Problem 50 - Chemistry The Science in Context

Question

Compounds \(A\) and \(B\) react to give a single product, \(C .\) Write the rate law for each of the following cases and determine the units of the rate constant by using the units \(M\) for concentration and s for time: a. The reaction is first order in \(A\) and second order in \(B\). b. The reaction is first order in \(A\) and second order overall. c. The reaction is independent of the concentration of \(\mathrm{A}\) and second order overall. d. The reaction is second order in both \(\mathrm{A}\) and \(\mathrm{B}\).

Step-by-Step Solution

Answer

a. First order in A and second order in B. b. First order in A and second order overall. c. Independent of A and second order overall. d. Second order in A and second order in B. Answer: a. Rate Law: Rate = k * [A]^1 * [B]^2 Units of Rate Constant: k = 1 / (M^2 * s) b. Rate Law: Rate = k * [A]^1 * [B]^1 Units of Rate Constant: k = 1 / (M * s) c. Rate Law: Rate = k * [B]^2 Units of Rate Constant: k = 1 / (M * s) d. Rate Law: Rate = k * [A]^2 * [B]^2 Units of Rate Constant: k = 1 / (M^3 * s)

Step 1: a. First order in A and second order in B

The rate law will be: Rate = k * [A]^1 * [B]^2 To find the units of the rate constant k, we can set up the equation: Rate = k * [A] * [B]^2 Since Rate has units M/s, [A] has units M, and [B]^2 has units M^2, we can write: M/s = k * M * M^2 So the units of k are: k = (M/s) / (M * M^2) = 1 / (M^2 * s)

Step 2: b. First order in A and second order overall

The rate law will be: Rate = k * [A]^1 * [B]^1 To find the units of the rate constant k, we can set up the equation: Rate = k * [A] * [B] Since Rate has units M/s, [A] has units M, and [B] has units M, we can write: M/s = k * M * M So the units of k are: k = (M/s) / (M * M) = 1 / (M * s)

Step 3: c. Independent of A and second order overall

The rate law will be: Rate = k * [B]^2 To find the units of the rate constant k, we can set up the equation: Rate = k * [B]^2 Since Rate has units M/s and [B]^2 has units M^2, we can write: M/s = k * M^2 So the units of k are: k = (M/s) / M^2 = 1 / (M * s)

Step 4: d. Second order in A and second order in B

The rate law will be: Rate = k * [A]^2 * [B]^2 To find the units of the rate constant k, we can set up the equation: Rate = k * [A]^2 * [B]^2 Since Rate has units M/s, [A]^2 has units M^2, and [B]^2 has units M^2, we can write: M/s = k * M^2 * M^2 So the units of k are: k = (M/s) / (M^2 * M^2) = 1 / (M^3 * s)

Key Concepts

Order of Reaction

The order of a reaction refers to the power to which the concentration of a reactant is raised in the rate law expression. It indicates how the rate of reaction depends on the concentration of the reactants. For each reactant in a chemical reaction, there is an order of reaction that is denoted by exponents in the rate law.

For example, if a reaction is first order in reactant A and second order in reactant B, the rate law can be expressed as:

• Rate = k \([A]^1\) \([B]^2\)

This means that the concentration of A impacts the rate linearly, and changing the concentration of B has a quadratic effect on the rate of the reaction.

The overall order of reaction is the sum of the orders with respect to each reactant. So, if a reaction is first order in A and second order in B, the overall reaction order is 3. Understanding the order helps predict how the reaction rate will change as reactant concentrations change.

Rate Constant Units

The rate constant, denoted by the symbol "k," is a critical element in the rate law that helps relate the concentration of reactants to the rate of reaction. The units of the rate constant vary depending on the overall order of the reaction.

To find the units of the rate constant, rearrange the rate law to solve for "k" and substitute the units:

• If the overall reaction is first order: Rate = k \([A]^1\)
• Units: \([\text{M/s}] = k [\text{M}]^1\)
• \(k = \text{M/s} / \text{M} = \text{s}^{-1}\)

For second order reactions, the units would typically be:

• For Rate = k \([A]^1[B]^1\), units of \(k = (\text{M/s}) / (\text{M}) = \text{M}^{-1}\text{s}^{-1}\)

Determining these units is essential to calculating and understanding reaction kinetics, and they provide important context on how the rate depends on concentrations.

Reaction Kinetics

Reaction kinetics is a branch of chemistry focused on studying the rates of chemical reactions. It considers how different conditions influence the speed of a reaction and helps us understand the steps or mechanism by which reactions occur.

Several factors affect reaction kinetics, including

• Concentration of reactants: Higher concentrations typically increase reaction rates.
• Temperature: Raising temperature generally speeds up reactions.
• Presence of catalysts: Catalysts lower activation energy, increasing reaction rate.
• Pressure: For reactions involving gases, higher pressure can increase reaction rates.

Studying reaction kinetics provides insight into how we can manipulate conditions to control the rate of a reaction. It also aids in economic and industrial processes where optimizing reaction speed can improve efficiency and cost-effectiveness.

Concentration

Concentration refers to the amount of a substance present in a certain volume of solution, typically expressed in molarity (M), which is moles of solute per liter of solution. In reaction kinetics, concentration plays a vital role since it directly affects the rate of reactions as indicated in the rate law expression.

For example, in the rate law expression Rate = k \([A]^n[B]^m\), the exponents \(n\) and \(m\) show how the concentration of reactants \([A]\) and \([B]\) affect the reaction rate.

A higher concentration of reactant usually increases the probability of collision between reactant molecules, which leads to an increased rate of reaction based on the order indicated. This relationship forms the basis for understanding chemical reactivity and designing experiments with desired reaction rates.