Problem 162 - A Complete Resource Book in Chemistry for JEE Main

Question

For a zero order reaction, the concentration of reactant after \(10 \mathrm{~s}\) is \(0.2 \mathrm{~mol} / /\). If \(\mathrm{k}_{0}\) is \(2 \times 10^{-2}\) mol \(l^{-1} \mathrm{~s}^{-1}\), the initial concentration of reactant is (a) \(0.6 \mathrm{~mol} / l\) (b) \(0.4 \mathrm{~mol} / /\) (c) \(0.8 \mathrm{~mol} / l\) (d) \(1 \mathrm{~mol} / l\)

Step-by-Step Solution

Answer

The initial concentration of the reactant is \(0.4\ \text{mol/L}\).

Step 1: Understand the Zero-Order Reaction

In a zero-order reaction, the rate of reaction is constant and is independent of the concentration of the reactant. The rate law for a zero-order reaction can be expressed as \( r = k_0 \), where \( r \) is the rate and \( k_0 \) is the rate constant.

Step 2: Write Down the Zero-Order Integrated Rate Equation

The integrated rate equation for a zero-order reaction is given by \( [A] = [A]_0 - k_0 \cdot t \), where \([A]\) is the concentration at time \( t \), \([A]_0\) is the initial concentration, and \( k_0 \) is the rate constant.

Step 3: Substitute the Known Values into the Equation

Let's substitute the known values into the integrated rate equation. The concentration of reactant after 10 seconds is given as \( 0.2 \ mol/L \), the rate constant \( k_0 \) is \( 2 \times 10^{-2} \ mol/L/s \), and time \( t \) is \( 10 \ s \). Substitute these into the equation:\[ 0.2 = [A]_0 - 2 \times 10^{-2} \cdot 10 \].

Step 4: Rearrange and Solve for Initial Concentration \([A]_0 \)

Rearrange the equation from Step 3 to solve for \([A]_0\):\[ [A]_0 = 0.2 + (2 \times 10^{-2} \cdot 10) \].

Step 5: Calculate the Initial Concentration

Perform the calculation from Step 4:\[ [A]_0 = 0.2 + 0.2 = 0.4 \ mol/L \]. Thus, the initial concentration of the reactant is \( 0.4 \ mol/L \).

Key Concepts

Integrated Rate Equation

A fundamental concept in chemical kinetics involves understanding the integrated rate equation. This equation is pivotal for zero-order reactions, dictating how reactant concentrations change over time. For a zero-order reaction, the integrated rate equation is represented as:
\[ [A] = [A]_0 - k_0 \cdot t \]
This equation shows that the concentration \([A]\) at any time \(t\) is equal to the initial concentration \([A]_0\) minus the product of the reaction rate constant \(k_0\) and the time elapsed \(t\).

• The integrated rate equation provides a linear relationship between concentration and time.
• It clearly exhibits the reduction in concentration as the reaction proceeds.

By utilizing this equation, you can predict how much of the reactant remains at any given point, making it easier to study the kinetics of zero-order reactions.

Reaction Rate Constant

The reaction rate constant, often symbolized as \(k_0\), plays a crucial role in determining the speed of a zero-order reaction. It quantifies the rate at which reactions occur, with the specific units of \(mol \, L^{-1} \, s^{-1}\) for zero-order rates.

• The rate constant indicates how fast the concentration of a reactant decreases per second.
• In zero-order reactions, \(k_0\) remains independent of reactant concentration, emphasizing the constancy in the reaction rate.

Understanding the reaction rate constant and its implications allows chemists to compare the kinetics of different reactions and ascertain the overall reaction dynamics.

Initial Concentration Calculation

Finding the initial concentration, \([A]_0\), involves manipulating the integrated rate equation. This process allows one to backtrack from known observations to determine the starting concentration of reactants.
In the provided exercise, with a known concentration after 10 seconds and a given \(k_0\), the initial concentration can be calculated as follows:

• Start by inputting known values into the integrated rate equation: \[ 0.2 = [A]_0 - 2 \times 10^{-2} \cdot 10 \]
• Rearrange the equation to solve for \([A]_0\): \[ [A]_0 = 0.2 + 0.2 \]
• Perform the arithmetic: \[ [A]_0 = 0.4 \, mol/L \]

This calculation not only confirms the reaction's initial conditions but also reinforces the practical application of the integrated rate equation for zero-order reactions.