Problem 9 - Chemistry and Chemical Reactivity

Question

The reaction between ozone and nitrogen dioxide at \(231 \mathrm{K}\) is first- order in both \(\left[\mathrm{NO}_{2}\right]\) and \(\left[\mathrm{O}_{3}\right]\) $$ 2 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) $$ (a) Write the rate equation for the reaction. (b) If the concentration of \(\mathrm{NO}_{2}\) is tripled (and \(\left[\mathrm{O}_{3}\right]\) is not changed , what is the change in the reaction rate? (c) What is the effect on reaction rate if the concentration of \(\mathbf{O}_{3}\) is halved (with no change in \(\left.\left[\mathrm{NO}_{2}\right]\right) ?\)

Step-by-Step Solution

Answer

(a) Rate = k[NO2][O3]; (b) Reaction rate triples; (c) Reaction rate is halved.

Step 1: Understanding the Rate Equation

The rate equation of a reaction gives the relationship between the rate of reaction and the concentrations of the reactants. For this reaction, the rate is first-order in both \([\mathrm{NO}_2]\) and \([\mathrm{O}_3]\). This means the rate equation can be written as \[\text{Rate} = k [\mathrm{NO}_2]^1 [\mathrm{O}_3]^1 = k [\mathrm{NO}_2][\mathrm{O}_3]\]where \(k\) is the rate constant.

Step 2: Evaluating the Effect of Tripling \([NO_2]\)

If the concentration of \([\mathrm{NO}_2]\) is tripled, we substitute \([\mathrm{NO}_2] = 3 \times [\mathrm{NO}_2]_0\) into the rate equation. The new rate will be:\[\text{Rate}_{\text{new}} = k [3 \times \mathrm{NO}_2]_0 [\mathrm{O}_3]_0 = 3k [\mathrm{NO}_2]_0[\mathrm{O}_3]_0\]Compared to the original rate:\[\text{Rate}_{\text{original}} = k [\mathrm{NO}_2]_0 [\mathrm{O}_3]_0\]The reaction rate will therefore triple.

Step 3: Evaluating the Effect of Halving \([O_3]\)

If the concentration of \([\mathrm{O}_3]\) is halved, we use \([\mathrm{O}_3] = \frac{1}{2} \times [\mathrm{O}_3]_0\) in the rate equation. The new rate will be:\[\text{Rate}_{\text{new}} = k [\mathrm{NO}_2]_0 \left(\frac{1}{2} [\mathrm{O}_3]_0\right) = \frac{1}{2} k [\mathrm{NO}_2]_0 [\mathrm{O}_3]_0\]This shows that the reaction rate is halved compared to its original value.

Key Concepts

First-order Reaction

A first-order reaction is characterized by its dependency on the concentration of a single reactant. In our case, the reaction between ozone ( O_3) and nitrogen dioxide ( NO_2) is first-order with respect to each reactant, meaning:

• The rate at which the reaction proceeds is directly proportional to the concentration of both NO_2 and O_3 .
• When you double the concentration of one reactant, the reaction rate doubles as well.

This directly influences how we construct the rate equation, which shows the interplay between the rate constant ( k ) and reactant concentrations. The simplicity of first-order reactions allows for straightforward calculations of changes in reaction conditions.

Concentration Change Effect

Concentration changes impact the reaction rate significantly in chemical kinetics. For this specific reaction:

• If the concentration of NO_2 is tripled, the reaction rate triples, as demonstrated in the step-by-step solution.
• Alternatively, halving the O_3 concentration decreases the reaction rate by half.

These effects stem from the first-order nature of the reaction concerning each reactant. Since both reactants individually affect the rate linearly, simple arithmetic changes in concentrations directly translate to proportional changes in the reaction rate.

Rate Constant k

The rate constant ( k ) is a crucial part of the reaction rate equation. It is specific to a particular reaction at a given temperature and provides the link between the concentrations of reactants and the rate of the reaction. In our current study:

• The value of k remains constant if temperature conditions remain unchanged.
• It helps predict how fast the reaction will occur for given concentrations of the reactants.

By understanding the role of k , chemists can compare the speeds of different reactions or the same reaction under varied conditions.

Chemical Kinetics

Chemical kinetics involves the study of reaction rates and how they change under different conditions. In our reaction:

• We analyze how first-order rates change with varying reactant concentrations.
• The principles of kinetics aid in determining the speed of transformation from reactants to products.

Understanding kinetics is vital for optimizing reactions in industrial applications, environmental modeling, and even biochemical pathways. It informs us not only about how quickly a reaction is proceeding but also about the mechanisms driving the transformation.