Question

At \(300 \mathrm{~K}\) the half-life of a sample of a gaseous compound initially at \(1 . \mathrm{atm}\) is \(100 \mathrm{sec}\). When the pressure is \(0.5\) atm the half-life is \(50 \mathrm{sec}\). The order of reaction is : (a) 0 (b) 1 (c) 2 (d) 3

Short answer

The reaction is first-order because the half-life changes proportionally with the initial concentration.

Step-by-step solution

Understand the Concept of Half-life and Reaction Order

The half-life of a reaction is the time it takes for the concentration of a reactant to decrease by half. For a zero-order reaction, the half-life depends on the initial concentration; for a first-order reaction, it is constant; for a second-order reaction, it depends inversely on the initial concentration; and for a third-order reaction, the relationship is more complex. Since the exercise provides us with half-lives at different concentrations, we can use this information to determine the order of the reaction.

Compare the Changes in Half-life with Changes in Concentration

According to the information given, when the initial pressure (acting as a proxy for concentration) is halved from 1 atm to 0.5 atm, the half-life also halves from 100 sec to 50 sec. This suggests that the half-life is inversely proportional to the initial concentration, which is characteristic of a first-order reaction.

Choose the Correct Reaction Order

Since the half-life of the reaction is directly proportional to the concentration (i.e., when the concentration is halved, the half-life is also halved), we conclude that the reaction is first-order. Therefore, the correct answer is (b) 1.

Key concepts

Half-Life in Chemical Reactions

The concept of half-life is crucial when studying the speed and extent of chemical reactions. In the context of chemical kinetics, the half-life of a substance is the time required for the concentration of that substance to reduce to half its initial value. This concept is not just theoretical; it has practical applications in various fields, including medicine, where it helps in determining the dosage and timing of drug administration.

Understanding half-life also assists in the analysis of reaction rates and the stability of substances over time. The half-life is a defining aspect of radioactive decay as well as other types of chemical reactions. It's a clear indicator of how quickly a reaction proceeds but is entirely dependent on the order of the reaction. For instance, in a first-order reaction, the half-life remains constant regardless of the concentration of reactants, which is not the case with zero or second-order reactions, where the half-life varies with concentration.

Chemical Kinetics Fundamentals

Chemical kinetics is the study of reaction rates and the factors that affect them. It aims to understand the speeds at which reactions occur and the steps by which they proceed. Factors that typically impact reaction rates include reactant concentrations, temperature, catalysts, and the physical state or surface area of the reactants. Kinetics enables chemists to not only describe and predict how quickly reactions will occur but also to determine the mechanics of the reaction at a molecular level - the reaction mechanism.

By analyzing the kinetics of a chemical reaction, chemists can optimize conditions to increase efficiency, reduce waste, produce desired products selectively, and ensure safety by controlling exothermic reactions. For students and chemists alike, mastering this branch of chemistry is essential, as it forms the foundation for understanding biochemical processes, environmental systems, and industrial applications.

First-Order Reactions

First-order reactions are a category within chemical kinetics where the rate of reaction is directly proportional to the concentration of one reactant. A characteristic property is that their half-life is independent of initial concentration, providing a simplified model for analyzing reaction kinetics. Mathematically, the rate of a first-order reaction is often expressed as \( \frac{d[A]}{dt} = -k[A] \), where \( [A] \) is the concentration of the reactant, \( dt \) indicates a small increment of time, and \( k \) is the rate constant specific to the reaction.

These reactions are commonplace in everyday life, for instance, the degradation of pharmaceuticals in the body or the radioactive decay of isotopes. In the context of the original exercise, the fact that the half-life of the reaction did not change with varying pressure (which is proportional to concentration for gases) indicates a first-order reaction. This kind of analysis is foundational for students as it enables them to predict how concentrations will change over time—a key aspect in the design of chemical processes and the safe handling of reactive materials.