First-Order Reactions

First-Order Reactions describe chemical processes where the transformation rate directly depends on the concentration of a single substance. We define first-order reactions as reactions where the initial concentration of the initial element doubles then its rates also doubles.

In this article, we will learn about, First-Order Reaction Definition, Differential Rate Law for the First-Order Reaction, Integrated Rate Law for a First-Order Reaction, a Graphical Representation of a First-Order Reaction, Pseudo First-Order Reaction, and others in detail.

What is a First-Order Reaction?

First-Order Reaction is a chemical process where the rate of transformation of a substance is directly proportional to its concentration. As the concentration decreases, the rate of the reaction diminishes at a corresponding rate. These reactions are vital in various natural and artificial phenomena, like radioactive decay and specific chemical changes.

Examples of First-Order Reactions

Some examples of the First-Order Reactions are,

• Radioactive Decay: Isotopes such as carbon-14 decay at a rate proportional to their concentration.
• Chemical Decomposition: Decomposition of hydrogen peroxide into water and oxygen follows first-order kinetics.

Some other examples of first order reactions are,

• 2N₂O₅ → O₂ + 4NO₂
• 2H₂O₂ → 2H₂O + O₂

Differential Rate Law for a First-Order Reaction

The differential rate law for a first-order reaction is expressed as r=k[A] signifying the rate(r) directly proportional to the concentration .

In a first-order reaction, the rate of change of the reactant concentration is directly proportional to the reactant concentration itself.

r = k[A]

Where,

• r Represents Rate of Reaction
• k Represents Rate Constant

Integrated Rate Law for a First-Order Reaction

Rate laws explain how fast a reaction happens based on how much of the substances (reactants) are present. They are figured out through experiments and tell us how the speed changes depending on the amounts of reactants. Integrated rate laws show how these amounts change over time during reactions, which helps us understand how reactions occur.

For a first-order reaction (r = k[A]), the integrated rate law (ln([A]₀/[A]) = -kt) helps predict reaction rates. It involves initial concentration ([A]₀), concentration at time t ([A]), rate constant (k), and time (t).

The integrated rate law for a first-order reaction is

ln([A]₀/[A]) = kt

Where,

• [A]_t is the Concentration at Time t
• [A]₀ is the Initial Concentration

This integrated rate law helps in understanding how the concentration of a reactant changes over time, allowing the determination of reactant concentrations at various time intervals.

Derivation Integrated Rate Law for a First-Order Reaction

For a first-order reaction Rate law equation,

r = k[A]

Integrated rate law: ln([A]₀/[A]) = -kt

To derive the integrated rate law for a first-order reaction, let’s integrate the rate expression with respect to time (t) from [A]₀ at time 0 to [A] at time t.

Starting with the rate law equation:

r = k[A]

This equation represents the rate of the reaction at any given time. It signifies that the rate of change of concentration of the reactant [A] is directly proportional to its concentration at that time.

Now, integrating the rate law equation for a first-order reaction,

∫(1/[A]) d[A] = -k ∫dt

Integrating from [A]₀ to [A] at time t and from 0 to t:

∫(1/[A]) d[A] = -k ∫dt

Integrating from [A]₀ to [A] and from 0 to t, respectively, we get:

ln([A]) – ln([A]₀) = -kt

Simplifying further

ln([A]/[A]₀) = -kt

Rearranging terms, we arrive at the integrated rate law for a first-order reaction:

ln([A]₀/[A]) = kt

⇒ [A]₀/[A] = e^kt

[A] = [A]_o e^-kt

This integrated rate law equation allows us to relate the concentration of the reactant [A] at any time (t) to its initial concentration [A]₀ and the rate constant (k) for the first-order reaction.

​Graphical Representation of a First-Order Reaction

When plotted on a graph, a first-order reaction yields a straight line when ln[A] is plotted against time. This graphical representation confirms the first-order nature of the reaction.

The image added below shows the First-Order Reactions,

Characteristics of First-Order Reactions

Various characteristics of First-Order Reactions are,

Constant Half-Life: First-order reactions maintain an unchanging half-life irrespective of the initial concentration of the reactant. This consistency implies that it takes the same time duration for the reactant’s concentration to decrease by half, regardless of the initial amount.

Sole Dependence on Single Reactant: The rate of the reaction in a first-order process is exclusively determined by the concentration of a sole reactant. As this concentration declines, the reaction rate decreases proportionately. This dependency on a single reactant’s concentration distinguishes first-order reactions from other reaction types, simplifying the analysis of their kinetics.

• Rate of Reaction is not inversely proportional to the reactant concentration.
• Rate of Reaction is proportional to the square of the reactant concentration.
• Square Root of Reactant concentration determines the rate of the reaction.

Pseudo First-Order Reaction

Pseudo first-order reactions might seem like they follow the simple rules of first-order reactions, even though they’re more complex. They act like they have just one step or one substance involved, similar to a regular first-order reaction. However, in reality, these reactions might have many steps or substances participating. This imitation of simple behavior happens because of special situations or when one substance is much more than the others. Understanding these reactions properly needs more detailed study to figure out how they actually work.

