Problem 6 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The reaction: \(\mathrm{A}(\mathrm{g})+2 \mathrm{~B}(\mathrm{~g}) \rightarrow \mathrm{C}(\mathrm{g})+\mathrm{D}(\mathrm{g})\) is an elementary process. In an experiment, the initial partial pressure of \(\mathrm{A}\) and \(\mathrm{B}\) are \(P_{\mathrm{A}}=0.60 \mathrm{~atm}\) and \(P_{\mathrm{B}}=0.80 \mathrm{~atm}\). When \(P_{\mathrm{B}}=0.20 \mathrm{~atm}\), the rate of reaction, relative to the initial rate is (a) \(\frac{1}{16}\) (b) \(\frac{1}{24}\) (c) \(\frac{1}{32}\) (d) \(\frac{1}{48}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{16}\)

Step 1 - Identify the Rate Law

Since the reaction is elementary, the stoichiometric coefficients can be used directly to determine the rate law. The rate, therefore, is proportional to the concentration of reactant A to the power of 1, and reactant B to the power of 2. The rate law can be written as: Rate = k[P_A][P_B]^2, where k is the rate constant.

Step 2 - Calculate Initial Rate

Using the initial conditions given for when the reaction starts, we can find the initial rate. Substitute the given initial partial pressures into the rate law to get the initial rate: Initial Rate = k(0.60 atm)(0.80 atm)^2.

Step 3 - Calculate Rate at Given Condition

When the partial pressure of B drops to 0.20 atm, we need to calculate the new rate. Plug in the partial pressures at this point into the rate law: New Rate = k(0.60 atm)(0.20 atm)^2.

Step 4 - Compare Rates to Find Relative Rate

To find the relative rate, divide the New Rate by the Initial Rate. Thus, Relative Rate = (k(0.60 atm)(0.20 atm)^2) / (k(0.60 atm)(0.80 atm)^2) = (0.20^2 / 0.80^2) = (0.04/0.64) = 1/16.

Step 5 - Select the Correct Answer

Based on the calculation in Step 4, the relative rate of the reaction when PB=0.20 atm compared to the initial rate is 1/16. Therefore, the correct answer is (a) 1/16.

Key Concepts

Rate Law Determination

Understanding how the rate of a chemical reaction is affected by the concentration of reactants is crucial for mastering chemical kinetics. The rate law expresses this relationship and is determined through experiments or, in the case of an elementary reaction, can be deduced directly from the reaction's stoichiometry.

For the given elementary reaction, \( \mathrm{A}(\mathrm{g})+2 \mathrm{B}(\mathrm{g}) \rightarrow \mathrm{C}(\mathrm{g})+\mathrm{D}(\mathrm{g}) \), the rate law can be written, using the stoichiometric coefficients, as Rate = k[\(P_\mathrm{A}\)][\(P_\mathrm{B}\)]^2 where k is the rate constant. Why squared for \(P_\mathrm{B}\)? It's because for every mole of \(A\) reacting, two moles of \(B\) are involved, indicating a second-order dependence on \(B\)'s concentration.

This direct relation allows us to assess how changes in concentration influence the rate without requiring complex calculations. Remember that this simplicity applies only to elementary reactions, which proceed in a single step, unlike more complex multi-step reactions that require a different approach to determine the rate law.

Elementary Reactions

An elementary reaction is one that occurs in a single step and represents a simple collision between reactant molecules resulting in the immediate formation of products. Because the reaction proceeds in one step, the reaction order directly corresponds to the stoichiometry of the balanced chemical equation. This assumes that the reaction mechanism is as straightforward as the overall reaction suggests.

Elementary reactions are the simplest types of reactions and often serve as steps within complex reactions known as reaction mechanisms. They help us understand the basic interactions that occur at a molecular level and provide a foundation for predicting reaction behavior under different conditions. Elementary reactions are invaluable for teaching purposes because they allow us to apply theoretical concepts directly to reaction situations without additional complexities of real-world scenarios.

Reaction Rate Calculation

Calculating the rate of a reaction involves determining how quickly reactants are consumed or products are formed over time. For the elementary reaction presented, the rate calculation is straightforward. Once the rate law is established, we simply plug in the relevant concentrations or, in this case, partial pressures.

To calculate the initial rate, we use the initial conditions provided: \(P_\mathrm{A}=0.60 \, \mathrm{atm}\) and \(P_\mathrm{B}=0.80 \, \mathrm{atm}\). With the rate law Rate = k[\(P_\mathrm{A}\)][\(P_\mathrm{B}\)]^2, the initial rate is k(0.60 atm)(0.80 atm)^2. When the partial pressure of \(B\) changes, the new rate can be found using the same rate law with the new partial pressure of \(B\): \(P_\mathrm{B}=0.20 \, \mathrm{atm}\) resulting in a different rate k(0.60 atm)(0.20 atm)^2.

Comparing the new rate to the initial rate gives us the relative rate or the fraction representing how much the reaction rate has changed due to the change in \(P_\mathrm{B}\). This step is essential in understanding the practical impacts of concentration changes on reaction rates, reinforcing the theoretical principles outlined in the rate law.