Problem 199 - A Complete Resource Book in Chemistry for JEE Main

Question

Units of rate constant for the first and zero-order reactions in terms of molarity \(\mathrm{M}\), units are respectively [2002] (a) \(\mathrm{s}^{-1}, \mathrm{Ms}^{-1}\) (b) \(\mathrm{s}^{-1}, \mathrm{M}\) (c) \(\mathrm{M} \mathrm{s}^{-1}, \mathrm{~s}^{-1}\) (d) \(\mathrm{M}, \mathrm{s}^{-1}\)

Step-by-Step Solution

Answer

The correct answer is (a) \(\mathrm{s}^{-1}, \mathrm{M} \mathrm{s}^{-1}\).

Step 1: Understand the Reaction Order

Identify the order of reactions given in the problem. The first-order reaction depends linearly on the concentration of one reactant, while the zero-order reaction is independent of the concentration of the reactants.

Step 2: Formula for Rate Constant Units

For a first-order reaction, the rate equation is \(r = k[A]\). Solving for \(k\), we have \(k = \frac{r}{[A]}\). For rate \(r\) with units \(\mathrm{M} \mathrm{s}^{-1}\) and concentration \([A]\) with units \(\mathrm{M}\), the units for \(k\) are \(\mathrm{s}^{-1}\). For a zero-order reaction, the rate equation is \(r = k\). Thus, \(k = r\) with units \(\mathrm{M} \mathrm{s}^{-1}\).

Step 3: Match Units to Answer Options

The units for the first-order reaction are \( \mathrm{s}^{-1} \), and the units for the zero-order reaction are \( \mathrm{M} \mathrm{s}^{-1} \). Compare these units with the provided options. Option (a) indicates the correct units: \(\mathrm{s}^{-1}\) for the first-order and \(\mathrm{M} \mathrm{s}^{-1}\) for the zero-order reactions.

Key Concepts

Reaction Order

The reaction order of a chemical reaction offers insights into how the rate of the reaction is affected by the concentration of its reactants. It is essentially a summary of how these concentrations impact the reaction speed. A reaction can be zero-order, first-order, second-order, or even fractional order. Each order reflects the exponent on the concentration of the reactant in the rate equation for the reaction.
In a zero-order reaction, the rate is independent of the concentration of the reactants. This means that changing the concentration does not affect the rate of the reaction. On the other hand, a first-order reaction is linearly dependent on the concentration of one reactant. This linear relationship implies that if you double the concentration, you would also double the rate of the reaction. Understanding the reaction order is crucial to predicting how a change in concentration will affect the rate of a reaction.

First-Order Reaction

A first-order reaction is characterized by its direct proportionality to the concentration of a single reactant. The rate equation for a typical first-order reaction is represented as \(r = k[A]\), where \(r\) is the reaction rate, \(k\) is the rate constant, and \([A]\) is the concentration of reactant A.
Since the rate is directly dependent on the concentration of one reactant, changes in this concentration significantly impact the reaction's speed. When solving for the rate constant \(k\), the relationship can be expressed as:

• \(k = \frac{r}{[A]}\)
• This shows that the units for \(k\) will be \(\mathrm{s}^{-1}\) when \(r\) is in the units of \(\mathrm{M} \mathrm{s}^{-1}\) (molarity per second) and \([A]\) in \(\mathrm{M}\) (molarity).

First-order reactions are common in processes such as radioactive decay, where the rate at which a substance decays is directly proportional to its current amount.

Zero-Order Reaction

In zero-order reactions, the rate of reaction does not depend on the concentration of the reactants. The rate equation for such a reaction is given by \(r = k\), implying that the rate is equal to the rate constant. This means the reaction proceeds at a constant rate, regardless of varying concentrations of the reactant involved.
When examining the units of the rate constant \(k\) for zero-order reactions, it becomes evident that:

• \(k\) has the units of \(\mathrm{M} \mathrm{s}^{-1}\).
• This is because the dimensions of the reaction rate \(r\) itself are \(\mathrm{M} \mathrm{s}^{-1}\), and since \(r = k\), the rate constant directly adopts this unit.

Zero-order reactions are rare and typically occur under conditions where the surface or catalyst is saturated by the reactant, leading to a reaction rate that remains unaffected by more reactant being added.

Rate Equation

The rate equation provides a crucial insight into the mechanics of a reaction by detailing how the concentration of reactants affects the reaction rate. It is a mathematical expression that relates the reaction rate to the concentration of reactants and includes the rate constant and reaction order.
The general form of a rate equation can be written as:

• \(r = k[A]^n\)

Here,

• \(r\) is the reaction rate,
• \(k\) is the rate constant,
• \([A]^n\) represents the concentration of reactant A raised to the power of its reaction order \(n\).

By analyzing the rate equation, one can determine how changes in reactant concentration impact the speed of the reaction and predict the behavior of the reaction over time. For example, knowing whether a reaction is zero, first, or second order helps in calculating the half-life or the time required for the reaction to complete a certain percentage. The rate equation is a cornerstone of reaction kinetics, allowing for complex measurements of reaction dynamics in chemical processes.