Problem 23 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The rate constant of a zero order reaction is \(2.0 \times 10^{-2} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\). If the concentration of the reactant after 25 seconds is \(0.5 \mathrm{M}\). What is the initial concentration? [Main Online April 23, 2013](a) \(0.5 \mathrm{M}\) (b) \(1.25 \mathrm{M}\) (c) \(12.5 \mathrm{M}\) (d) \(1.0 \mathrm{M}\)

Step-by-Step Solution

Answer

The initial concentration is \(1.0\,\mathrm{M}\) (option d).

Step 1: Understand the Zero Order Reaction

In a zero-order reaction, the rate of reaction is independent of the concentration of the reactant, and the rate law formula can be expressed as \[ [A]_t = [A]_0 - kt \] where \([A]_t\) is the concentration of the reactant at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time.

Step 2: Identify the Known Values

From the problem, we are given the rate constant \(k = 2.0 \times 10^{-2} \text{ mol L}^{-1} \text{ s}^{-1}\), the concentration at 25 seconds \([A]_t = 0.5 \text{ M}\), and the time \(t = 25 \text{ s}\). The task is to find the initial concentration \([A]_0\).

Step 3: Substitute Values into the Equation

Using the zero-order reaction formula, substitute the known values: \[ 0.5 = [A]_0 - (2.0 \times 10^{-2}) \times 25 \]

Step 4: Solve for Initial Concentration

Rearrange the formula to solve for \([A]_0\): \[ [A]_0 = 0.5 + (2.0 \times 10^{-2} \times 25) \] Calculate the expression: \(2.0 \times 10^{-2} \times 25 = 0.5\). Thus, \([A]_0 = 0.5 + 0.5 = 1.0 \text{ M}\).

Key Concepts

Rate Constant

In a zero-order reaction, the rate constant is a crucial parameter that helps us understand how quickly the reaction occurs regardless of the concentration of reactants. For a zero-order reaction, the reaction rate is independent of the concentration, meaning that it remains constant throughout. The rate constant, often denoted as \(k\), has units of \( \text{mol L}^{-1}\text{s}^{-1} \). In the context of this exercise, the rate constant is given as \(2.0 \times 10^{-2} \text{ mol L}^{-1} \text{ s}^{-1}\).The rate constant is important because it provides the rate at which the concentration of a reactant decreases over time. In practical terms, the rate constant tells us how many moles of a substance are reacted in a liter every second. Since it remains constant, it offers a stable prediction of the reaction's progress over a period.

• Zero-order reactions result in a linear decrease in reactant concentration over time.
• The rate constant informs us of the consistent rate of reaction.

Initial Concentration

The initial concentration \([A]_0\) is a key starting point in any chemical reaction, especially in a zero-order reaction. This value represents the concentration of a reactant at the moment the reaction begins. Knowing the initial concentration allows us to predict how much reactant will remain after a certain period and calculate the time required to reach a desired concentration.In this exercise, the task is to determine the initial concentration given the concentration at 25 seconds. Using the zero-order reaction formula \( [A]_t = [A]_0 - kt \), where \([A]_0\) is the initial concentration, \([A]_t\) is the concentration after time \(t\), and \(k\) is the rate constant. We can rearrange the equation to solve for \([A]_0\) using the known values:

1. Rate constant \(k = 2.0 \times 10^{-2} \text{ mol L}^{-1} \text{ s}^{-1}\).
2. Concentration at 25 seconds \([A]_t = 0.5 \text{ M}\).
3. Time \(t = 25 \text{ s}\).

This allows us to find \([A]_0\), which is crucial for predicting future reaction behavior.

Reaction Rate Law

The reaction rate law provides a mathematical description of how the rate of a reaction depends on the concentration of its reactants. In zero-order reactions, this relationship is particularly straightforward. For a zero-order reaction, the rate law is expressed as: \[ r = -\frac{d[A]}{dt} = k \]Here, \(r\) is the rate of reaction, \([A]\) is the concentration of the reactant, and \(k\) is the rate constant. The negative sign reflects the fact that the concentration decreases over time. The zero-order rate law demonstrates that the reaction rate is constant and does not depend on the concentration of the reactant. This implies:

• The reaction proceeds at a steady rate.
• The concentration decreases at a uniform rate over time.

For the zero-order reaction involving the problem at hand, the equation \([A]_t = [A]_0 - kt\) can be effectively used to calculate the time-dependent concentration by simply subtracting the product of rate constant and time from the initial concentration. Understanding these mechanics allows one to predict and control the longevity and progression of the chemical reaction.