Question

The chemical kinetics of the reaction \(\mathrm{aA}+\mathrm{bB} \rightarrow\) \(\mathrm{C}\) at \(298 \mathrm{~K}\) were followed. The initial rates were recorded rates were recorded under different initial conditions and are summarized as follows. \begin{tabular}{lll} \hline Initial conc. \([\mathrm{A}]_{0}(\mathrm{~mol} / \mathrm{L})\) & Initial conc. \([\mathrm{B}]_{0}(\mathrm{~mol} / \mathbf{L})\) & Initial rate \((\mathrm{mol} / \mathrm{L} \mathrm{s})\) \\ \hline \(0.1\) & \(0.1\) & \(2.4 \times 10^{-3}\) \\ \(0.2\) & \(0.1\) & \(4.8 \times 10^{-3}\) \\ \(0.4\) & \(0.1\) & \(9.7 \times 10^{-3}\) \\ \(0.1\) & \(0.2\) & \(9.6 \times 10^{-3}\) \\ \(0.1\) & \(0.4\) & \(3.8 \times 10^{-2}\) \\ \hline \end{tabular} Which of the following statements is incorrect? (a) The rate constant \(\mathrm{k}\) is governed by the activation energy of the reaction (b) Reaction rate \(=\mathrm{k}[\mathrm{A}][\mathrm{B}]^{2}\) (c) In the chemical equation of \(a \mathrm{~A}+\mathrm{bB} \rightarrow \mathrm{C}, \mathrm{a}\) is 0 and \(b\) is 3 . (d) The overall order of reaction is third order.

Short answer

Statement (c) is incorrect as it wrongly claims zero reaction order for \( A \).

Step-by-step solution

Write the rate equation

The general rate equation for the reaction is \( \text{Rate} = k [A]^m [B]^n \), where \( m \) and \( n \) represent the orders of reaction with respect to \( A \) and \( B \), respectively. Our task is to determine these values.

Determine order with respect to A

Compare the first and second experimental conditions: \([A]_0 = 0.1\) to \([A]_0 = 0.2\) while \([B]_0 = 0.1\) is constant, leading to doubling of rate. Use the formula: \[ \frac{(4.8 \times 10^{-3})}{(2.4 \times 10^{-3})} = \left( \frac{0.2}{0.1} \right)^m \]. \( m = 1 \) because the rate doubles as \([A]_0\) is doubled.

Determine order with respect to B

Compare the first and fourth experiments: \([B]_0 = 0.1\) to \([B]_0 = 0.2\) while \([A]_0 = 0.1\) is constant, leading to quadrupling of rate. Use the formula: \[ \frac{(9.6 \times 10^{-3})}{(2.4 \times 10^{-3})} = \left( \frac{0.2}{0.1} \right)^n \]. \( n = 2 \) because the rate quadruples as \([B]_0\) is doubled.

Establish the rate equation

Based on the orders found, the rate equation is given by \( \text{Rate} = k [A]^1 [B]^2 \). Thus, statement (b) suggests the correct expression for the reaction rate.

Identify the reaction orders

Since the order with respect to \( A \) is 1 and with respect to \( B \) is 2, the overall reaction order is 1 + 2 = 3.

Analyze statements

Evaluate each statement: (a) is correct as rate constant \( k \) is influenced by activation energy through the Arrhenius equation. (b) is correct as it matches our derived rate law. (c) incorrectly states that \( a \) is 0 and \( b \) is 3; reactions aren't typically zero-order for reactants. (d) is correct as the overall order is third order. Statement (c) is incorrect.

Key concepts

Rate Constant

The rate constant, denoted as \( k \), is a fundamental parameter in chemical kinetics that determines the speed of a reaction under given conditions. It is a unique value for each reaction and is influenced by factors such as temperature and the presence of a catalyst. The rate constant appears in the rate equation and it essentially links the concentration of the reactants to the rate of the reaction.

To further elaborate, think of the rate constant as a proportionality factor. In the rate equation \( \text{Rate} = k [A]^m [B]^n \), \( k \) determines how quickly the reactants \( A \) and \( B \) are converted into products.

• It has units that depend on the overall order of the reaction. For a first-order reaction, it's expressed in \( \,s^{-1} \), while for a second-order reaction, the units are \( \,mol^{-1}L\,s^{-1} \).
• The temperature's effect on \( k \) is described by the Arrhenius equation, \( k = A e^{-E_a/RT} \), where \( A \) is the pre-exponential factor and \( E_a \) is the activation energy.

Understanding the role of the rate constant and the factors affecting it is crucial for predicting how changes in conditions can alter the reaction rate.

Reaction Order

Reaction order refers to the power to which the concentration of a reactant is raised in the rate equation. It indicates how the rate of reaction is affected by the concentration of that reactant. The overall order of reaction is the sum of the orders with respect to all reactants present in the reaction.

In the provided exercise, the reaction order with respect to \( A \) is 1, and with respect to \( B \) is 2, making the overall reaction order 3.

• The individual order with respect to a reactant tells you how sensitive the reaction rate is to changes in that particular reactant's concentration.
• Zero-order reactions mean the rate is independent of reactant concentration, while first-order implies a linear relationship, and a second-order indicates a quadratic relationship.

To summarize: The reaction order is crucial for understanding and predicting how varying reactant concentrations affect the reaction rate. Different order reactions behave differently under changing conditions.

Rate Equation

The rate equation, also known as the rate law, is a mathematical expression that correlates the rate of a chemical reaction to the concentration of its reactants. Each concentration term is raised to a power, corresponding to the reaction order with respect to that reactant.

In the exercise, the rate equation derived was \( \text{Rate} = k [A][B]^2 \). This tells us several important things:

• The rate depends on the concentration of reactant \( A \) to the first power and \( B \) to the second power.
• If you double the concentration of \( A \), the rate doubles. But if you double \( B \), the rate increases four times.

The rate equation is essential because it provides a direct way to predict how the rate will change when concentrations are varied. Additionally, it can help identify mechanisms and intermediates during the detailed study of reaction pathways.

In conclusion, mastering the concept of the rate equation allows chemists to understand reaction dynamics at a deep level and to design chemical processes more efficiently.