Problem 49 - Chemistry The Science in Context

Question

Write rate laws and determine the units of the rate constant (using the units \(M\) for concentration and s for time) for the following reactions: a. The reaction of oxygen atoms with \(\mathrm{NO}_{2}\) is first order in both reactants. b. The reaction between \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) is second order in NO and first order in \(\mathrm{Cl}_{2}\). c. The reaction between \(\mathrm{Cl}_{2}\) and chloroform \(\left(\mathrm{CHCl}_{3}\right)\) is first order in \(\mathrm{CHCl}_{3}\) and one-half order in \(\mathrm{Cl}_{2}\) "d. The decomposition of ozone \(\left(\mathrm{O}_{3}\right)\) to \(\mathrm{O}_{2}\) is second order in \(\mathrm{O}_{3}\) and an order of -1 in \(\mathrm{O}\) atoms.

Step-by-Step Solution

Answer

Answer: a) Rate Law: Rate = k[O][NO₂]; Units of k: M⁻¹s⁻¹ b) Rate Law: Rate = k[NO]²[Cl₂]; Units of k: M⁻²s⁻¹ c) Rate Law: Rate = k[CHCl₃][Cl₂]^(1/2); Units of k: M⁻¹/₂s⁻¹ d) Rate Law: Rate = k[O₃]²[O]⁻¹; Units of k: M⁻¹s⁻¹

Step 1: Reaction (a) Rate Law and Rate Constant Units

For the reaction of oxygen atoms with \(\mathrm{NO}_{2}\), which is first order in both reactants, the rate law is written as follows: Rate = k[O][NO\(_2\)] To determine the units of the rate constant k, we will use the dimensional analysis: Units of k = Units of Rate / {[O] * [NO\(_2\)]} Units of k = M/s / [M * M]; Units of k = M\(^{-1}\)s\(^{-1}\)

Step 2: Reaction (b) Rate Law and Rate Constant Units

For the reaction between \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\), which is second order in \(\mathrm{NO}\) and first order in \(\mathrm{Cl}_{2}\), the rate law is: Rate = k[NO]\(^2\)[Cl\(_2\)] To determine the units of the rate constant k, we will use the dimensional analysis: Units of k = Units of Rate / {[NO]\(^2\) * [Cl\(_2\)]} Units of k = M/s / [M\(^2\) * M] Units of k = M\(^{-2}\)s\(^{-1}\)

Step 3: Reaction (c) Rate Law and Rate Constant Units

For the reaction between \(\mathrm{Cl}_{2}\) and chloroform \(\left(\mathrm{CHCl}_{3}\right)\), which is first order in \(\mathrm{CHCl}_{3}\) and one-half order in \(\mathrm{Cl}_{2}\), the rate law is: Rate = k[CHCl\(_3\)][Cl\(_2\)]\(^{1/2}\) To determine the units of the rate constant k, we will use the dimensional analysis: Units of k = Units of Rate / {[CHCl\(_3\)] * [Cl\(_2\)]\(^{1/2}\)} Units of k = M/s / [M * (M)\(^{1/2}\)] Units of k = M\(^{-1/2}\)s\(^{-1}\)

Step 4: Reaction (d) Rate Law and Rate Constant Units

For the decomposition of ozone \(\left(\mathrm{O}_{3}\right)\) to \(\mathrm{O}_{2}\), which is second order in \(\mathrm{O}_{3}\) and an order of -1 in \(\mathrm{O}\) atoms, the rate law is: Rate = k[O\(_3\)]\(^2\)[O]\(^{-1}\) To determine the units of the rate constant k, we will use the dimensional analysis: Units of k = Units of Rate / {[O\(_3\)]\(^2\) * [O]\(^{-1}\)} Units of k = M/s / [M\(^2\) * M\(^{-1}\)] Units of k = M\(^{-1}\)s\(^{-1}\)

Key Concepts

Rate Laws

In chemical kinetics, a rate law expresses the relationship between the concentration of reactants and the rate of a chemical reaction. The rate law takes the general form: \[ \text{Rate} = k [A]^x [B]^y \] where \(k\) is the rate constant, \([A]\) and \([B]\) are the concentrations of the reactants, and \(x\) and \(y\) are the reaction orders with respect to each reactant. The rate law provides insight into how the concentration of the reactants influences the speed of the reaction. It's crucial to differentiate between the rate law, which is determined experimentally, and the balanced chemical equation, which only provides stoichiometric relationships.

Reaction Orders

Reaction order is a key concept in determining how the concentration of reactants affects the rate of reaction. Each reactant in a chemical reaction has an associated order, which can be zero, first, second, or even fractional or negative.

• **Zero Order:** The reaction rate is independent of the concentration of the reactant.
• **First Order:** The rate is directly proportional to the concentration of the reactant.
• **Second Order:** The rate is proportional to the square of the concentration of the reactant.
• **Fractional Order:** Indicates a more complex relationship often tied to mechanisms involving intermediates.
• **Negative Order:** Suggests that increasing the concentration decreases the reaction rate, often indicative of an inhibitory effect.

Understanding these orders helps predict how changes in concentrations can affect reaction speeds, making them invaluable tools in laboratory and industrial settings.

Rate Constant Units

The units of the rate constant \(k\) vary depending on the overall order of the reaction. Rate constant units are derived using dimensional analysis based on the expression of the rate law:

• The basic unit of rate is usually concentration per time, often \(M/s\).
• For a first-order reaction, the units of \(k\) are \(s^{-1}\) because the rate is solely dependent on time and a single concentration term cancels out with that of rate.
• For a second-order reaction, the units of \(k\) become \(M^{-1}s^{-1}\), as the concentration terms are squared in the rate law expression.
• For reactions of different orders or when dealing with fractional orders, such as one-half order, the dimensional analysis ensures that multiplying across terms converts \(k\) to suitable units, like \(M^{-1/2}s^{-1}\).

Accurate determination of these units is crucial for applying rate constants correctly in calculations and understanding the kinetics of a reaction.

Dimensional Analysis

Dimensional analysis serves as a powerful tool to ensure that equations, particularly in chemical kinetics, are dimensionally consistent, meaning the units on both sides of a reaction equation match up. It's a method used extensively to accurately determine the units of rate constants by analyzing the units of each component in rate equations. To perform dimensional analysis:

• Start by writing down the units of 'rate,' typically given in \(M/s\), where \(M\) stands for molarity.
• Next, consider the units of reactants involved, each represented in molarity \(M\).
• Insert these units into the rate law equation and rearrange to solve for \(k\).

For example, if the rate law is \(\text{Rate} = k[A]^x[B]^y\), the units of \(k\) derive from:\[ \text{Units of } k = \frac{M/s}{(M)^x \, (M)^y} \]Through this approach, consistent unit determination across various reaction orders ensures both precision in kinetic analysis and simplicity in applying these concepts to real-world problems.