Examples of Pseudo First Order Reaction

Hydrolysis of Ethyl Accetate is a Pseudo First Order reaction is,

• CH₃COOC₂H₅ + H₂O → CH₃COOH + C₂H₅OH

Other examples of pseudo first order reactions is Inversion of Cane Sugar

• C₁₂H₂₂O₁₁ + H₂O → C₆H₁₂O₆ + C₆H₁₂O₆

Half-Life of First-Order Reaction

It is the time taken for the concentration of a reactant to reduce to half its initial value. First-order reactions exhibit a constant half-life, irrespective of the initial concentration of the reactant. The half life of the reaction is represnted as

t_1/2 = ln 2 /k

Now, ln 2 = 0.693

t_1/2 = 0.693 /k

Where

• t _1/2 Represnts Half Life
• k Represents Rate Constant

Read More,

• Factors Affecting Rate of a Chemical Reaction
• Integrated Rate Equations
• Temperature Dependence of the Rate of a Reaction

Examples on First Order Reaction

Example 1: In a first-order reaction (A -> B), the concentration of the reactant decreases from 0.1 M to 0.025 M in 60 minutes. Calculate the rate constant for this reaction.

Solution:

Given,

• Initial concentration ([A]₀) = 0.1 M
• Final concentration ([A]) = 0.025 M
• Time taken (t) = 60 minutes = 3600 seconds

Rate constant (k) = ln([A]₀/[A]) / t

k = ln (0.025/0.1) / 3600

k ≈ 1.93 × 10⁻⁴ s⁻¹

Example 2: The half-life of a first-order reaction is 40 minutes. If the initial concentration of the reactant is 0.4 M, what will be the concentration after 120 minutes?

Solution:

Given,

• Half-life (t₁/₂) = 40 minutes
• Initial concentration ([A]₀) = 0.4 M
• Time (t) = 120 minutes

Rate constant (k) = ln(2) / t₁_/₂

k = 0.693/40

k ≈ 0.0173 min⁻¹

Concentration ([A]) = [A]₀ × e^(-kt)

[A] ≈ 0.1 M

Example 3: If a first-order reaction has a rate constant of 5.6 × 10⁻³ s⁻¹, calculate its half-life.

Solution:

Given,

• Rate Constant (k) = 5.6 × 10⁻³ s⁻¹

Half-life (t₁_/₂) = ln(2) / k

t₁_/₂ ≈ 124 s

Example 4: A reaction follows first-order kinetics with a rate constant of 0.02 min⁻¹. If the initial concentration is 1.5 M, what will be the concentration after 60 minutes?

Solution:

Given,

• Rate Constant (k) = 0.02 min⁻¹
• Initial Concentration ([A]_o) = 1.5 M
• Time (t) = 60 minutes

Concentration ([A]) = [A]₀ × e^(-kt)

[A] = 1.5 × e^-(0.02)(60)

[A] ≈ 0.556 M

Example 5: A sample undergoes a first-order reaction, and after 30 seconds, its concentration decreases to 20% of the initial concentration. What is the reaction’s rate constant?

Solution:

Given,

• Time (t) = 30 s
• Percentage of Initial Concentration (20%) = 0.20

Rate Constant (k) = -ln(0.2) / t

k = – (-1.609437)/30

k ≈ 0.046 s⁻¹

Practice Problems on First Order Reaction

Problem 1: For a first-order reaction, if the concentration of the reactant decreases from 0.4 M to 0.1 M in 50 minutes, calculate the rate constant for this reaction.

Problem 2: The half-life of a first-order reaction is 30 minutes. If the initial concentration of the reactant is 0.2 M, what will be the concentration after 90 minutes?

Problem 3: A substance decays following first-order kinetics. If its initial concentration is 2.0 M and after 60 seconds, the concentration reduces to 1.0 M, determine the rate constant.

Problem 4: If a first-order reaction has a rate constant of 8.0 × 10⁻⁴ s⁻¹, what will be the concentration of the reactant after 5 minutes if the initial concentration was 0.5 M?

Problem 5: In a first-order reaction, the concentration of the reactant decreases from 0.8 M to 0.2 M in 40 minutes. Calculate the time required for the concentration to reduce from 0.6 M to 0.15 M.

First Order Reaction-FAQs

1. What is the Definition of a First-Order Reaction?

First order reactions are reaction where the rate is proportional to the concentration of a single reactant.

2. How does Rate Change in a First-Order Reaction?

A first-order reaction is a chemical reaction where the rate is directly proportional to the concentration of a single reactant.

3. What is Characteristic of Half-Life in a First-Order Reaction?

Half life of a first order reaction remains constant, irrespective of the initial concentration.

4: What are Examples of First-Order Reactions in Daily-Life?

Radioactive Decay, Drug Degradation, and Chemical Decomposition of Various Compounds are common examples of First Order Reactions.

5. Does Temperature Affect the Rate Constant in a First-Order Reaction?

Yes, the rate constant typically increases with temperature due to enhanced molecular collisions.

6. Is First-Order Reaction Time-Dependent?

Yes, first-order reactions exhibit a time-dependent decay of reactants.

7. What are Pseudo-First-Order Reactions?

Pseudo-first-order reactions are complex reactions that appear to follow first-order kinetics under certain conditions.