Question

Automobile Engine The following reaction takes place in an automobile’s engine and exhaust system. $$\mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})$$ 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 \(\left[\mathrm{NO}_{2}\right]=0.0048 \mathrm{mol} / \mathrm{L} ?\)

Short answer

The initial, instantaneous rate of the reaction when \([NO_2] = 0.0048 \frac{mol}{L}\) is \(1.152 \times 10^{-5} \frac{L}{s}\).

Step-by-step solution

Understand the given information.

From the exercise, we are given the rate law of the reaction: Rate \(= 0.50 \frac{L}{mol \cdot s} [NO_2]^2\), where [NO2] represents the concentration of NO2 in moles per liter (mol/L). We are also given the initial concentration of NO2: \([NO_2] = 0.0048 \frac{mol}{L}\).

Plug in the given concentration into the rate law.

Now, simply plug in the given concentration of NO2 into the rate law: Rate \(= 0.50 \frac{L}{mol \cdot s} \left(0.0048 \frac{mol}{L}\right)^2\)

Calculate the initial, instantaneous reaction rate.

Finally, calculate the rate by squaring the concentration of NO2 and multiplying it by the rate constant (0.50 L/(mol·s)). Rate \(= 0.50 \frac{L}{mol \cdot s} \left(0.0048 \frac{mol}{L}\right)^2 = 0.50 \frac{L}{mol \cdot s} \times \left(2.304 \times 10^{-5} \frac{mol^2}{L^2}\right)\) Rate \(= 1.152 \times 10^{-5} \frac{L}{s}\) Thus, the initial, instantaneous rate of the reaction when \([NO_2] = 0.0048 \frac{mol}{L}\) is \(1.152 \times 10^{-5} \frac{L}{s}\).

Key concepts

Rate Law

Understanding the rate law in chemistry is essential for determining how quickly a reaction progresses. In chemistry, the **rate law** is an equation that connects the reaction rate with the concentrations of reactants. For the given reaction, the rate law is defined as: \( \text{Rate} = k [\text{NO}_2]^2 \)where \( k \) is the rate constant, specific to a reaction at a given temperature. This indicates that the rate of reaction is directly proportional to the square of the concentration of \( \text{NO}_2 \). These details reveal an important aspect called the "order of reaction," which in this instance is second-order in respect to \( \text{NO}_2 \) because it is raised to the power of two.
The power to which the concentration term is raised is known as the **reaction order** with respect to that reactant. Knowing the reaction order is important because it tells us how changes in the reactant concentration will affect the reaction rate. If, for example, the concentration of \( \text{NO}_2 \) doubles, the rate of reaction will increase by four times because it is second-order.

Concentration

Concentration plays a pivotal role in determining the speed at which chemical reactions occur. **Concentration** refers to the amount of a substance within a particular volume of a solution, often measured in moles per liter (mol/L). In the given chemical context, the concentration affects how frequently reacting particles collide and interact to form products.
In rate laws, as seen in the equation Rate \( = 0.50 \frac{L}{mol \cdot s} [\text{NO}_2]^2\), the concentration of \( \text{NO}_2 \) is crucial. A higher initial concentration of \( \text{NO}_2 \), such as 0.0048 mol/L, will influence how fast the reaction begins. As you increase the concentration, you find that the rate of reaction alters based on the power it is raised to. In this case, since it's squared, the impact on the rate is significant.
For reactions involving gases or solutions, understanding concentration effects is quite important. Scientists calculate changes in rate by using these concentration terms, predicting how changes in concentrations impact the overall reaction over time.

Chemical Kinetics

The field of **chemical kinetics** explores the speed or rate at which chemical reactions take place. It's the part of chemistry that deals with dynamics of reactions and how various conditions such as temperature, pressure, and concentration affect them. Chemical kinetics provides insights into reaction mechanisms, helping to understand step-by-step processes that occur as reactants transform into products.
Within chemical kinetics, the concept offered by rate laws is foundational. It combines with experimental data to determine reaction rates under controlled conditions. Practically, this helps chemists refine processes in industries like pharmaceuticals, where understanding of reaction rates is crucial to producing compounds efficiently.
Furthermore, by analyzing kinetic data, predictions can be made about reaction behaviors under different conditions. Factors such as catalysts, which lower the activation energy, can increase reaction rate without altering the rate law. Thus, kinetics helps not only in understanding existing reactions but also in designing new reactions or improving current ones for enhanced efficiency and productivity.