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

Question

A graph plotted between concentration of reactant, consumed at any time \((\mathrm{x})\) 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: Understand the reaction order and graph relation

When a graph of concentration of reactant consumed \((x)\) versus time \(t\) is linear and passes through the origin, it indicates a zero-order reaction. In a zero-order reaction, the rate of reaction is independent of the concentration of the reactant.

Step 2: Identify equation of zero-order reaction

The rate law for a zero-order reaction is given by: \[ \text{Rate} = -\frac{dx}{dt} = k \] where \(k\) is the rate constant. The integrated form of this equation is: \[ [A]_t = [A]_0 - kt \]

Step 3: Graph relationship explanation for zero-order

Rearranging the integrated form, we get: \[ x = kt \] where \(x\) is the concentration of reactant consumed. This linear relationship \(x = kt\) results in a straight line through the origin, with a slope equal to \(k\).

Key Concepts

Graph Interpretation

Understanding the relationship between graphs and chemical reactions is vital for analyzing reaction orders. In chemistry, interpreting a graph where the x-axis represents time and the y-axis represents the concentration of reactant consumed is crucial. If the graph produces a straight line that passes through the origin, it indicates that the relationship between these variables is linear. This linearity is a key indicator of the reaction order.

In such cases, the graph’s slope provides valuable information. It tells us about the rate at which the reactant is consumed relative to time. Specifically, the slope is equal to the rate constant \(k\) of the reaction. Clear interpretation of this kind of graph helps in identifying and confirming the characteristics of the reaction order.

• A straight line graph denotes a simple, predictable relationship between time and consumption of the reactant.
• Understanding this graphical data can reveal whether the reaction order is zero or something else.

Clearly being able to read and interpret such graphs is a fundamental skill in chemistry when determining reaction orders.

Zero-Order Reaction

A zero-order reaction is characterized by a rate of reaction that is independent of the concentration of the reactants. This means that the concentration of the reactants does not impact how quickly the reaction proceeds. Instead, the reaction rate is constant over time, leading to a straightforward rate law and graph representation.

The rate law for a zero-order reaction is expressed as:
\[ \text{Rate} = -\frac{dx}{dt} = k \]
Here, \(k\) is the rate constant, and it remains constant as the reaction proceeds. The integrated form of the zero-order rate law is:
\[ [A]_t = [A]_0 - kt \]
This formula relates the initial concentration \([A]_0\) to the concentration at any time \([A]_t\), illustrating how the concentration decreases linearly over time. The equation rearranges to \(x = kt\), confirming why a graph of concentration versus time produces a straight line through the origin.

• Key feature: The amount of reactant consumed linearly depends on time.
• No impact from the changes in reactant concentration on the reaction rate.

This makes predicting changes in concentration straightforward as the reaction progresses.

Rate Law

The rate law is a mathematical expression that describes the rate of a chemical reaction as a function of the concentration of its reactants. It is crucial for understanding how different factors influence the speed of a reaction. In the case of a zero-order reaction, the rate law is quite simple and is depicted as:
\[ \text{Rate} = k \]
This signifies that the rate is constant, not varying with any change in the concentration of reactants. The rate constant \(k\) is unique to this type of reaction and is indicative of the reaction’s volatility or speed.

For reactions of other orders, such as first or second-order, the rate law becomes more complex, incorporating reactant concentrations raised to a power which represents the order of the reaction. However, for zero-order, it remains straightforward with no dependency on concentration.

• Rate is unaffected by how much reactant is present.
• Focus is on the reaction proceeding at a constant rate \(k\).

Understanding the rate law is fundamental for predicting how the reactant concentration changes over time and for planning industrial chemical processes efficiently.