Problem 34 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For a reaction \(2 \mathrm{~A}+\mathrm{B}+3 \mathrm{C} \rightarrow \mathrm{D}+3 \mathrm{E}\), the following date is obtained: $$ \begin{array}{ccccc} \hline \text { Reaction } & \multicolumn{2}{c} {\text { Concentration in }} & \text { Initial rate of } \\ & \multicolumn{2}{c} {\text { mole per litre }} & \text { formation of } \\ & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D}\left(\text { torr } \mathbf{s}^{-1}\right) \\ \hline 1 & 0.01 & 0.01 & 0.01 & 2.5 \times 10^{-4} \\ 2 & 0.02 & 0.01 & 0.01 & 1.0 \times 10^{-3} \\ 3 & 0.01 & 0.02 & 0.01 & 2.5 \times 10^{-4} \\ 4 & 0.01 & 0.02 & 0.02 & 5.0 \times 10^{-4} \\ \hline \end{array} $$ The order with respect to \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are, respectively, (a) \(0,1,2\) (b) \(2,0,1\) (c) \(1,0,2\) (d) \(2,1,1\)

Step-by-Step Solution

Answer

The order with respect to A is 2, as the rate increases by a factor of 4 when its concentration is doubled. The order with respect to B is 0, as the rate does not change when B's concentration is doubled. The order with respect to C is 1, as the rate doubles when its concentration is doubled. The correct answer is (d) 2,1,1.

Step 1: Analyze Changes between Experiments 1 and 2

Compare the concentration of reactants and initial rate of formation of D between Experiment 1 and Experiment 2 to determine the order of reaction with respect to A. Only the concentration of A is doubled, while the concentrations of B and C remain the same.

Step 2: Calculate the Order with Respect to A

By comparing the initial rates of reaction in Experiment 1 and Experiment 2, we can see that when the concentration of A is doubled, the rate increases by a factor of 4 (from 2.5 x 10^-4 to 1.0 x 10^-3 torr s^-1). According to the rate law, this indicates that the reaction is second order with respect to A.

Step 3: Analyze Changes between Experiments 1 and 3

Now, compare the concentration of reactants and initial rate of formation of D between Experiment 1 and Experiment 3 to determine the order of reaction with respect to B. The concentration of B is doubled, while the concentrations of A and C remain the same; the rate does not change.

Step 4: Analyze Changes between Experiments 3 and 4

Compare the concentration of reactants and initial rate of formation of D between Experiment 3 and Experiment 4 to determine the order of reaction with respect to C. The concentration of C is doubled, while the concentrations of A and B remain constant.

Step 5: Calculate the Order with Respect to C

The rate of reaction doubles (from 2.5 x 10^-4 to 5.0 x 10^-4 torr s^-1) when the concentration of C is doubled. This implies that the reaction is first order with respect to C.

Key Concepts

Rate Law

Understanding the rate law is crucial for determining how the concentration of reactants affects the speed of a chemical reaction. The rate law is a mathematical expression that links the rate of a reaction to the concentration of its reactants. It usually takes the form of \( Rate = k[A]^m[B]^n \) where \( k \) is the rate constant, \( A \) and \( B \) are the concentrations of reactants, and \( m \) and \( n \) represent the reaction orders with respect to each reactant.

In the given exercise, analyzing how varying concentrations of reactants \( A \) and \( C \) affect the initial rate of formation of product \( D \) allows us to deduce their specific order in the reaction. Through careful observation and calculation based on the provided data, one can apply the rate law to unravel these reaction order dependencies, which is pivotal in understanding how the reaction proceeds.

Chemical Kinetics

Chemical kinetics deals with the speed or rate at which chemical reactions occur and the factors influencing this speed. It's a field of study in chemistry that focuses on reaction rates, mechanisms, and the pathway from reactants to products. The rate of a reaction is influenced by various factors, including reactant concentration, temperature, presence of a catalyst, and surface area. The principles of kinetics enable chemists to predict and control the speed of chemical processes.

For instance, in the exercise example, chemical kinetics principles are used to understand how the change in concentration affects the rate of reaction, helping to unravel the complexities underlying the reaction's behavior. Determining how fast a reaction occurs under different conditions can be essential, especially in industries where time and reaction efficiency are key.

Reaction Rates

The reaction rate is defined as the speed at which reactants transform into products over time. It's typically expressed in terms of concentration change per unit time. A fast reaction has a high rate, meaning the concentration of reactants decreases rapidly, and products form quickly. Conversely, a slow reaction has a low rate, with little change in reactant concentration over a longer time period.

In the exercise, we see the rate analyzed through changes in pressure (torr) per second, which indicates how swiftly product \( D \) forms from the reactants \( A \) and \( C \). By observing the initial rates and how they change with varying concentrations of reactants, we gain insights into the speed of the reaction and the efficiency of reactant conversion under different conditions.

Concentration Effects on Rate

In chemical kinetics, the effect of concentration on the rate of a reaction is fundamental. The rate usually increases when the concentration of a reactant increases, provided that other conditions remain constant. This is because more reactant particles are present, leading to a higher chance of successful collisions that can form products.

The exercise demonstrates this principle by showing changes in the reaction rate when concentrations of reactants \( A \) and \( C \) are altered. We observe that doubling \( A \) causes a fourfold increase in the rate, indicating a second-order dependence on \( A \) according to the rate law. Doubling \( C \) leads to doubling the rate, revealing a first-order dependence. However, doubling \( B \) does not affect the rate, suggesting zero-order dependence on \( B \) for this reaction.