Problem 21 - Chemistry The Molecular Science

Question

The reaction of \(\mathrm{CO}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{~g})\) is second-order in \(\mathrm{NO}_{2}\) and zeroth-order in \(\mathrm{CO}\) at temperatures less than \(500 \mathrm{~K}\). (a) Write the rate law for the reaction. (b) Determine how the reaction rate changes if the \(\mathrm{NO}_{2}\) concentration is halved. (c) Determine how the reaction rate changes if the concentration of CO is doubled.

Step-by-Step Solution

Answer

(a) \(\text{Rate} = k[\mathrm{NO}_2]^2\). (b) Rate decreases by factor of 4. (c) Rate remains unchanged.

Step 1: Understand the Orders of Reaction

The given reaction is second-order with respect to \(\mathrm{NO}_2\) and zeroth-order with respect to \(\mathrm{CO}\). This means the rate of reaction depends on \([\mathrm{NO}_2]^2\) and is independent of \([\mathrm{CO}]\).

Step 2: Write the Rate Law

Since the reaction is second-order in \(\mathrm{NO}_2\) and zeroth-order in \(\mathrm{CO}\), the rate law can be written as follows:\[\text{Rate} = k[\mathrm{NO}_2]^2[\mathrm{CO}]^0\]Where \(k\) is the rate constant. The term \([\mathrm{CO}]^0\) equals 1, reflecting zero dependence on \(\mathrm{CO}\), thus simplifying the rate law to:\[\text{Rate} = k[\mathrm{NO}_2]^2\]

Step 3: Effect of Halving \(\mathrm{NO}_2\) Concentration

If the concentration of \(\mathrm{NO}_2\) is halved, \([\mathrm{NO}_2]\) becomes \(\frac{1}{2}[\mathrm{NO}_2]\). Substitute into the rate law:\[\text{New Rate} = k\left(\frac{1}{2}[\mathrm{NO}_2]\right)^2 = k\frac{1}{4}[\mathrm{NO}_2]^2\]Thus, the reaction rate decreases by a factor of 4.

Step 4: Effect of Doubling \(\mathrm{CO}\) Concentration

Since the reaction is zeroth-order in \(\mathrm{CO}\), the rate does not depend on \([\mathrm{CO}]\). Therefore, doubling the \(\mathrm{CO}\) concentration does not change the reaction rate. The rate remains \(k[\mathrm{NO}_2]^2\).

Key Concepts

Rate Law

In the world of chemistry, a rate law is an equation that links the rate of a chemical reaction to the concentration of reactants. It's important because it tells us how the quantity of reactants influences the speed of the reaction. The rate law can be written in the form \( \text{Rate} = k [A]^x [B]^y \), where:

• \( k \) is the rate constant, a unique value for every reaction at a given temperature.
• \([A]\) and \([B]\) are the concentrations of the reactants.
• \(x\) and \(y\) are the orders of the reaction with respect to each reactant.

Example: Reaction of \( \mathrm{CO} \text{ and } \mathrm{NO}_2 \)

For the reaction \( \mathrm{CO}(\mathrm{g}) + \mathrm{NO}_2(\mathrm{g}) \), we have a rate law: \( \text{Rate} = k[\mathrm{NO}_2]^2[\mathrm{CO}]^0 \). Notice how \([\mathrm{CO}]^0\) equals 1, simplifying the rate law to \( \text{Rate} = k[\mathrm{NO}_2]^2 \). This means that the reaction rate is dependent on the square of the concentration of \( \mathrm{NO}_2 \) and is unaffected by the concentration of \( \mathrm{CO} \).
Understanding the rate law helps us predict how changes in concentration affect the speed of the reaction.

Reaction Order

The reaction order is a key concept in reaction kinetics that tells us the power to which the concentration of a reactant is raised in the rate law. Each reactant in a reaction can have a different order, and the overall reaction order is the sum of these individual orders.

• A zeroth-order reaction means that changes in concentration do not affect the rate. For \( \mathrm{CO} \) in our reaction, since it is zeroth-order, even doubling \([\mathrm{CO}]\) concentration will not change the rate.
• A first-order reaction means that if the concentration of a reactant doubles, the rate doubles.
• A second-order reaction, like for \( \mathrm{NO}_2 \) in this example, means the rate is proportional to the square of the concentration. Thus, if \([\mathrm{NO}_2]\) is halved, the reaction rate drops by a factor of four.

Knowing the order of each reactant is essential for accurately determining how changes in concentration will impact the reaction's speed.

Rate Constant

The rate constant, denoted as \( k \), is a value that is specific to a particular chemical reaction at a given temperature. The magnitude of \( k \) is affected by factors such as temperature and the presence of a catalyst. The rate constant is crucial because it is a part of the rate law equation \( \text{Rate} = k [A]^x [B]^y \). It tells us the proportional relationship between the concentrations and the rate at which the reaction occurs.

Characteristics of the Rate Constant

• A large \( k \) suggests a fast reaction since the rate is high even at low concentration of reactants.
• A small \( k \) indicates a slow reaction, so higher concentrations are needed to achieve the same rate as a fast reaction.
• The units of \( k \) vary depending on the overall order of the reaction. For a second-order reaction, the units are typically \( \text{L/mol} \cdot \text{s} \).

Understanding the role of the rate constant helps us anticipate and control the speed of reactions in both experimental and industrial settings by adjusting conditions such as temperature or concentration.