Problem 14 - Chemistry in 30 days

Question

The half-life period of reaction increases four-fold when the initial concentration is increased to four times its value. The order of the reaction is (a) 1 (b) 4 (c) 0 (d) \(\overline{2}\)

Step-by-Step Solution

Answer

The order of the reaction is 0 (option c).

Step 1: Understanding the Problem

In this exercise, we are asked to determine the order of a reaction based on how its half-life changes with initial concentration. We are given that the half-life quadruples when the initial concentration also quadruples.

Step 2: Identifying the Half-Life Dependency on Concentration

We know that for a reaction of order 'n', the half-life \( t_{1/2} \) is dependent on the initial concentration \( [A]_0 \) as follows: \( t_{1/2} \propto \) \( [A]_0^{1-n} \. \) We need to use this relationship to find 'n' when \( t_{1/2} \) increases by a factor of 4 when \( [A]_0 \) increases by a factor of 4.

Step 3: Setting Up the Equation

We know \( t_{1/2}' \propto \) \( (4[A]_0)^{1-n} \). From the condition given, \( t_{1/2}' = 4t_{1/2} \. \) Substituting gives us \( 4[A]_0^{1-n} = 4[A]_0^{1-n} \). This simplifies to \( 4 = 4^{1-n} \).

Step 4: Solving for 'n'

We simplify \( 4 = 4^{1-n} \). Taking logarithm base 4 on both sides or comparing the powers directly, we get \( 1 = 1-n \). Thus, \( n = 0 \).

Step 5: Conclusion

The calculated order of reaction is zero. Therefore, the correct answer is (c) 0.

Key Concepts

Half-Life Period

The half-life period, commonly denoted as \( t_{1/2} \), is the time it takes for the concentration of a reactant in a chemical reaction to decrease to half of its initial value. It is a crucial concept in reaction kinetics because it provides a measure of how fast a reaction proceeds. For reactions of different orders, the half-life behaves differently:

• For a first-order reaction, the half-life is constant and independent of the concentration.
• For a zero-order reaction, the half-life decreases as the concentration decreases.
• For reactions of other orders, it varies based on the specific relationship between concentration and rate.

Understanding half-life helps predict how long it will take for a reaction to proceed or for a substance to degrade.

Chemical Concentration

Chemical concentration refers to the amount of a substance present in a given volume of solution or mixture. It's commonly expressed in units like moles per liter (M) in chemistry. The concentration of a reactant can significantly impact the rate of a reaction.
Different concentrations can either speed up or slow down a reaction, depending on the order of the reaction. Concentration affects the hypothetical rate at which reactants collide and react: the more particles present, the more likely they will hit each other, leading to more frequent reactions. Understanding how concentration affects chemical reactions is fundamental to fields like biochemistry and pharmacology, where reaction rates are critical.

Initial Concentration

The initial concentration \( [A]_0 \) of a reactant is the concentration at the start of the reaction. It plays a pivotal role in determining the reaction rate and the half-life of chemical reactions.
In zero-order reactions, changing the initial concentration directly affects the half-life, as seen in the exercise, where quadrupling the concentration also quadruples the half-life. In contrast, for first-order reactions, the initial concentration has no impact on half-life; it remains a constant value.
The initial concentration determines how full the system is with reactants at the outset, influencing the subsequent path and speed of the reaction.

Chemical Reaction Kinetics

Chemical reaction kinetics is the branch of chemistry that studies the speed or rates at which chemical reactions occur. This area of study involves understanding how various conditions such as temperature, concentration, and catalysts affect the reaction rate. Reaction kinetics also classifies reactions by their order, which describes how the concentration of reactants affects the rate. For instance, the exercise showed how changing initial concentration altered the half-life of a reaction, leading us to deduce the reaction order.

• A zero-order reaction has a rate independent of the concentration of reactants.
• A first-order reaction has a rate directly proportional to the concentration of one reactant.
• A second-order reaction depends on either the concentration of two reactants or the square of the concentration of one reactant.

Understanding reaction kinetics is essential for predicting and controlling reaction behavior in various chemical processes.