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 \(\left[\mathrm{O}_{3}\right]\) $$2 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{3}(\mathrm{g}) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g})$$ (a) Write the rate equation for the reaction. (b) If the concentration of \(\mathrm{NO}_{2}\) is tripled, what is the change in the reaction rate? (c) What is the effect on reaction rate if the concentration of \(\mathrm{O}_{3}\) is halved?

Step-by-Step Solution

Answer

(a) Rate = k[NO₂][O₃]. (b) Rate triples. (c) Rate halves.

Step 1: Identifying Reaction Order

The reaction is first order in both \([\mathrm{NO}_{2}]\) and \([\mathrm{O}_{3}]\). This means the rate of the reaction depends linearly on each of these concentrations.

Step 2: Write the Rate Equation

For a reaction that is first order in two reactants, the rate equation is given by: \[\text{Rate} = k[\mathrm{NO}_{2}][\mathrm{O}_{3}]\]where \(k\) is the rate constant.

Step 3: Effect of Tripling \(\mathrm{NO}_{2}\) Concentration

If the concentration of \(\mathrm{NO}_{2}\) is tripled, the rate equation becomes:\[\text{New Rate} = k(3[\mathrm{NO}_{2}])[\mathrm{O}_{3}] = 3k[\mathrm{NO}_{2}]([\mathrm{O}_{3}])\]This implies that the rate of the reaction will be tripled.

Step 4: Effect of Halving \(\mathrm{O}_{3}\) Concentration

If the concentration of \(\mathrm{O}_{3}\) is halved, the rate equation becomes:\[\text{New Rate} = k[\mathrm{NO}_{2}](0.5[\mathrm{O}_{3}]) = 0.5k[\mathrm{NO}_{2}][\mathrm{O}_{3}]\]Therefore, the rate of the reaction will be halved.

Key Concepts

Reaction Order

In chemical kinetics, reaction order is a critical concept that tells us how the concentration of reactants affects the rate of a chemical reaction. For the given exercise, the reaction between ozone ( O_3 ) and nitrogen dioxide ( NO_2 ) is first order with respect to both reactants. This means that the rate of the reaction is directly proportional to the concentration of each reactant.

When a reaction is first order in a particular reactant, doubling that reactant's concentration will result in doubling the reaction rate. In our exercise, since the reaction is first order in both [ O_3 ] and [ NO_2 ] , it showcases a straightforward linear relationship:

• If one reactant triples, the rate triples.
• If one reactant is halved, the rate is halved.

Understanding reaction order helps in predicting how changes in conditions can alter the speed of a reaction.

Rate Equation

The rate equation is a mathematical expression that links the rate of reaction to the concentrations of reactants. For the specific reaction at hand, which is first order in both [NO_2] and [O_3], the rate equation becomes:

\[\text{Rate} = k[\mathrm{NO_2}][\mathrm{O_3}]\]

Here, \(k\) represents the rate constant, a factor affected by temperature and the presence of a catalyst among other conditions. This equation tells us that the rate of the reaction is proportional to the product of the concentrations of [NO_2] and [O_3].

The strength of the rate equation lies in its ability to predict how changes in concentrations of reactants will impact the rate, which is invaluable when designing industrial and laboratory processes.

Concentration Effect

In chemical reactions, the concentration of reactants significantly impacts the rate at which the reaction proceeds. In our ozone and nitrogen dioxide reaction, alterations in concentration carry notable effects. For instance:

- Tripling the concentration of [NO_2] multiplies the rate by three. This is due to the first-order nature of the reaction concerning [NO_2].
- On the contrary, halving the concentration of [O_3] will halve the reaction rate.

This predictable pattern is rooted in the rate equation \[\text{Rate} = k[\mathrm{NO_2}][\mathrm{O_3}]\]. By modifying the concentrations, we directly influence the rate as defined by this equation, offering control over the speed of the chemical process. The concentration effect is a powerful tool in chemical engineering and other applications where precise reaction control is desired.

Rate Constant

The rate constant, symbolized as \(k\), is integral to understanding the dynamics of a chemical reaction. It serves as a proportionality factor in the rate equation, \[\text{Rate} = k[\mathrm{NO_2}][\mathrm{O_3}]\], and its value is determined by a variety of factors including temperature.

Unlike reactant concentrations, the rate constant is unique to each reaction at a given temperature and doesn't change with the concentration. Instead, it reflects the reaction's specific conditions including the energy barrier (activation energy) and molecular collisions that lead to successful transformations.

To note:

• The units of \(k\) in this first-order, two-reactant system, would be \((L/mol\cdot s)\).
• Its particular value implicates how quickly a reaction can occur under set circumstances.

Grasping the implications of the rate constant helps in fine-tuning reactions in both experimental and industrial settings.