Problem 81 - Chemistry Matter and Change

Question

Automobile Engine The following reaction takes place in an automobile's engine and exhaust system. $$\quad \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})$$ The reaction's rate law at a particular temperature is Rate \(=0.50 \mathrm{L} /(\mathrm{mol} \cdot \mathrm{s})\left[\mathrm{NO}_{2}\right]^{2}\) . What is the reaction's initial, instantaneous rate when \(\left[\mathrm{NO}_{2}\right]=0.0048 \mathrm{mol} / \mathrm{L} ?\)

Step-by-Step Solution

Answer

The instantaneous rate is 0.00001152 mol/L ⋅ s.

Step 1: Identify Given Information

The reaction rate law is provided as \(\text{Rate} = 0.50 \text{ L}/(\text{mol} \cdot \text{s}) \left[\text{NO}_2\right]^2\). We are also given that \(\left[\text{NO}_2\right] = 0.0048 \text{ mol/L}\).

Step 2: Apply the Rate Law Formula

Substitute the given concentration of \(\text{NO}_2\) into the rate law equation. This gives us \(\text{Rate} = 0.50 \cdot (0.0048)^2\).

Step 3: Perform the Calculation

Calculate \((0.0048)^2\) which is \(0.00002304\). Then multiply this value by 0.50 to find the rate. Therefore, \(\text{Rate} = 0.50 \times 0.00002304 = 0.00001152\text{ mol/L} \cdot \text{s}\).

Step 4: Interpret the Solution

The initial, instantaneous rate of the reaction, with the given \([\text{NO}_2]\), is \(0.00001152\text{ mol/L} \cdot \text{s}\). This value represents how quickly the concentration of products forms or reactants are consumed.

Key Concepts

Rate Law

The rate law is a mathematical equation that describes the relationship between the rate of a chemical reaction and the concentration of its reactants. It provides valuable insights into the reaction kinetics. The generic form of a rate law is expressed as:
\[\text{Rate} = k [A]^m [B]^n ...\]Here, \(k\) is the rate constant, which varies with temperature, \([A]\) and \([B]\) represent the concentrations of the reactants, and \(m\) and \(n\) denote the reaction orders with respect to each reactant.

• Rate laws are determined experimentally.
• They can only be predicted by knowing the specific conditions under which the reaction takes place.
• Unlike stoichiometric coefficients, reaction orders do not necessarily match the coefficients in the balanced chemical equation.

In the given exercise, the rate law is Rate \(=0.50 \text{ L}/(\text{mol} \cdot \text{s})\left[\text{NO}_2\right]^{2}\), showing the reaction is second order with respect to \(\text{NO}_2\), meaning the reaction rate is directly proportional to the square of its concentration.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical processes and the factors that affect them. It helps us understand how fast a reaction occurs and how we can control these rates.

• Kinetics involves measuring the rate of reaction and modeling it mathematically.
• It provides an understanding of reaction mechanisms, which detail step-by-step how bonds are broken and formed.
• The study of kinetics can help optimize conditions for chemical reactions, making industrial processes efficient.

In our example, the study of chemical kinetics includes understanding how the rate of reaction in an automobile engine is affected by variables such as temperature, pressure, and concentration of \(\text{NO}_2\). This knowledge can improve engine designs for better fuel efficiency and reduced emission of pollutants.

Concentration

Concentration is a quantitative measure of the amount of a solute present in a given amount of solvent or solution. It influences the rate of chemical reactions significantly.

• The most common unit for concentration in rate laws is molarity (\(\text{mol/L}\)).
• Higher concentrations of reactants typically increase the rate of a reaction because there are more molecules available to collide and react.
• Rate laws make it possible to calculate how changes in concentration can change the reaction rate.

In the automobile engine example, the initial concentration of \(\text{NO}_2\) given is \(0.0048\, \text{mol/L}\). Using this in the rate law, it was possible to calculate the initial instantaneous rate of reaction as \(0.00001152\, \text{mol/L} \cdot \text{s}\). This shows that even a small concentration of \(\text{NO}_2\) can significantly affect the rate of reaction.

Stoichiometry

Stoichiometry involves calculating the quantities of reactants and products involved in a chemical reaction. It is based on the balanced chemical equation and the conservation of mass.

• Stoichiometry provides the ratios in which reactants combine and products form.
• A balanced equation ensures that the same number of atoms of each element is present on both sides of the equation.
• Stoichiometric coefficients can serve as a simple guide but might differ from reaction orders in the rate law.

In the given reaction \(\text{NO}_2 + \text{CO} \rightarrow \text{NO} + \text{CO}_2\), stoichiometry tells us that these reactants and products react and form in a 1:1 ratio. However, when analyzing reaction rates, we rely on the rate law, not stoichiometry, to understand how changes in concentrations affect the rate. This distinction is crucial for accurately predicting how a reaction proceeds under different conditions.