Problem 16 - Chemistry The Molecular Science

Question

For the reaction $$ \mathrm{O}_{3}(\mathrm{~g})+\mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{O}_{2}(\mathrm{~g}) $$ make qualitatively correct plots of the concentrations of \(\mathrm{O}_{3}(\mathrm{~g}), \mathrm{O}(\mathrm{g}),\) and \(\mathrm{O}_{2}(\mathrm{~g})\) versus time. Draw all three graphs on the same axes, assume that you start with \(\mathrm{O}_{3}(\mathrm{~g})\) and \(\mathrm{O}(\mathrm{g})\), each at a concentration of \(1.0 \mu \mathrm{mol} / \mathrm{L}\). Explain how you would determine, from these plots, (a) the initial rate of the reaction. (b) the final rate (that is, the rate as time approaches infinity).

Step-by-Step Solution

Answer

Start with slopes at 1.0, the initial rate by tangent at \(t=0\), final rate is zero when slopes flatten.

Step 1: Understand the initial concentrations

We start the reaction with concentrations of 1.0 \(\mu\mathrm{mol/L}\) for both \(\mathrm{O}_{3}\) and \(\mathrm{O}\). No \(\mathrm{O}_{2}\) is present initially, so its concentration starts at 0 \(\mu\mathrm{mol/L}\).

Step 2: Identify changes over time

As the reaction progresses, \(\mathrm{O}_{3}\) and \(\mathrm{O}\) will decrease in concentration because they are used up to form \(\mathrm{O}_{2}\). The concentration of \(\mathrm{O}_{2}\) will increase as the reaction proceeds.

Step 3: Sketch the concentration plot

On the y-axis, plot concentration \((\mu\mathrm{mol/L})\), and on the x-axis, plot time. The plot for \(\mathrm{O}_{3}\) will start at 1.0, slope downwards, and approach zero. Similarly, the plot for \(\mathrm{O}\) will start at 1.0 and also slope downwards towards zero. The plot for \(\mathrm{O}_{2}\) will start at 0 and rise towards a final value as \(\mathrm{O}_{3}\) and \(\mathrm{O}\) are converted into \(\mathrm{O}_{2}\).

Step 4: Determine the initial rate

The initial rate of the reaction corresponds to the steepest slope at time \(t=0\) on the plots for \(\mathrm{O}_{3}\) and \(\mathrm{O}\). You can determine it by finding the tangent line to these plots at \(t=0\).

Step 5: Determine the final rate

As time approaches infinity, the concentrations of \(\mathrm{O}_{3}\) and \(\mathrm{O}\) level off to zero, indicating that the reaction is complete, and \(\mathrm{O}_{2}\) has no change. The slope of these plots becomes zero, reflecting the final rate is zero.

Key Concepts

Reaction Rates

In chemical kinetics, understanding reaction rates is key for comprehending how reactions occur and change over time. The rate of a reaction essentially tells us how fast reactants are converted into products. It is typically measured as the change in the concentration of a reactant or product per unit time.

Factors affecting reaction rates include:

• Concentration of Reactants: A higher concentration usually increases the rate since there are more particles to collide and react.
• Temperature: Raising temperature generally increases the reaction rate as particles move faster and collide more often.
• Presence of a Catalyst: Catalysts provide an alternative pathway with a lower activation energy, speeding up the reaction without being consumed.
• Surface Area: In reactions involving solids, a larger surface area can lead to a faster rate.

Observing how fast \(\mathrm{O}_{3} \text{ and } \mathrm{O}\) are consumed and \(\mathrm{O}_{2}\) is formed in the given reaction directly involves understanding these principles.

Concentration vs Time Graphs

Concentration vs time graphs are a practical way to visually represent how the concentration of reactants and products in a chemical reaction changes over time. These graphs give visual cues about the speed and progress of the reaction.

On the y-axis, we plot concentration (in \(\mu\mathrm{mol/L}\)), while the x-axis represents time. In the given chemical reaction involving ozone \((\mathrm{O}_{3})\), atomic oxygen \((\mathrm{O})\), and molecular oxygen \((\mathrm{O}_{2})\), we can sketch:

• Both \(\mathrm{O}_{3}\) and \(\mathrm{O}\) will start at their initial concentration values and fall towards zero as they are used up.
• The \(\mathrm{O}_{2}\) starts at 0 \(\mu\mathrm{mol/L}\) and its concentration increases over time, reflecting the formation of the product.

From such a graph, you can determine the speed of reaction by observing how steep the slopes are at different points.

Initial Rate of Reaction

The initial rate of a reaction refers to the speed at which reactants are converted into products at the very beginning of the reaction, specifically at time \(t=0\). This is typically the fastest rate in the reaction because reactant concentrations are at their highest at the start.

For the given reaction, the initial rate can be found by determining the slope of the tangent line to the concentration vs time plots for \(\mathrm{O}_{3}\) and \(\mathrm{O}\) at \(t=0\). The steeper this slope, the higher the initial rate.

Understanding the initial rate is particularly important because it provides insight into the effectiveness of reactants under specific conditions, allowing chemists to compare different experimental setups or predict the behavior of reactions under different initial concentrations.

Final Rate of Reaction

As a reaction proceeds towards completion, the rate usually decreases unless affected by external factors. The final rate of a reaction is considered when the concentrations of the reactants no longer change as time approaches infinity. This means the reaction has reached a point of equilibrium or has gone to completion.

In the case of our reaction involving \(\mathrm{O}_{3}\), \(\mathrm{O}\), and \(\mathrm{O}_{2}\):

• As time progresses, the reactant concentrations decrease toward zero, and the curve of \(\mathrm{O}_{3}\) and \(\mathrm{O}\) levels off.
• The line becomes flat, reflecting a slope of zero, which indicates the final rate is zero, signifying the reaction has stopped.

The final rate helps us understand the total extent of a reaction and provides information on the stability of the products formed.