Problem 28 - Chemistry The Molecular Science

Question

For each of these rate laws, state the reaction order with respect to the hypothetical substances \(\mathrm{A}\) and \(\mathrm{B}\), and give the overall order. (a) Rate \(=k[\mathrm{~A}][\mathrm{B}]^{3}\) (b) Rate \(=k[\mathrm{~A}][\mathrm{B}]\) (c) Rate \(=k[\mathrm{~A}]\) (d) Rate \(=k[\mathrm{~A}]^{3}[\mathrm{~B}]\)

Step-by-Step Solution

Answer

(a) A: 1, B: 3, overall: 4; (b) A: 1, B: 1, overall: 2; (c) A: 1, B: 0, overall: 1; (d) A: 3, B: 1, overall: 4.

Step 1: Identify Reaction Orders for Substance A

For each given rate law, determine the power to which the concentration of substance A, \( [\mathrm{A}] \), is raised. This value corresponds to the order of the reaction with respect to A. (a) Here, \( [\mathrm{A}] \) has a power of 1, so the order with respect to \( \mathrm{A} \) is 1.(b) \( [\mathrm{A}] \) also has a power of 1, making the order with respect to \( \mathrm{A} \) also 1. (c) Again, \( [\mathrm{A}] \) has a power of 1, so the order with respect to \( \mathrm{A} \) is 1.(d) In this case, \( [\mathrm{A}] \) is raised to the power of 3, so the order with respect to \( \mathrm{A} \) is 3.

Step 2: Identify Reaction Orders for Substance B

Similarly, for each rate law, determine the power to which the concentration of substance B, \( [\mathrm{B}] \), is raised. This value corresponds to the order of the reaction with respect to B.(a) \( [\mathrm{B}] \) is raised to the power of 3, so the order with respect to \( \mathrm{B} \) is 3.(b) \( [\mathrm{B}] \) is raised to the power of 1, making the order with respect to \( \mathrm{B} \) 1.(c) \( [\mathrm{B}] \) does not appear in the rate law, meaning the order with respect to \( \mathrm{B} \) is 0.(d) \( [\mathrm{B}] \) is raised to the power of 1, so the order with respect to \( \mathrm{B} \) is 1.

Step 3: Calculate the Overall Reaction Order

To find the overall reaction order, add together the order for \( \mathrm{A} \) and the order for \( \mathrm{B} \) for each rate law.(a) The overall order is \(1 + 3 = 4\).(b) The overall order is \(1 + 1 = 2\).(c) The overall order is \(1 + 0 = 1\).(d) The overall order is \(3 + 1 = 4\).

Key Concepts

Rate Law

Rate law in chemical reactions describes how the concentration of reactants affects the rate at which a chemical reaction proceeds. It is usually expressed in the form of a mathematical equation that relates the rate of reaction to the concentrations of reactants involved. The general form is:
\[ \text{Rate} = k[A]^m[B]^n \]
where:

• \( k \) is the rate constant, a proportionality factor that is specific to a particular reaction at a given temperature.
• \([A]\) and \([B]\) are the concentrations of reactants A and B.
• \(m\) and \(n\) represent the orders of the reaction with respect to A and B, respectively.

The exponents \(m\) and \(n\) indicate how the rate depends on the concentration of each reactant. These are determined experimentally and do not necessarily correspond to the stoichiometric coefficients in the balanced chemical equation. Understanding the rate law is crucial for predicting how changes in concentration affect the speed of a reaction.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the speed or rate of a chemical reaction and the factors that influence this rate. It not only tells us how fast a reaction occurs, but also provides insight into the mechanism by which the reaction proceeds.
Several factors can influence the rate of reaction:

• Concentration of reactants: Higher concentration often leads to increased reaction rates as there are more particles available to collide and react.
• Temperature: Generally, increasing the temperature raises reaction rates as particles move faster and collide more often with greater energy.
• Presence of a catalyst: Catalysts lower the activation energy required for a reaction, increasing the rate without being consumed in the process.

Chemical kinetics helps us understand the complexities of reaction mechanisms—step-by-step sequences of elementary reactions—thereby allowing chemists to develop better ways of controlling reaction conditions to optimize product formation.

Reaction Rate Expressions

Reaction rate expressions are mathematical representations that describe how the rate varies with the concentration of the reacting species. They offer a way to quantitatively express the relationship between the reaction rate and the concentrations of the reactants.
For example, given the rate laws from the exercise:

• (a) \( \text{Rate} = k[A][B]^3 \)
• (b) \( \text{Rate} = k[A][B] \)
• (c) \( \text{Rate} = k[A] \)
• (d) \( \text{Rate} = k[A]^3[B] \)

we see that the expressions directly communicate how doubling the concentration of a reactant, for example, will affect the rate of reaction. If the order of reactant B is 3, as seen in expression (a), tripling the concentration of B would increase the rate by a factor of \(3^3 = 27\).
Reaction rate expressions are powerful tools in both theoretical and applied chemistry, providing insights needed in industrial applications, such as the design of reactors and the production of key chemical products.