First-Order Reaction Kinetics: Complete Guide

A first-order reaction is a chemical reaction where the rate of transformation of a substance is directly proportional to its concentration. As concentration decreases, the rate diminishes correspondingly. These reactions are vital in natural and artificial phenomena, including radioactive decay (e.g., carbon-14 dating), certain chemical decompositions, and drug metabolism. Understanding first-order kinetics allows scientists to predict how fast reactants disappear and to design efficient chemical processes.

where \(k\) is the rate constant (units: time⁻¹, e.g., s⁻¹).

📘 Differential and Integrated Rate Laws

The differential rate law expresses how the rate depends on concentration at any instant. For a first-order reaction, the rate is linearly dependent on \([A]\). To relate concentration to time, we integrate:

The integrated rate law shows exponential decay. A plot of \(\ln[A]\) versus time gives a straight line with slope \(-k\), confirming first-order behaviour.

⌛ Half-Life of a First-Order Reaction

The half-life \(t_{1/2}\) is the time required for the concentration to fall to half its original value. Substituting \([A] = \frac{1}{2}[A]_0\) into the integrated law:

Key insight: For a first-order reaction, the half-life is constant and independent of the initial concentration. This unique property is exploited in radiocarbon dating and pharmacokinetics.

📊 Graphical Representations

🔬 Examples of First-Order Reactions

• Radioactive decay: \( ^{14}C \to ^{14}N + e^- + \bar{\nu}_e \) – used in archeological dating.
• Decomposition of hydrogen peroxide: \( 2H_2O_2 \to 2H_2O + O_2 \) (often first-order in H₂O₂).
• Decomposition of dinitrogen pentoxide: \( 2N_2O_5 \to 4NO_2 + O_2 \).
• Hydrolysis of sucrose (inversion of cane sugar): \( C_{12}H_{22}O_{11} + H_2O \to C_6H_{12}O_6 + C_6H_{12}O_6 \) (pseudo-first-order under excess water).

🧪 Pseudo-First-Order Reactions

A pseudo-first-order reaction appears to follow first-order kinetics even though the true rate law is more complex. This occurs when one reactant is present in large excess, keeping its concentration essentially constant. The observed rate constant \(k_{\text{obs}}\) incorporates the constant concentration of the excess reactant. Examples:

• Hydrolysis of ethyl acetate: \( CH_3COOC_2H_5 + H_2O \to CH_3COOH + C_2H_5OH \). In aqueous solution, water is in large excess → rate = \(k'[ester]\).
• Inversion of cane sugar: excess water makes the reaction pseudo-first-order.
• Many drug degradation reactions in the body (plasma concentration often much lower than enzyme/metabolite concentrations).

📚 Full Derivation of Integrated Rate Law

Starting from the differential form:

Separate variables: \(\displaystyle -\frac{d[A]}{[A]} = k\,dt\).

Integrate from \([A]_0\) at \(t=0\) to \([A]\) at time \(t\):

Exponentiating both sides gives \([A] = [A]_0 e^{-kt}\).

For half-life: set \([A] = [A]_0/2\) → \(\ln 2 = k t_{1/2}\). Hence \(t_{1/2} = \ln 2 / k\).

📊 Characteristics of First-Order Reactions

• Constant half-life: Independent of initial concentration.
• Sole dependence on one reactant: The rate depends only on the concentration of a single species.
• Linear relationship: \(\ln[A]\) vs time is a straight line.
• Units of rate constant: time\(^{-1}\) (e.g., s\(^{-1}\), min\(^{-1}\)).