# First Order Reaction

## What is a First-order Reaction

A first-order reaction is a chemical reaction where the reaction rate depends linearly on the reactant’s concentration. In other words, if the concentration is doubled, then the reaction rate is also doubled. A first-order reaction can have one reactant, as with the decomposition reaction, or two reactants ^{[1-6]}.

## First-order Reaction Rate Law

The kinetics of a chemical equation can be studied using rate law, which gives a relationship between the reaction rate and the concentration of the reactants. The rate law is expressed in differential and integral forms ^{[1-7]}.

### Differential Rate Law

The differential rate law gives the derivative of the reactant’s concentration with time. For a first-order reaction, it is given as,

*R = – d[A]/dt = k [A]*

Where,

*R* is the reaction rate

*[A]* is the concentration of the reactant A

*k* is the rate constant

The term *d[A]/dt* is the derivative of *[A]* with time.

**Units**

The unit for concentration is **moles**, the unit for the reaction rate is **moles/second**, and that for the rate constant is **1/second**. Therefore, the numerical value of *k* for a first-order reaction is independent of the unit in which concentration is expressed.

### Integrated Rate Law

The integrated rate law is beneficial in predicting how much of a reactant will remain at some time or how long it will take for the concentration to drop to some fixed amount (e.g., one-half). In order to derive the integrated form of the first-order reaction, the differential form must be integrated over concentration and time.

The integral form is given by,

*[A] = [A] _{o }exp (-kt)*

Where,

*[A] _{o}* is the concentration at time

*t = 0*

*[A]* is the concentration at time *t*

From the above expression, the concentrations can be expressed in terms of their natural logarithm.

ln *[A] = *ln* [A] _{o }– kt*

The above equation is that of a straight line and can be applied to determine the rate constant.

**How to Determine the First-order Reaction**

In order to determine if it a reaction is a first-order reaction, plot the natural logarithm of the concentration versus time and see whether the graph is linear. If the graph is linear and has a negative slope, then the reaction must be first-order.

**Half-life of a First-order Reaction ^{[9]}**

The half-life of a reaction is defined as the time taken for the reactant’s concentration to reduce to one-half. Therefore,

*[A] = [A] _{o }/ 2*

The half-life is given by,

*t _{1/2} = 0.693 / k*

## Characteristics of First-order Reaction

Here are some facts and characteristics of the first-order reaction.

- Reaction rate is proportional to the reactant’s concentration
- Half-life is constant and independent of the reactant’s concentration

## Examples of First-order Reaction

Here are some examples of first-order chemical reactions.

1. Trimethyl bromomethane ((CH_{3})_{3}CBr) reacts with sodium hydroxide (NaOH) to give trimethyl methanol ((CH_{3})_{3}COH) and sodium bromide (NaBr) ^{[8]}.

(CH_{3})_{3}CBr + NaOH → (CH_{3})_{3}COH + NaBr

2. Decomposition of hydrogen peroxide (H_{2}O_{2}) into water (H_{2}O) and oxygen (O_{2})

2 H_{2}O_{2} (l) → H_{2}O (l) + O_{2} (g)

3. Hydrolysis of methyl acetate (CH_{3}COOCH_{3}) in the presence of mineral acids

CH_{3}COOCH_{3} (aq.) + H_{2}O (l) → CH_{3}COOH (aq. + CH_{3}OH (aq.)

4. Decomposition of ammonium nitrite (NH_{4}NO_{2}) in aqueous solution

NH_{4}NO_{2} (aq.) → N_{2} (g) + 2 H_{2}O (l)

5. Hydrogenation of ethene (C_{2}H_{4})

C_{2}H_{4} (g) + H_{2} (g) → C_{2}H_{6} (g)

## Pseudo First-order Reaction

In some chemical reactions, the concentration of one of the reactants is so high that it does not change with time. In other words, the concentration remains constant throughout the reaction.

Consider the following reaction,

A + B → Products

The rate is given by,

*R = k [A] [B]*

This equation is the rate law for a second-order reaction. Suppose the concentration of B remains constant throughout the reaction. Then, it can be incorporated into the rate constant term *k,* and the rate *R* is given by,

*R = k’ [A]*

Where,

*k’ = k [B]* is the new rate constant.

Therefore, the pseudo first-order reaction is a second-order reaction made to behave like a first-order reaction.

**Example**: Hydrolysis of sucrose (C_{12}H_{22}O_{11}) gives glucose (C_{6}H_{12}O_{6}) and fructose (C_{6}H_{12}O_{6}).

C_{12}H_{22}O_{11} + H_{2}O → C_{6}H_{12}O_{6 }+ C_{6}H_{12}O_{6}

## Zero, First, and Second-order Reactions

The following table lists the differences between the zero, first, and second-order reactions ^{[10]}.

Zero Order | First Order | Second Order | |
---|---|---|---|

Definition | Reaction rate is constant over time and independent of the reactants’ concentration | Reaction rate is proportional to the first power of the reactant’s concentration | Reaction rate is proportional to the second power of the reactant’s concentration or product of two reactants’ concentration |

Rate equation | R = constant | R = k [A] | R = k [A]or ^{2 }R = k [A] [B] |

Concentration | [A] = kt | [A] = [A]_{o }exp (-kt) | 1 / [A] = 1 / [A]_{o} + kt |

Half-life | [A]_{o }/ 2k | 0.693 / k | 1 / k[A]_{o} |

Example | Decomposition of nitrous oxide | Decomposition of hydrogen peroxide | Decomposition of nitrogen dioxide |

**References**

- Chem.libretexts.org
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