# Second-Order Reactions

## A second-order reaction depends on concentration to the power of two, the rate law is: $r=-\frac{dA}{dt}=k[A]^{2}$.

#### Key Points

• A second-order reaction depends on the concentrations of one second-order reactant or two first-order reactants.

• If we know the rate constant and the initial concentration, we can find the concentration [A] at any time of interest during the reaction.

• In the rate equation, conflating the 2 inside the constant for the first derivative form will only make it required in the second integrated form. The option of keeping the 2 out of the constant in the derivative form is considered more correct.

#### Terms

• A second-order reaction depends on the concentrations of one second-order reactant or two first-order reactants.

• the step by step sequence of elementary transformations by which overall chemical change occurs

• a law which states that the entropy of an isolated system never decreases, because isolated systems spontaneously evolve towards thermodynamic equilibrium

#### Figures

1. ##### Nitrogen Dioxide

Nitrogen Dioxide undergoes a second order reaction to become nitric oxide and oxygen.

For a generic reaction: $aA + bB \rightarrow C$ with no intermediate steps in its reaction mechanism (that is, an elementary reaction), the rate is given by:

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

where [A] and [B] express the concentration of the species A and B, respectively (usually in moles per liter (molarity, M)); x and y are the respective stoichiometric coefficients of the balanced equation; they must be determined experimentally. k is the rate coefficient or rate constant of the reaction. The value of this coefficient k depends on conditions such as temperature, ionic strength, surface area of the adsorbent or light irradiation. For elementary reactions, the rate equation can be derived from first principles using collision theory. Again, x and y are not always derived from the balanced equation.

A reaction is said to be second order when the overall order is two. The rate of a second-order reaction may be proportional to one concentration squared, or (more commonly) to the product of two concentrations. For a second order reaction, its reaction rate is given by:

$-\frac{d[A]}{dt}=2k{[A]}^{2}$

or

$-\frac{d[A]}{dt} = k[A][B]$

or

$-\frac{d[A]}{dt} = 2k[B]^2$

In several popular kinetics books, the definition of the rate law for second-order reactions is:

$-\frac{d[A]}{dt}=k[A]^{2}$

Conflating the 2 inside the constant for the first derivative form will only make it required in the second integrated form. The option of keeping the 2 out of the constant in the derivative form is considered more correct because it is almost always used in peer-reviewed literature, tables of rate constants, and simulation software.

Consider then a second order reaction, such as butadiene dimerization. The general second order reaction A→products has the rate law:

$Rate=−\frac{d[A]}{dt}$

$=k{[A]}^{2}$

We can use Calculus to find the function [A](t) from the above equation. The result is most easily written as:

$\frac{1}{[A]}=\frac{1}{{[A]}_{0}}+k(t)$

Note that, as t increases, $\frac{1}{[A]}$ increases and therefore [A] decreases.  This equation reveals that, for a reaction which is second order in the reactant A, we can plot ${t}_{\frac{1}{2}}=\frac{1}{k{[A]}_{0}}$

This shows that, unlike a first order reaction, the half-life for a second order reaction depends on how much material we start with. The more concentrated the reactant is, the shorter the half-life.

An example of an second order reaction is the process of nitrogen dioxide turning into nitric oxide and oxygen Figure 1.

#### Key Term Glossary

balanced equation
The law of conservation of mass dictates the quantity of each element does not change in a chemical reaction. Thus, each side of the chemical equation must represent the same quantity of any particular element. Similarly, the charge is conserved in a chemical reaction. Therefore, the same charge must be present on both sides of the balanced equation.
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coefficient
A constant by which an algebraic term is multiplied.
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collision theory
Any theory that relates collisions among particles to reaction rate; reaction rate depends on such factors as concentration, surface area, temperature, stirring, and the presence of either a catalyst or an inhibitor.
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Collision Theory
Collision theory qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions.
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concentration
the proportion of a substance in a mixture
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constant
Consistently recurring over time; persistent
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ionic
of, relating to, or containing ions
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kinetics
The branch of chemistry that is concerned with the rates of chemical reactions.
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liter
a non-SI metric system unit of volume equal to 1 cubic decimeter (dm^3), 1,000 cubic centimeters (cm^3) or 1/1,000 cubic meter (m^3)
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molarity
The concentration of a substance in solution, expressed as the number moles of solute per litre of solution.
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mole
In the International System of Units, the base unit of the amount of substance; the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. Symbol: mol. The number of atoms in a mole is known as Avogadro’s number.
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nitrogen
a chemical element (symbol N) with an atomic number of 7 and atomic weight of 14.0067
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oxide
a binary chemical compound of oxygen with another chemical element
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oxygen
a chemical element (symbol O) with an atomic number of 8 and relative atomic mass of 15.9994
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product
a chemical substance formed as a result of a chemical reaction
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Rate law
The rate law or rate equation for a chemical reaction is an equation that links the reaction rate with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders).
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reactant
Any of the participants present at the start of a chemical reaction. Also a molecule before it undergoes a chemical change.
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reaction mechanism
the step by step sequence of elementary transformations by which overall chemical change occurs
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reaction rate
how fast or slowly a reaction takes place
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stoichiometric coefficient
The number of molecules of a given component that participate in the reaction as written.
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temperature
A measure of cold or heat, often measurable with a thermometer.