Problem 49 - Chemistry: An Atoms-Focused Approach

Question

Write rate laws and determine the units of the rate constant (by 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

Question: Determine the units of the rate constant for each reaction: 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. Answer: a. The units of the rate constant for the reaction of oxygen atoms with \(\mathrm{NO}_{2}\) are \(\dfrac{1}{M \cdot s}\). b. The units of the rate constant for the reaction between \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) are \(\dfrac{1}{M^{2} \cdot s}\). c. The units of the rate constant for the reaction between \(\mathrm{Cl}_{2}\) and chloroform \(\left(\mathrm{CHCl}_{3}\right)\) are \(\dfrac{1}{M^{3/2} \cdot s}\). d. The units of the rate constant for the decomposition of ozone \(\left(\mathrm{O}_{3}\right)\) to \(\mathrm{O}_{2}\) are \(\dfrac{1}{M \cdot s}\).

Step 1: Write the rate law for the reaction

Using the general form of the rate law, we have: \(Rate=k[\mathrm{O}][\mathrm{NO}_{2}]\)

Step 2: Determine the units of the rate constant

To find the units of the rate constant, k, we can rearrange the rate law equation: \(k = \dfrac{Rate}{[\mathrm{O}][\mathrm{NO}_{2}]}\) Since the rate has units of M/s, and the concentrations of O and NO2 are both in M, we get: \(k = \dfrac{M/s}{M \cdot M} = \dfrac{1}{M \cdot s}\) b. The reaction between \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) is second order in NO and first order in \(\mathrm{Cl}_{2}\).

Step 3: Write the rate law for the reaction

Using the general form of the rate law, we have: \(Rate=k[\mathrm{NO}]^{2}[\mathrm{Cl}_{2}]\)

Step 4: Determine the units of the rate constant

To find the units of the rate constant, k, we can rearrange the rate law equation: \(k = \dfrac{Rate}{[\mathrm{NO}]^2[\mathrm{Cl}_{2}]}\) Since the rate has units of M/s, the concentrations of NO and Cl2 are both in M, we get: \(k = \dfrac{M/s}{M^2 \cdot M} = \dfrac{1}{M^2 \cdot s}\) 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 \(\mathrm{Cl}_{2}\)

Step 5: Write the rate law for the reaction

Using the general form of the rate law, we have: \(Rate=k[\mathrm{CHCl}_{3}][\mathrm{Cl}_{2}]^{1/2}\)

Step 6: Determine the units of the rate constant

To find the units of the rate constant, k, we can rearrange the rate law equation: \(k = \dfrac{Rate}{[\mathrm{CHCl}_{3}][\mathrm{Cl}_{2}]^{1/2}}\) Since the rate has units of \(M/s\), the concentrations of CHCl3 and Cl2 are both in M, we get: \(k = \dfrac{M/s}{M \cdot M^{1/2}} = \dfrac{1}{M^{3/2} \cdot s}\) 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 7: Write the rate law for the reaction

Using the general form of the rate law, we have: \(Rate=k[\mathrm{O}_{3}]^{2}[\mathrm{O}]^{-1}\)

Step 8: Determine the units of the rate constant

To find the units of the rate constant, k, we can rearrange the rate law equation: \(k = \dfrac{Rate}{[\mathrm{O}_{3}]^2[\mathrm{O}]^{-1}}\) Since the rate has units of M/s, the concentrations of O3 and O are both in M, we get: \(k = \dfrac{M/s}{M^2 \cdot M^{-1}} = \dfrac{1}{M \cdot s}\)

Key Concepts

Reaction Order

Understanding reaction order is crucial in chemical kinetics, helping us determine how the concentration of reactants affects the rate of a chemical reaction. The term "order" of a reaction represents the power to which the concentration of a reactant is raised in the rate law expression. For example, in a reaction where the rate law is expressed as \(Rate = k[A]^m[B]^n\), the reaction is said to be of order \(m\) with respect to reactant \(A\) and order \(n\) with respect to reactant \(B\). The overall order of the reaction is the sum of these orders, \(m+n\). This sum indicates how the rate changes when the concentration of all reactants is changed proportionally. Key Points to Remember

• A first-order reaction means the rate is directly proportional to the concentration of one reactant.
• A second-order reaction indicates the rate is proportional to the square of the concentration of a single reactant or the concentrations of two different reactants multiplied together.
• Fractional orders and negative orders are also possible, indicating more complex relationships between concentration and rate.

Grasping the concept of reaction order enables prediction and control over the speed of reactions, crucial for many industrial and lab processes.

Rate Constant

The rate constant, symbolized as \(k\), is an essential component of the rate law equation and plays a significant role in describing the speed of a chemical reaction. It is specific to a given reaction at a specific temperature, and it provides insight into how fast or slow a reaction occurs. In mathematical terms, the rate law for a reaction is expressed as \(Rate = k[A]^m[B]^n\), where \([A]\) and \([B]\) are concentrations of the reactants. Here, k remains constant under the same conditions, and it carries unique units which depend on the overall order of the reaction. Understanding Units of the Rate Constant:

• For a zero-order reaction, the units of \(k\) are \(M/s\).
• For a first-order reaction, the units are \(1/s\).
• For a second-order reaction, the units are \(1/(M \cdot s)\).
• In general, for an n-th order reaction, the units of \(k\) are \(1/(M^{n-1} \cdot s)\).

Understanding the rate constant and its units helps in determining the speed and feasibility of reactions, essential for practical applications in synthesis, pharmacology, and environmental science.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions and their mechanisms. By examining how fast reactions proceed, scientists can better understand the conditions that affect these rates and the steps involved in the reactions. Key Elements of Chemical Kinetics:

• **Reaction Rate:** It describes how quickly a reactant is consumed or a product is formed. It is often measured as a change in concentration over time.
• **Rate Laws:** They express the rate of a reaction in terms of the concentration of reactants, each raised to a specific power (the reaction order).
• **Collision Theory:** This theory explains how the rate of a chemical reaction depends on the collisions between reactant molecules, which must have sufficient energy and the correct orientation.
• **Activation Energy:** The minimum energy needed for reactants to successfully collide and form products.

Chemical kinetics not only provides a deeper understanding of reaction dynamics but also enables optimization of conditions to maximize yield and efficiency, important in industries such as pharmaceuticals, agriculture, and manufacturing.