Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

A graph plotted between concentration of reactant, consumed at any time \((x)\) and time ' \(\mathrm{t}\) ' is found to be a straight line passing through the origin. The reaction is of (a) first-order (b) zero-order (c) third-order (d) second-order

Step-by-Step Solution

Answer

The reaction is of zero-order.

Step 1: Understanding the Graph Description

The problem describes a graph where the concentration of the reactant consumed, given as \( x \), is plotted against time \( t \). It states that the graph is a straight line passing through the origin.

Step 2: Determining the Reaction Order

In kinetics, if a graph of concentration vs. time results in a straight line through the origin, it signifies a zero-order reaction. This is because, in a zero-order reaction, the consumption of reactants is constant (linear with time) irrespective of the concentration of the reactant.

Step 3: Verifying with Zero-Order Kinetics Definition

For a zero-order reaction, the change in concentration of reactant \( [A] \) with respect to time can be expressed as \( [A] = [A]_0 - kt \), where \( k \) is the rate constant and \( t \) is time. This linear relationship, \( x = kt \), explains why the graph is a straight line through the origin.

Key Concepts

Zero-Order Reaction

In chemical kinetics, a zero-order reaction is unique in its characteristics and behavior. Unlike other reaction orders, the rate of a zero-order reaction is constant and does not depend on the concentration of the reactant. This means that the reaction proceeds at a steady rate until the reactant is depleted. The mathematical representation for a zero-order reaction is given by the formula \[[A] = [A]_0 - kt\]where

• \([A]\) is the concentration of the reactant at a time \(t\).
• \([A]_0\) is the initial concentration of the reactant.
• \(k\) is the rate constant.

All these elements combined help make zero-order reactions quite predictable. In the context of a graph plotting concentration versus time, a zero-order reaction yields a straight line, as the formula indicates a linear relationship between concentration and time.

Concentration Versus Time

When we observe the progression of a reaction over time, understanding how concentration changes is crucial. In a zero-order reaction, as time progresses, the concentration of the reactant steadily decreases in a manner independent of its initial amount. This behavior is represented by a simple linear equation, showcasing how the changes in concentration correspond to elapsed time. The concentration-time graph for a zero-order reaction typically features:

• A straight line with a negative slope.
• The slope equates to the negative rate constant \(-k\).
• The line passing through the origin indicates that if the starting concentration is zero, the reactant is also absent.

To visualize, imagine a scenario where paint dries on a surface at a constant rate, irrespective of how much paint is left. This exemplifies how zero-order kinetics works in simple, real-world processes.

Rate Constant

A key parameter in chemical kinetics is the rate constant, often symbolized as \(k\) in reaction rate equations. For zero-order reactions, the rate constant quantifies the steady rate at which reactants are consumed. It's determined experimentally by analyzing concentration versus time data to ascertain consistent reaction rates. The rate constant in zero-order reactions directly relates to the slope of the line in a concentration-time graph. Importantly, the magnitude of the rate constant can give insights into the speed of a reaction under specified conditions. The formula \[[A] = [A]_0 - kt\]where

• \([-k]\) is the slope of the line (which is directly proportional to \(k\)).
• The negative sign indicates a decrease in concentration over time.

Understanding the rate constant's role allows for predictions about reaction timeframes, facilitating key decisions in industrial and laboratory settings alike. Zero-order kinetics often appear in scenarios with limited reactant, enzyme saturation, or surface-reactant reactions.