Problem 56 - Chemistry: An Atoms-Focused Approach

Question

The reaction between \(\mathrm{N}_{2} \mathrm{O}_{5}\) and water is a source of nitric acid in the atmosphere: $$\mathrm{N}_{2} \mathrm{O}_{5}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightarrow 2 \mathrm{HNO}_{3}(g)$$ a. The reaction is first order in each reactant. Write the rate law for the reaction. b. When \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]=0.132 \mathrm{m} \mathrm{M}\) and \(\left[\mathrm{H}_{2} \mathrm{O}\right]=230 \mathrm{m} M,\) the rate of the reaction is \(4.55 \times 10^{-4} \mathrm{mM} / \mathrm{min} .\) What is the rate constant for the reaction?

Step-by-Step Solution

Answer

Answer: The rate constant for the reaction is approximately 1.41 x 10^{-2} min^{-1}.

Step 1: Writing the rate law

To write the rate law for the given reaction, we will first take into account that the reaction is first order in each reactant. Thus, the rate law for the reaction can be written as: Rate = k[N2O5][H2O] where k is the rate constant, and [N2O5] and [H2O] are the concentrations of N2O5 and H2O, respectively.

Step 2: Finding the rate constant

Given the values of concentrations of reactants and the rate of the reaction, we can now plug these values into the rate law equation to solve for the rate constant, k. Rate = 4.55 x 10^{-4} mM/min [N2O5] = 0.132 mM [H2O] = 230 mM We have Rate = k[N2O5][H2O] So, 4.55 x 10^{-4} = k(0.132)(230) Now, divide both sides by (0.132)(230) to solve for k: k = (4.55 x 10^{-4}) / (0.132 * 230) Now, we compute the value of k: k ≈ 1.41 x 10^{-2} min^{-1} Thus, the rate constant for the reaction is approximately 1.41 x 10^{-2} min^{-1}.

Key Concepts

Rate Law

In chemical kinetics, the rate law is a mathematical expression that describes the rate of a reaction concerning the concentration of its reactants. For the reaction between \( \mathrm{N}_{2} \mathrm{O}_{5} \) and water, the rate law can be written as:

• Rate = \( k [\mathrm{N}_{2} \mathrm{O}_{5}][\mathrm{H}_{2} \mathrm{O}] \)

This equation tells us how the speed of the reaction changes with varying concentrations of \( \mathrm{N}_{2} \mathrm{O}_{5} \) and \( \mathrm{H}_{2} \mathrm{O} \). Here, \( k \) is known as the rate constant, a crucial factor in determining the speed of the reaction.

By examining the rate law, students can understand how each reactant's concentration influences the overall reaction rate. It's vital in predicting how fast a reaction will proceed under certain conditions. This is especially significant in atmospheric chemistry, where reactions need to be analyzed at different concentrations and temperatures.

Reaction Order

The reaction order indicates the power to which the concentration of a reactant is raised in the rate law. For any given reaction, each reactant can have a different reaction order. In this exercise, the reaction is first order with respect to both \( \mathrm{N}_{2} \mathrm{O}_{5} \) and \( \mathrm{H}_{2} \mathrm{O} \).

Thus, the overall reaction order is the sum of the orders for each reactant, which in this case is:

• First order in \( \mathrm{N}_{2} \mathrm{O}_{5} \)
• First order in \( \mathrm{H}_{2} \mathrm{O} \)

This gives us a total reaction order of two.

Understanding the reaction order is crucial as it influences the rate law and helps predict how changes in concentrations affect the reaction speed. A first-order reaction means that if the concentration of a reactant doubles, the reaction rate also doubles. The reaction order can be determined experimentally and provides valuable insight into the molecular dynamics of the reaction.

Rate Constant

The rate constant, symbolized as \( k \), is a unique number that characterizes the speed of a reaction. It provides a quantitative measure of how quickly or slowly a reaction proceeds.

For the reaction

• \( \mathrm{N}_{2} \mathrm{O}_{5} + \mathrm{H}_{2} \mathrm{O} \rightarrow 2 \mathrm{HNO}_{3} \)

with the given rate law, \( k \) can be calculated using the formula:\[ k = \frac{\text{Rate}}{[\mathrm{N}_{2} \mathrm{O}_{5}][\mathrm{H}_{2} \mathrm{O}]} \]By plugging in known values, like the reaction rate and the concentrations of the reactants, we found:\[ k \approx 1.41 \times 10^{-2} \text{ min}^{-1} \]The units of \( k \) depend on the order of the reaction; in this case, it's inverse minutes, \( \text{min}^{-1} \). This constant is essential in various applications, including engineering and environmental modeling, where understanding the speed of reactions is paramount.

Atmospheric Chemistry

Atmospheric chemistry involves the chemical compositions and reactions occurring in the Earth's atmosphere. Reactions like the one between \( \mathrm{N}_{2} \mathrm{O}_{5} \) and water are vital because they produce nitric acid, a component of acid rain.

These chemical processes have substantial impacts on the environment, notably affecting air quality and the ecosystem.
Understanding atmospheric reactions helps:

• Predict weather and climate patterns
• Assess environmental impact and pollution levels
• Formulate policies to reduce harmful emissions

Atmospheric chemistry also includes the study of how pollutants interact with natural atmospheric components, how their concentrations change over time, and how they contribute to phenomena such as global warming and ozone depletion. This branch of chemistry is critical for ensuring a sustainable and healthy environment for future generations.