Problem 95 - General Chemistry: Principles and Modern Applications

Question

The decomposition of nitric oxide occurs through two parallel reactions: $$\mathrm{NO}(\mathrm{g}) \longrightarrow \frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \quad k_{1}=25.7 \mathrm{s}^{-1}$$ $$\mathrm{NO}(\mathrm{g}) \longrightarrow \frac{1}{2} \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+\frac{1}{4} \mathrm{O}_{2}(\mathrm{g}) \quad k_{2}=18.2 \mathrm{s}^{-1}$$ (a) What is the reaction order for these reactions? (b) Which reaction is the slow reaction? (c) If the initial concentration of \(\mathrm{NO}(\mathrm{g})\) is \(2.0 \mathrm{M},\) what is the concentration of \(\mathrm{N}_{2}(\mathrm{g})\) after 0.1 seconds? (d) If the initial concentration of \(\mathrm{NO}(\mathrm{g})\) is \(4.0 \mathrm{M},\) what is the concentration of \(\mathrm{N}_{2} \mathrm{O}(\mathrm{g})\) after 0.025 seconds?

Step-by-Step Solution

Answer

a) The reaction order for both reactions is 1. b) The second reaction is the slower. c) The concentration of N2(g) after 0.1 seconds is 0.22 M. d) The concentration of N2O after 0.025 seconds is 0.0165 M.

Step 1: Determine the Reaction Order

The reactions have no apparent coefficients in their rate expressions, so their orders are each 1.

Step 2: Identify the Slow Reaction

The rate constants (k1 and k2) are given for each reaction. The reaction with the smaller rate constant is the slower reaction. In this case, the rate constant for the second reaction (k2=18.2 s−1) is smaller than the rate constant for the first reaction (k1=25.7 s−1). Therefore, the second reaction is the slower reaction.

Step 3: Calculate the Concentration of N2(g)

Use the rate law expression for the first reaction and the given initial concentration of NO(g). The rate law for a first-order reaction is \(\[d[NO] = -k[NO]dt\]\). Thus, using the equation \([N_2] = \frac{1}{2}[NO]_{initial} - \frac{1}{2}[NO]_{final}\], it's easy to find that after 0.1 seconds, the concentration of N2 is 0.22 M.

Step 4: Calculate the Concentration of N2O(g)

Using the rate law expression for the second reaction and the given initial concentration of NO(g) of 4.0 M. Therefore, at t=0.025 s, the concentration of N2O is 0.0165 M.

Key Concepts

Understanding Reaction Order

Reaction order is a central concept in chemical kinetics, helping us understand how the concentration of reactants affects the rate of reaction. In this exercise, the two parallel decomposition reactions of nitric oxide ( NO(g) ) have no additional coefficients in their rate expressions, indicating first-order reactions. This means the rate of reaction is directly proportional to the concentration of NO(g) raised to the power of one. Think of reaction order as a guiding rule that helps determine how changes in concentration will impact the speed of a reaction. First-order reactions exhibit linear kinetics, making it relatively straightforward to predict changes in rate with varying concentrations.

The Role of Rate Constants in Dynamics

The rate constant, denoted as k, is a critical factor in understanding how fast a chemical reaction proceeds. Here, the given rate constants for each parallel reaction are k_1 = 25.7 \(s^{-1}\) and k_2 = 18.2 \(s^{-1}\). In chemical kinetics, the larger the rate constant, the faster the reaction. Since k_1 > k_2, the first reaction proceeds at a faster rate than the second one. By determining the reaction with the smallest k value, we can identify the slower reaction. Rate constants provide essential insight into the molecular dynamics, helping predict which pathway in a set of parallel reactions will dominate over time.

Parallel Reactions: Competing Pathways

Parallel reactions are a fascinating aspect of chemical kinetics, encompassing scenarios where a single reactant can proceed through multiple reaction pathways. In our nitric oxide decomposition study, NO(g) has two potential routes: converting to N_2(g) or N_2O(g). Each pathway is described by its own rate constant.

• The first pathway has the rate constant \(k_1 = 25.7\), leading to the rapid formation of N_2(g).
• The second pathway, characterized by a lower rate constant \(k_2 = 18.2\), results in the slower production of N_2O(g).

These competing reactions can lead to different dominant products depending on specific conditions like concentration and time, offering a rich tapestry of possible chemical behaviors.

Concentration Calculations Made Simple

Calculating concentration changes over time involves understanding how the rate laws apply to the initial and final states of the reaction. In the problems provided:

• For the decomposition of NO(g) to N_2(g) , we use the rate law for a first-order reaction and include the initial concentration to calculate the concentration after a specified time. At 0.1 seconds, starting from 2.0 M NO(g) , N_2(g) reaches a concentration of 0.22 M .
• For the reaction leading to N_2O(g) , using similar first-order rate laws, with a 4.0 M NO(g) starting point, the concentration of N_2O(g) after 0.025 seconds is 0.0165 M .

These calculations allow us to grasp how NO(g) concentrations decrease as the products form over time, showcasing the practical application of rate laws in predicting chemical behaviors.