Third Order Reactions

Third order reactions are chemical reactions where the rate is proportional to the cube of the concentration of one reactant or the product of the concentrations of three reactants. These reactions are less common than first or second order reactions, but they are important in the study of chemical kinetics.

Rate Law for Third Order Reactions

The rate law for a third order reaction can be written in the general form:

$$ \text{Rate} = k [A]^m [B]^n [C]^p $$

where ( k ) is the rate constant, and ( m ), ( n ), and ( p ) are the orders with respect to reactants ( A ), ( B ), and ( C ), respectively. For a third order reaction, the sum of ( m + n + p ) must be equal to 3.

Specific Cases

1. One reactant: If the reaction involves a single reactant, the rate law is:

$$ \text{Rate} = k [A]^3 $$

1. Two reactants: If the reaction involves two reactants, the rate law could be:

$$ \text{Rate} = k [A]^2 [B] $$

or

$$ \text{Rate} = k [A] [B]^2 $$

1. Three reactants: If the reaction involves three different reactants, the rate law is:

$$ \text{Rate} = k [A] [B] [C] $$

Integrated Rate Law for Third Order Reactions

Unlike first and second order reactions, there is no simple integrated rate law for third order reactions. However, for a reaction with a single reactant, the integrated rate law can be derived and is given by:

$$ \frac{1}{[A]^2} = 2kt + \frac{1}{[A]_0^2} $$

where ( [A]_0 ) is the initial concentration of the reactant ( A ), and ( [A] ) is the concentration of ( A ) at time ( t ).

Half-Life of Third Order Reactions

The half-life of a third order reaction with a single reactant is given by:

$$ t_{1/2} = \frac{1}{k[A]_0^2} $$

This indicates that the half-life of a third order reaction depends on the initial concentration of the reactant.

Examples of Third Order Reactions

An example of a third order reaction is the reaction between nitric oxide and oxygen to form nitrogen dioxide:

$$ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) $$

The rate law for this reaction is:

$$ \text{Rate} = k [NO]^2 [O_2] $$

Differences Between Reaction Orders

Here is a table summarizing the differences between first, second, and third order reactions:

Feature	First Order	Second Order	Third Order
Rate Law	( k [A] )	( k [A]^2 ) or ( k [A][B] )	( k [A]^3 ), ( k [A]^2[B] ), or ( k [A][B][C] )
Integrated Rate Law	( \ln[A] = -kt + \ln[A]_0 )	( \frac{1}{[A]} = kt + \frac{1}{[A]_0} )	No simple form for multiple reactants
Half-Life	Constant	Depends on initial concentration	Depends on initial concentration squared
Plot for Linear Relationship	( \ln[A] ) vs. ( t )	( \frac{1}{[A]} ) vs. ( t )	( \frac{1}{[A]^2} ) vs. ( t )

Conclusion

Third order reactions are an important part of chemical kinetics, providing insight into the mechanisms of more complex reactions. Understanding the rate laws, integrated rate laws, and half-lives of these reactions is crucial for predicting the behavior of chemical systems and for designing chemical processes. While they are less common and more complex than first and second order reactions, third order reactions can still be analyzed and understood through careful experimentation and mathematical analysis.