First-Order Reaction Calculator

What is a First-Order Reaction?

In chemistry, a first-order reaction is a type of chemical reaction where the rate (speed) at which the reaction happens depends directly on the concentration of just one reactant. If you double the concentration of that reactant, the reaction rate also doubles. These reactions are very common and important in many natural and industrial processes.

The rate law for a first-order reaction is expressed as:

Here, 'Rate' is how fast the reaction proceeds, '[A]' is the concentration of the reactant, and 'k' is the rate constant, which tells us how quickly the reaction converts reactants into products.

The Integrated Rate Law: Predicting Concentration Over Time

While the rate law tells us the instantaneous speed, the integrated rate law allows us to predict the concentration of a reactant at any given time during a first-order reaction, or to find the time it takes for a concentration to change. It's derived from the rate law and is often used in its logarithmic form:

Where:

• [A] is the concentration of the reactant at time 't'
• [A]₀ is the initial concentration of the reactant (at time t=0)
• k is the rate constant
• t is the time elapsed

Half-Life (t½): A Constant for First-Order Reactions

The half-life (t½) of a reaction is the time it takes for the concentration of a reactant to decrease to half of its initial value. A remarkable feature of first-order reactions is that their half-life is constant and does not depend on the initial concentration of the reactant. This means it takes the same amount of time for the concentration to drop from 100% to 50% as it does from 50% to 25%, and so on.

The formula for the half-life of a first-order reaction is:

Since ln(2) is approximately 0.693, this simplifies to: t₁/₂ ≈ 0.693/k. This constant half-life makes first-order reactions particularly easy to analyze and predict.

Common Examples of First-Order Reactions

First-order reactions are found in many important chemical and physical processes:

• Radioactive Decay: The breakdown of unstable atomic nuclei (e.g., Carbon-14 dating) follows first-order kinetics. This is a classic example where the half-life concept is widely used.
• Decomposition Reactions: Many reactions where a single compound breaks down into simpler substances (e.g., the decomposition of N₂O₅).
• Enzyme-Catalyzed Reactions: Under certain conditions, some biological reactions catalyzed by enzymes can exhibit first-order behavior.
• Drug Metabolism: The elimination of many drugs from the body often follows first-order kinetics, which is crucial in pharmacology.
• Environmental Processes: The degradation of pollutants in the environment can sometimes be modeled as first-order reactions.

Graphical Analysis of First-Order Reactions

One way to confirm if a reaction is first-order and to determine its rate constant is through graphical analysis. If you plot the natural logarithm of the reactant's concentration (ln[A]) against time (t), a first-order reaction will yield a straight line.

• The slope of this straight line will be equal to -k (negative of the rate constant).
• The y-intercept of the line will be ln[A]₀ (the natural logarithm of the initial concentration).