Rate equation of a first order reaction

Rate Equation of a First-Order Reaction

A first-order reaction is one in which the rate of the reaction depends linearly on the concentration of a single reactant. These reactions are common in chemical kinetics and are especially important in understanding radioactive decay, enzyme kinetics, and certain types of decomposition reactions.

Definition

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

General form: A → Products

Rate law: Rate = k[A]

Where:

• Rate is the rate of the reaction.
• k is the rate constant.
• [A] is the concentration of the reactant A.

Integrated Rate Equation

The integrated form of the rate equation for a first-order reaction is:

ln[A] = ln[A]₀ – kt

or

[A] = [A]₀e^–kt

Where:

• [A]₀ is the initial concentration of reactant.
• [A] is the concentration of reactant at time t.
• k is the rate constant.
• t is the time elapsed.

Half-Life (\(t_{\frac{1}{2}}\)) of a First-Order Reaction

The rate equation for a first-order reaction, when integrated, is expressed as:

\(t_{\frac{1}{2}}\) = 0.693 / k

This means that the time required for the concentration of the reactant to reduce to half its original value remains constant throughout the reaction.

Graphical Representation

• A plot of ln[A] vs. time gives a straight line with a slope of –k.
• A plot of [A] vs. time gives an exponential decay curve.

Applications

• Radioactive decay
• Decomposition of nitrogen dioxide (NO₂)
• Pharmacokinetics of certain drugs