Problem 12 - Chemistry: Principles and Reactions

Question

A reaction has two reactants \(\mathrm{X}\) and \(\mathrm{Y}\). What is the order with respect to each reactant and the overall order of the reaction described by the following rate expressions? (a) rate \(=k_{1}[\mathrm{X}]^{2} \times[\mathrm{Y}]\) (b) rate \(=k_{2}[\mathrm{X}]\) (c) rate \(=k_{3}[\mathrm{X}]^{2} \times[\mathrm{Y}]^{2}\) (d) rate \(=k_{4}\)

Step-by-Step Solution

Answer

Question: Determine the order with respect to each reactant and the overall order of the reaction for the following rate expressions: a) rate \(=k_{1}[\mathrm{X}]^{2} \times[\mathrm{Y}]\) b) rate \(=k_{2}[\mathrm{X}]\) c) rate \(=k_{3}[\mathrm{X}]^{2} \times[\mathrm{Y}]^{2}\) d) rate \(=k_{4}\) Answer: a) Order with respect to X: 2, Order with respect to Y: 1, Overall order: 3. b) Order with respect to X: 1, Order with respect to Y: 0, Overall order: 1. c) Order with respect to X: 2, Order with respect to Y: 2, Overall order: 4. d) Order with respect to X: 0, Order with respect to Y: 0, Overall order: 0.

Step 1: a) Determine the order for X and Y in the first rate expression

For the rate expression rate \(=k_{1}[\mathrm{X}]^{2} \times[\mathrm{Y}]\), we can see that the concentration of X is raised to the power of 2, while the concentration of Y is raised to the power of 1. Thus, the order with respect to X is 2, and the order with respect to Y is 1.

Step 2: a) Determine the overall order of the reaction for the first rate expression

To determine the overall order of the reaction, we sum the orders of each reactant. In this case, that is \(2(\text{for X}) + 1(\text{for Y})\). So, the overall order of the reaction is 3.

Step 3: b) Determine the order for X in the second rate expression

For the rate expression rate \(=k_{2}[\mathrm{X}]\), the concentration of X is raised to the power of 1, and there is no dependence on the concentration of Y. Thus, the order with respect to X is 1, and the order with respect to Y is 0.

Step 4: b) Determine the overall order of the reaction for the second rate expression

The overall order of the reaction is the sum of the orders of each reactant, which is \(1(\text{for X}) + 0(\text{for Y})\). Therefore, the overall order of the reaction is 1.

Step 5: c) Determine the order for X and Y in the third rate expression

For the rate expression rate \(=k_{3}[\mathrm{X}]^{2} \times[\mathrm{Y}]^{2}\), the concentration of X is raised to the power of 2, and the concentration of Y is raised to the power of 2 as well. Thus, the order with respect to X is 2, and the order with respect to Y is 2.

Step 6: c) Determine the overall order of the reaction for the third rate expression

To determine the overall order of the reaction, we sum the orders of each reactant, which is \(2(\text{for X}) + 2(\text{for Y})\). Thus, the overall order of the reaction is 4.

Step 7: d) Determine the order and overall order for the fourth rate expression

For the rate expression rate \(=k_{4}\), there is no dependence on the concentration of X or Y. Thus, the order with respect to X is 0, and the order with respect to Y is 0.

Step 8: d) Calculate the overall order of the reaction for the fourth rate expression

The overall order of the reaction is the sum of the orders of each reactant, which is \(0(\text{for X}) + 0(\text{for Y})\). Therefore, the overall order of the reaction is 0.

Key Concepts

Rate Expression

Rate expression is a mathematical equation that shows how the rate of a chemical reaction depends on the concentration of its reactants. It is usually expressed in the form of: \[\text{rate} = k [A]^m [B]^n\]where:- \(k\) is the rate constant, a unique value for each reaction at a given temperature- \([A]\) and \([B]\) represent the concentrations of the reactants- \(m\) and \(n\) are the reaction orders with respect to reactants A and B, respectivelyThe form of the rate expression provides valuable insight into how different reactants influence the rate of a reaction. These exponents, \(m\) and \(n\), can be determined experimentally and may not coincide with the stoichiometric coefficients from the balanced chemical equation.
By analyzing the rate expression, one can predict reaction behavior under various conditions.

Reaction Kinetics

Reaction kinetics is the study of the speed of chemical reactions and the factors that affect this speed. It explores how different conditions, such as concentration, temperature, and catalysts, influence reaction rates. Factors affecting reaction kinetics:

• Concentration: Higher concentrations often lead to higher rates as there are more frequent reactant collisions.
• Temperature: Increasing temperature generally increases reaction rates because particles move faster and collide more energetically.
• Catalysts: These substances speed up reactions without being consumed by providing an alternative pathway with a lower activation energy.

Understanding these factors helps chemists control reaction speeds in industrial processes, ensuring efficiency and economic viability. Reaction kinetics is essential in interpreting rate expressions and predicting how fast a reaction will proceed to the intended equilibrium.

Overall Reaction Order

The overall reaction order is the sum of the powers to which the concentration terms in a rate expression are raised. It indicates how the rate of reaction changes with changes in reactant concentrations. To calculate the overall reaction order, simply add the individual orders of reactants as presented in the rate expression. For instance, using the expression:\[\text{rate} = k [X]^a [Y]^b\]The overall order is \(a + b\).
For the following rate expressions, the overall orders are:

• rate = \(k_1 [X]^2 [Y]\), overall order = 2 + 1 = 3
• rate = \(k_2 [X]\), overall order = 1 + 0 = 1
• rate = \(k_3 [X]^2 [Y]^2\), overall order = 2 + 2 = 4
• rate = \(k_4\), overall order = 0 + 0 = 0 (constant rate)

Knowing the overall reaction order helps to understand how sensitive the rate is to changes in reactant concentration, which is crucial for the design and control of chemical processes.