Question

Consider the reaction: $$ \mathrm{NO}_{2}(g) \longrightarrow \mathrm{NO}(g)+\frac{1}{2} \mathrm{O}_{2}(g) $$ The tabulated data were collected for the concentration of \(\mathrm{NO}_{2}\) as a function of time: $$ \begin{array}{cc} \text { Time (s) } & {\left[\mathrm{NO}_{2}\right] \text { (M) }} \\ \hline 0 & 1.000 \\ \hline 10 & 0.951 \\ \hline 20 & 0.904 \\ \hline 30 & 0.860 \\ \hline 40 & 0.818 \\ \hline 50 & 0.778 \\ \hline 60 & 0.740 \\ \hline 70 & 0.704 \\ \hline 80 & 0.670 \\ \hline 90 & 0.637 \\ \hline 100 & 0.606 \\ \hline \end{array} $$ a. What is the average rate of the reaction between 10 and 20 s? Between 50 and 60 s? b. What is the rate of formation of \(\mathrm{O}_{2}\) between 50 and \(60 \mathrm{~s}\) ?

Short answer

The average rate of reaction between 10 and 20 seconds is 0.0047 M/s. The average rate of reaction between 50 and 60 seconds is 0.0038 M/s. The rate of formation of O\(_2\) between 50 and 60 seconds is 0.0019 M/s.

Step-by-step solution

Determine Change in NO\(_2\) Concentration between 10 and 20 seconds

First, find the difference in concentration for NO\(_2\) at 10 s and 20 s using the given data. Concentration at 10 s is 0.951 M and at 20 s is 0.904 M.

Calculate the Average Rate of Reaction between 10 and 20 seconds

The average rate of a reaction is calculated using the change in concentration over the change in time. Apply the formula: \(\text{Average Rate} = \frac{\text{Change in Concentration}}{\text{Change in Time}}\). For the interval between 10 and 20 seconds, use the following equation to calculate the average rate: \(\text{Average Rate}_{10-20s} = \frac{[NO_2]_{10s} - [NO_2]_{20s}}{20s - 10s}\).

Determine Change in NO\(_2\) Concentration between 50 and 60 seconds

Similarly, find the difference in concentration for NO\(_2\) at 50 s and 60 s using the given data. Concentration at 50 s is 0.778 M and at 60 s is 0.740 M.

Calculate the Average Rate of Reaction between 50 and 60 seconds

Use the same formula for the average rate. For the interval between 50 and 60 seconds, use the following equation: \(\text{Average Rate}_{50-60s} = \frac{[NO_2]_{50s} - [NO_2]_{60s}}{60s - 50s}\).

Calculate Rate of Formation of O\(_2\) between 50 and 60 seconds

For every 2 moles of NO\(_2\) decomposed, 1 mole of O\(_2\) is formed (from the stoichiometry of the equation). To find the rate of formation of O\(_2\), divide the average rate of disappearance of NO\(_2\) from the step above by 2. Use the following equation: \(\text{Rate of Formation of O}_2 = \frac{\text{Average Rate of disappearance of NO}_2}{2}\).

Key concepts

Reaction Rate

The reaction rate quantifies how quickly reactants turn into products in a chemical reaction. Understanding this rate is crucial because it allows chemists to control and optimize reactions for various applications. In simple terms, the reaction rate can be calculated by measuring the change in concentration of a reactant or product over a specific time interval.

For the given reaction, \[ \mathrm{NO}_{2}(g) \longrightarrow \mathrm{NO}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \], the reaction rate is measured in terms of the decrease in concentration of \( \mathrm{NO}_{2} \) gas. The average rate for a specific time interval is determined by the formula:
\[ \text{Average Rate} = \frac{\text{Change in } [\mathrm{NO}_{2}]}{\text{Change in Time}} \].

This formula allows students to understand the speed of the reaction at a glance, providing insights into how quickly \( \mathrm{NO}_{2} \) is being consumed in the reaction. The average rate will differ at various intervals, indicating how reaction rate can change over the course of the reaction.

Concentration-Time Relationship

The concentration-time relationship in chemical kinetics plays a fundamental role in understanding how reactant concentrations change as a reaction progresses. This relationship is usually represented graphically, but can also be expressed mathematically, and helps in determining the order of the reaction and its rate constant.

By plotting the provided concentration data for \( \mathrm{NO}_{2} \) against time, one would typically observe that the concentration decreases over time, which is characteristic of a reaction where a reactant is being used up. Calculating the slope at different intervals on this graph provides the reaction rate at those specific times. Through this relationship, students can derive the kinetics of the reaction and begin to predict how long it will take for a reactant to reach a certain concentration. This is vital in industrial processes where reaction timing is essential.

Stoichiometry

Stoichiometry is the aspect of chemistry that relates the quantities of reactants and products in a chemical reaction. It is derived from the Greek words stoicheion (element) and metron (measure), hence it involves the measurement of elements. Stoichiometry is based on the principle that matter is conserved in a reaction, enforcing a balance between the reactants and products.

In the stoichiometry of the given reaction, for every two moles of \( \mathrm{NO}_{2} \) that decompose, one mole of \( \mathrm{O}_{2} \) is formed. This is a crucial concept when calculating the rate of formation of \( \mathrm{O}_{2} \) from the rate of disappearance of \( \mathrm{NO}_{2} \). By understanding the stoichiometric coefficients, one can determine the relative rates of formation and consumption of different species in a chemical reaction, which is essential for predicting how much product will form from given amounts of reactants.