Question

The reaction $$\mathrm{NO}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)$$ was studied by performing two experiments. In the first experiment the rate of disappearance of NO was followed in the presence of a large excess of \(\mathrm{O}_{3}\). The results were as follows \(\left(\left[\mathrm{O}_{3}\right]\right.\) remains effectively constant at \(1.0 \times 10^{14}\) molecules/cm \(^{3}\) ): In the second experiment [NO] was held constant at \(2.0 \times 10^{14}\) molecules/cm \(^{3}\). The data for the disappearance of \(\mathbf{O}_{3}\) are as follows: a. What is the order with respect to each reactant? b. What is the overall rate law? c. What is the value of the rate constant from each set of experiments? $$\text { Rate }=k^{\prime}[\mathrm{NO}]^{x} \quad \text { Rate }=k^{\prime \prime}\left[\mathrm{O}_{3}\right]^{y}$$ d. What is the value of the rate constant for the overall rate law? $$\text { Rate }=k[\mathrm{NO}]^{\mathrm{x}}\left[\mathrm{O}_{3}\right]^y$$

Short answer

The order with respect to each reactant is \(x = 1\) for NO and \(y = 1\) for O3. The overall rate law is \(\text{Rate} = k[\text{NO}][\text{O}_{3}]\). The value of the rate constants from each set of experiments are \(k_1 = 1.5 \times 10^{-12}\, \text{cm}^3 \text{/molecule} \cdot \text{s}\) and \(k_2 = 7.5 \times 10^{-13}\, \text{cm}^3 \text{/molecule} \cdot \text{s}\). The value of the rate constant for the overall rate law is \(k = 1.5 \times 10^{-12}\, \text{cm}^3 \text{/molecule} \cdot \text{s}\).

Step-by-step solution

The data from the first experiment refers to the disappearance of NO in the presence of a large excess of O3. The concentration of O3 remains constant at \(1.0 \times 10^{14}\) molecules/cm^3. #Step 2: Write down the rate equation for the first experiment#

Based on the data from the first experiment, our rate equation would be: \[\text{Rate}_1 = k_1 [\text{NO}]^x\] since the concentration of O3 remains constant. #Step 3: Identify the data from the second experiment#

The data from the second experiment refers to the disappearance of O3 when the concentration of NO is held constant at \(2.0 \times 10^{14}\) molecules/cm^3. #Step 4: Write down the rate equation for the second experiment#

Based on the data from the second experiment, our rate equation would be: \[\text{Rate}_2 = k_2 [\text{O}_{3}]^y\] since the concentration of NO remains constant. #Step 5: Determine the order of the reaction with respect to each reactant#

Analyze the data given for both experiments and determine the order x for NO and the order y for O3. Compare the change in concentrations in relation to the change in the disappearance rates. #Step 6: Write the overall rate law#

Using the orders x and y found in Step 5, write down the overall rate law: \[\text{Rate} = k [\text{NO}]^x [\text{O}_{3}]^y\] #Step 7: Determine the value of the rate constant from each set of experiments#

Use the rate equations for both experiments and the data given to solve for the rate constants \(k_1\) and \(k_2\): \[\text{Rate}_1 = k_1 [\text{NO}]^x\] \[\text{Rate}_2 = k_2 [\text{O}_{3}]^y\] #Step 8: Determine the value of the rate constant for the overall rate law#

Combine the rate constants \(k_1\) and \(k_2\) to find the constant k for the overall rate law: \[\text{Rate} = k [\text{NO}]^x [\text{O}_{3}]^y\]

Key concepts

Reaction Order

The reaction order is a crucial concept in chemical kinetics that describes the power to which the concentration of a reactant is raised in the rate law equation. Each reactant in a chemical reaction can have its own order, which may vary independently.

To determine the reaction order with respect to a specific reactant, we analyze how the change in its concentration affects the rate of reaction, while other reactant concentrations are kept constant. In the exercise provided, two separate experiments were conducted to find this order.

• In the first experiment, the concentration of \([\mathrm{O}_{3}]\) was held constant, allowing scientists to determine the order with respect to \([\mathrm{NO}]\).
• In the second experiment, the concentration of \([\mathrm{NO}]\) was kept constant to ascertain the order in relation to \([\mathrm{O}_{3}]\).

Through such experiments, we identify the exponents in the rate law for each reactant. These exponents are known as the reaction order, and they help in understanding how sensitively the rate of reaction responds to changes in reactant concentrations.

Rate Law

The rate law is an equation that expresses the rate of a chemical reaction as a function of the concentration of its reactants. The general form of the rate law is:
\[ \text{Rate} = k [A]^m [B]^n \]where \([A]\) and \([B]\) represent the concentrations of reactants A and B, m and n are their respective reaction orders, and k is the rate constant.

For the reaction given in the exercise,\[ \mathrm{NO}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) \]The rate law combines the effects of both reactants — NO and O3. Through the experiments, we determine:

• The order of reaction with respect to NO
• The order of reaction with respect to O3

Thus, the overall rate law will be:\[ \text{Rate} = k [\mathrm{NO}]^x [\mathrm{O}_{3}]^y \]Understanding the rate law is vital in predicting how changes in concentrations affect the speed of a reaction and can be used to optimize conditions in industrial processes.

Rate Constant

The rate constant, denoted as k, is a proportionality constant in the rate law equation that links the rate of reaction with the concentrations of reactants, modified by their respective reaction orders. It provides insight into the speed of a reaction under specified conditions.

The value of k is unique for every reaction and depends on factors like temperature and the presence of a catalyst. In the context of this exercise, distinct experiments helped estimate the rate constant for each reactant separately by holding one reactant constant while varying the other.

• From the first experiment, the rate constant can be determined for the disappearance of NO.
• The second experiment allows for the determination of the rate constant concerning the disappearance of O3.

After calculating these constants, they are combined to find the overall rate constant for the reaction. Determining the rate constant is crucial as it helps in calculating the reaction rate at any given concentration, allowing chemists to control and predict the behavior of a reaction in various settings.