Problem 64 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Which is correct about first-order reaction? (a) \(t_{0.5}=50 \mathrm{~s}, t_{0.75}=100 \mathrm{~s}\) (b) \(t_{0.5}=50 \mathrm{~s}, t_{0,75}=75 \mathrm{~s}\) (c) \(t_{0.5}=50 \mathrm{~s}, t_{0,75}=50 \mathrm{~s}\) (d) \(t_{0.5}=50 \mathrm{~s}, t_{0,25}=25 \mathrm{~s}\)

Step-by-Step Solution

Answer

Option (c) is correct as it shows that the time for 75% of the reaction to complete (\(t_{0.75}=50 \text{s}\text{} s\text{s}{s}\text{s}\text{s}{s} s\mathrm{s} \ 0Y)Y to \ 0Y)Y s\mathrm{s} \ 0Y)Y s\mathrm{s} \ 0Y)Y to \ 0Y)Y s\mathrm{s} \ 0Y} \ 0Y)Y s\mathrm{s}) is the same as the half-life period, which is a characteristic of first-order reactions.

Step 1: Understanding First-Order Reactions

For a first-order reaction, the half-life (the time it takes for half of the reactant to be used up) is constant and is not dependent on the initial concentration of the reactant. This means that regardless of the amount of reactant remaining, the time it takes for half of it to react and form product is the same at any point during the reaction.

Step 2: Analyzing the Options

We need to assess which option correctly demonstrates the property of a first-order reaction that the half-life period remains the same. The symbol \(t_{0.5}\) represents the half-life, meaning the time it takes for the concentration of the reactant to decrease by half. The symbol \(t_{0.75}\) or \(t_{0,25}\) is used for the time it takes for 75% or 25% of the reactant to be consumed, respectively.

Step 3: Eliminating Incorrect Options

For a first-order reaction, once the half-life \(t_{0.5}\) has passed, another half-life period will cause the reaction to be 75% complete, not 100% or 25%. Option (a), (b), and (d) suggest incorrect relationships between \(t_{0.5}\), \(t_{0.75}\), and \(t_{0,25}\), so they can be eliminated.

Step 4: Confirming the Correct Option

Option (c) indicates that \(t_{0.5}\) is the same as \(t_{0,75}\). This is correct for a first-order reaction because after one half-life period, 50% of the reactant remains, and after another identical half-life period, an additional half of the remaining 50%, or 25%, is used up, totaling 75% consumption of the original reactant. Therefore, option (c) correctly represents the characteristic of a first-order reaction.

Key Concepts

Reaction Kinetics

Reaction kinetics is a subfield of physical chemistry focusing on understanding the rates of chemical processes. It involves the study of how fast reactions occur and the factors that can influence these rates, such as temperature, concentration of reactants, and catalysts. For a first-order reaction, which is a common type of chemical reaction, the rate depends linearly on the concentration of one reactant. In simpler terms, if you were to double the concentration of the reactant in a first-order reaction, the rate of the reaction would also double. This linear relationship makes first-order reactions particularly interesting and useful in a variety of chemical applications.

For educational purposes, it's crucial to grasp that the concept of a reaction's order is a mathematical description of the dependence between the rate and concentration. A first-order reaction implies that the rate law, which governs how the chemical reaction proceeds, can be described by an equation where the rate of reaction is directly proportional to the concentration of the reactant.

Chemical Half-Life

The chemical half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. In the realm of first-order reactions, the half-life is constant, meaning that it does not change regardless of the concentration of the reactant. This is a unique property of first-order reactions and serves as an essential concept for understanding how these reactions proceed over time.

Half-life is commonly represented by the symbol \(t_{0.5}\) and is a pivotal concept in various fields, including pharmacology, where it helps to determine how frequently a drug should be administered to maintain its therapeutic effect. It's important for students to memorize that for first-order reactions, no matter when you start timing (i.e., irrespective of the initial concentration), it always takes the same amount of time for half of the remaining reactant to be used up.

Rate of Reaction

The rate of reaction is a measure of how quickly the reactants are converted into products in a chemical reaction. For first-order reactions, the rate is proportional to the concentration of the single reactant. This is expressed mathematically by the rate law, which for a first-order reaction is written as \(rate = k[A]\), where \(k\) is the rate constant and \([A]\) is the concentration of the reactant A.

Understanding the rate of a reaction is fundamental for calculating reaction timings in various applications, such as industrial synthesis and environmental chemistry. A high rate of reaction might indicate a fast process, suitable for quick manufacturing, while a slower rate could apply to processes that need to be controlled over longer periods, like the release of medication into the body.

Physical Chemistry

Physical chemistry is the branch of chemistry that deals with the physical structure of chemical compounds and the energetic changes accompanying chemical reactions. It intersects with physics to explain chemical phenomena at the molecular and atomic levels. Within physical chemistry, reaction kinetics, thermal dynamics, quantum chemistry, and electrochemistry are key areas of focus.

First-order reactions are a vital topic in physical chemistry because they exemplify the intersection of the chemical kinetics sub-discipline with real-world processes. As students delve into physical chemistry, they explore how principles like enthalpy, entropy, and free energy contribute to our understanding of chemical reactions. By connecting these concepts to the study of reaction rates and orders, physical chemistry helps bring a deeper understanding to how and why chemical reactions happen the way they do.