Problem 6 - A Complete Resource Book in Chemistry for JEE Main

Question

For a first-order reaction, the half-life period is independent of (a) initial concentration (b) cube root of initial concentration (c) first power of final concentration (d) square root of final concentration

Step-by-Step Solution

Answer

The half-life is independent of all provided options.

Step 1: Understanding Reaction Order

In a first-order reaction, the rate depends on the concentration of only one reactant. The rate law for this reaction is \( r = k[A]^1 \), where \( A \) is the concentration of the reactant, and \( k \) is the rate constant.

Step 2: Determining Half-life Dependency

For a first-order reaction, the half-life \( t_{1/2} \) is given by the formula \( t_{1/2} = \frac{0.693}{k} \). This shows that the half-life is independent of the concentration of the reactant.

Step 3: Analyzing Each Option

Since the half-life formula for a first-order reaction does not include concentration terms, the half-life is independent of any power or root of initial or final concentration. Thus, the correct answer is it is independent of all the given options.

Step 4: Choosing the Correct Option

All mentioned options imply a dependency on some form of concentration, but for a first-order reaction's half-life, concentration does not affect it. Therefore, it is independent of all the options provided.

Key Concepts

Reaction Rate Law

The reaction rate law provides a mathematical description of how the concentration of reactants affects the rate of a chemical reaction. For first-order reactions, which involve just one reactant affecting the rate, the rate law is simply expressed as:
\[ r = k[A]^1 \]
Here, \( r \) represents the reaction rate, \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant. Because the exponent is 1, it indicates a linear relationship: as the concentration of \( A \) increases, the reaction rate increases proportionally. This simplicity makes first-order reactions straightforward to analyze, as you can directly predict the change in rate with changes in concentration.

Half-life Calculation

Half-life, denoted as \( t_{1/2} \), is the time required for the concentration of a reactant in a first-order reaction to decrease to half of its initial concentration. One of the key features of first-order reactions is that their half-life is constant and independent of the initial concentration of the reactants. This is encapsulated in the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
In this equation, \( 0.693 \) is the natural logarithm of 2, reflecting that half of the substance reacts. The constant \( k \) is the rate constant. The fact that the half-life remains unchanged regardless of how much reactant you start with is a unique characteristic of first-order reactions, making it very reliable for predicting reaction times.

Concentration Dependency

Concentration dependency refers to how the concentration of reactants affects both the rate of reaction and the duration it takes for half of the reactants to be consumed (half-life). For first-order reactions, this relationship takes a special form.

• The reaction rate is directly proportional to the concentration, as shown by the expression \( r = k[A]^1 \).
• However, unlike other reaction orders, the half-life does not depend on the initial or final concentration. This is evident from the half-life formula \( t_{1/2} = \frac{0.693}{k} \), which does not include a concentration term.

Therefore, no matter how much or how little of the reactant you start with, the half-life will remain constant, a peculiarity that simplifies the study and application of first-order kinetics.