Question

11\. 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 \(=k_{1}[\mathrm{X}][\mathrm{Y}]^{2}\) (b) rate \(=k_{2}[\mathrm{X}]^{2}\) (c) rate \(=k_{3}[\mathrm{X}][\mathrm{Y}]\) (d) rate \(=k_{4}\)

Short answer

Question: Determine the order with respect to each reactant (X and Y) and the overall order of the reaction for each given rate expression. a. Rate = \(k_{1}[X][Y]^{2}\) b. Rate = \(k_{2}[X]^{2}\) c. Rate = \(k_{3}[X][Y]\) d. Rate = \(k_{4}\)

Step-by-step solution

(a) Finding the order with respect to X and Y for rate = \(k_{1}[X][Y]^{2}\)

For this rate expression, the order with respect to X is 1 and the order with respect to Y is 2, as seen from the exponents of their respective concentrations.

(a) Determining the overall order

The overall order of the reaction is the sum of the orders with respect to each reactant. In this case, the overall order is 1 (X's order) + 2 (Y's order) = 3.

(b) Finding the order with respect to X for rate = \(k_{2}[X]^{2}\)

For this rate expression, the order with respect to X is 2. There is no Y term in the expression; therefore, the order with respect to Y is 0.

(b) Determining the overall order

The overall order of the reaction is the sum of the orders with respect to each reactant. In this case, the overall order is 2 (X's order) + 0 (Y's order) = 2.

(c) Finding the order with respect to X and Y for rate = \(k_{3}[X][Y]\)

For this rate expression, the order with respect to X is 1 and the order with respect to Y is 1, as seen from the exponents of their respective concentrations.

(c) Determining the overall order

The overall order of the reaction is the sum of the orders with respect to each reactant. In this case, the overall order is 1 (X's order) + 1 (Y's order) = 2.

(d) Finding the order with respect to X and Y for rate = \(k_{4}\)

For this rate expression, there are no concentration terms. Therefore, the order with respect to X is 0 and the order with respect to Y is 0.

(d) Determining the overall order

The overall order of the reaction is the sum of the orders with respect to each reactant. In this case, the overall order is 0 (X's order) + 0 (Y's order) = 0.

Key concepts

Rate Law

Understanding the rate law is crucial for mastering chemical kinetics. It expresses the relationship between the rate of a chemical reaction and the concentration of the reactants. In simple terms, it tells us how changes in reactant concentrations affect the speed of the reaction.

For any given reaction, the rate law is written as: \[\text{rate} = k [\text{A}]^{m}[\text{B}]^{n}\cdots\] where \(k\) represents the rate constant, and \(m\) and \(n\) are the orders of the reaction with respect to reactants A and B, respectively. These orders are typically whole numbers, but can also be fractions or zero. Importantly, the reaction order cannot be inferred from the stoichiometry of the balanced equation; it must be determined experimentally.

Using the original exercises as examples, we see that for part (a), the rate law is given by \[\text{rate} = k_1[\text{X}][\text{Y}]^2\] signifying that the reaction is first-order with respect to X and second-order with respect to Y. The overall reaction order is the sum of these individual orders, which in this case, totals to third-order.

Chemical Kinetics

The study of reaction rates and the factors affecting them falls under the umbrella of chemical kinetics. This area of chemistry is concerned not only with how fast reactions occur, but also with the steps (or mechanisms) that take place from reactants to products.

In the context of the given exercise, understanding kinetics is essential to grasp the reasoning behind the derivation of the order of reactants from rate expressions. Factors impacting the reaction rate include reactant concentrations, temperature, presence of a catalyst, and the physical state of the reactants.

Moving from theory to practice, look at part (b): when we have a rate law of the form \[\text{rate} = k_2[\text{X}]^2\] the absence of Y indicates that the rate of reaction is not influenced by the concentration of Y, hence it is zero-order with respect to Y and second-order with respect to X. Chemical kinetics is what guides us in understanding these intricacies.

Reaction Rates

Reaction rates refer to the speed at which reactants are converted into products in a chemical reaction. They can be expressed in terms of how quickly a reactant is depleted or how fast a product is formed. Rates are typically dependent on the concentration of reactants, which is why we study rate laws when analyzing reaction kinetics.

For example, examining part (c) of our exercise, the rate law \[\text{rate} = k_3[\text{X}][\text{Y}]\] means that both reactants, X and Y, influence the rate at which the reaction proceeds and both are first-order. If one reactant’s concentration is doubled, the rate of the reaction would also double, indicating a direct proportionality. On the other hand, part (d) presents a zero-order reaction with the rate law \[\text{rate} = k_4\], suggesting that the rate is independent of the concentration of the reactants; here, the reaction rate is constant as long as the reactants are present.