Question

How is the half-life of a first-order reaction affected by the initial concentration of the reactant?

Short answer

The half-life of a first-order reaction is independent of the initial concentration of the reactant.

Step-by-step solution

Understanding First-Order Reactions

First-order reactions are chemical reactions whose rate depends linearly on only one reactant concentration. Their rate law can be expressed as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant. The half-life of a first-order reaction (t_1/2) is the time required for the concentration of the reactant to decrease to half its initial value.

Deriving Half-Life for First-Order Reactions

The mathematical representation of the half-life for a first-order reaction is given by the formula t_1/2 = (ln(2))/k, where ln(2) is the natural logarithm of 2. This equation shows that the half-life of a first-order reaction depends only on the rate constant k and is independent of the initial concentration of the reactant.

Analyzing the effect of Initial Concentration on Half-Life

Since the half-life equation for a first-order reaction does not include the initial concentration of the reactant, changes in the initial concentration do not affect the half-life. The half-life remains constant as long as the reaction mechanism and temperature (which can affect the rate constant) are unchanged.

Key concepts

First-Order Reactions

First-order reactions are a cornerstone concept in chemical kinetics, which is the study of the rates of chemical processes. In a first-order reaction, the rate at which the reaction occurs is directly proportional to the concentration of a single reactant. In simple terms, this means that if you have a larger quantity of the reactant, the reaction will proceed faster, and if you have less, it'll go slower.

One of the key features of first-order reactions is their distinctive half-life, which is the time it takes for half of the reactant to be used up in the reaction. Unlike other orders of reactions, the half-life of a first-order reaction doesn't depend on the initial amount of reactant; it's a fixed value that is characteristic of the particular reaction at a given temperature. This is a useful property, as it means you can easily predict how long it'll take for half of the reactant to react, regardless of how much you started with.

Reaction Rate Law

The reaction rate law is a mathematical way to describe the speed of a chemical reaction. It tells us how the rate of the reaction is affected by the concentration of the reactants involved. For first-order reactions, the rate law expression has a specific form.

The rate law for a first-order reaction is often written as \( rate = k[A] \), where:\

\
• \( rate \) is the speed at which the reactant concentration decreases over time.
\
• \( k \) is the rate constant, a number that incorporates various factors influencing the reaction rate, including temperature.
\
• \( [A] \) represents the concentration of the reactant 'A'.
\

This equation tells us that the reaction rate is equal to the rate constant times the concentration of the reactant. When the concentration of the reactant is higher, the rate of the reaction is greater. This fundamental relationship allows for calculations of how the reaction will proceed under different conditions.

Rate Constant

The rate constant, symbolized as \( k \), is a vital part of the reaction rate law. It is a unique number for every chemical reaction at a particular temperature and provides valuable information about the reaction's speed. The rate constant connects the concentration of the reactants to the rate of the reaction. For a first-order reaction, it's especially interesting because it remains constant regardless of the concentration. This means that, under constant conditions, the rate at which the reaction proceeds is directly proportional to the reactant's concentration at any given moment.

The value of the rate constant is influenced by various factors, including the presence of a catalyst, the temperature of the reaction environment, and even the solvent the reaction occurs in. However, it is not affected by the initial concentration of reactants, setting the stage for how half-life is a consistent measurement in first-order reactions.

Effect of Initial Concentration on Half-Life

In the context of first-order reactions, half-life represents a key concept—predictability. It is defined as the time required for the concentration of the reactant to fall to half its initial value. For a first-order reaction, the half-life is given by the equation \( t_{1/2} = \frac{\ln(2)}{k} \), where \( \ln(2) \) is the natural logarithm of 2 and \( k \) is the rate constant.

Notably, the initial concentration of the reactant does not appear in the half-life expression, which is somewhat unusual. Regardless of how much reactant you start with, the half-life stays the same. This independence from initial concentration is highly valuable as it simplifies calculations and makes half-life a reliable and consistent indicator of reaction progress in a first-order reaction context.