate equations for elementary reactions

#### Key Points

- For a generic reaction [latex]aA + bB \rightarrow C[/latex] with no intermediate steps in its reaction mechanism (that is, an elementary reaction), the rate is given by: [latex]r=k[A]^{x}[B]^{y}[/latex] .
- For elementary reactions, the rate equation can be derived from first principles using collision theory.
- The rate equation of a reaction with a multi-step mechanism cannot, in general, be deduced from the stoichiometric coefficients of the overall reaction; it must be determined experimentally.

#### Term

- Rate lawAn equation relating the rate of a chemical reaction to the concentrations or partial pressures of the reactants.

The rate law for a chemical reaction is an equation that relates the reaction rate with the concentrations or partial pressures of the reactants. For the general reaction[latex]aA + bB \rightarrow C[/latex] with no intermediate steps in its reaction mechanism, meaning that it is an elementary reaction, the rate law is given by:

[latex]r=k[A]^{x}[B]^{y}[/latex]

In this equation, [A] and [B] express the concentrations of A and B, respectively, in units of moles per liter. The exponents *x* and *y* vary for each reaction, and they must be determined experimentally; they are *not* related to the stoichiometric coefficients of the chemical equation. Lastly, *k* is known as the rate constant of the reaction. The value of this coefficient *k* will vary with conditions that affect reaction rate, such as temperature, pressure, surface area, etc. A smaller rate constant indicates a slower reaction, while a larger rate constant indicates a faster reaction.

## Reaction Order

To reiterate, the exponents *x* and *y* are not derived from the balanced chemical equation, and the rate law of a reaction must be determined experimentally. These exponents may be either integers or fractions, and the *sum* of these exponents is known as the overall reaction order. A reaction can also be described in terms of the order of each reactant. For example, the rate law [latex]Rate=k[NO]^2[O_2][/latex] describes a reaction which is second-order in nitric oxide, first-order in oxygen, and third-order overall. This is because the value of *x* is 2, and the value of *y* is 1, and 2+1=3.

## Example 1

A certain rate law is given as [latex]Rate=k[H_2][Br_2]^\frac{1}{2}[/latex]. What is the reaction order?

[latex]x=1,\;y=\frac{1}{2}[/latex]

[latex]reaction\;order=x+y=1+\frac{1}{2}=\frac{3}{2}[/latex]

The reaction is first-order in hydrogen, one-half-order in bromine, and [latex]\frac{3}{2}[/latex]-order overall.

## Example 2

The reaction between nitric oxide and ozone, [latex]NO(g) + O_3(g)\rightarrow NO_2(g) + O_2(g)[/latex] , is first order in both nitric oxide and ozone. The rate law equation for this reaction is: [latex]Rate = k[NO]^{1}[O_{3}]^{1}[/latex]. The overall order of the reaction is 1 + 1 = 2.