Problem 42 - General Chemistry: Principles and Modern Applications

Question

Ammonia decomposes on the surface of a hot tungsten wire. Following are the half-lives that were obtained at \(1100^{\circ} \mathrm{C}\) for different initial concentrations of \(\mathrm{NH}_{3}:\left[\mathrm{NH}_{3}\right]_{0}=0.0031 \mathrm{M}, t_{1 / 2}=7.6 \mathrm{min} ; 0.0015 \mathrm{M}\) \(3.7 \mathrm{min} ; 0.00068 \mathrm{M}, 1.7 \mathrm{min.}\) For this decomposition reaction, what is (a) the order of the reaction; (b) the rate constant, \(k ?\)

Step-by-Step Solution

Answer

The order of reaction is 2nd order, and the rate constant \(k\) approximates to 43.0 M-1min-1.

Step 1: Identifying the reaction order

The initial concentrations of NH3 and their corresponding half-life times are given. Observe the relationship between the concentration of NH3 and its half-life. If the half-life is constant and does not change with concentration, the reaction is first order. However, if the half-life is variable and changes with concentration, the reaction is more likely to be of second or zero order. Here, it can be seen that as the concentration decreases, the half-life also decreases, indicating that the reaction is probably not first order as its half-life is not constant and depends on concentration.

Step 2: Confirming Reaction Order

To further confirm, you can look at the relationship between the half-life and the concentration. If it's a direct relationship (increased concentration leads to increased half-life), the reaction is second order. If it's an inverse relationship (increased concentration leads to decreased half-life), the reaction is zero order. Here, as the concentration increases, the half-life also increases, indicating that this reaction is second-order.

Step 3: Finding the Rate Constant, \(k\)

For a second order reaction, the half-life is given by \( t_{1/2} = \frac{1}{{k[N]_0}} \). So, it can be rearranged to find k: \( k = \frac{1}{{t_{1/2} [N]_0}} \). Using the given concentration and half-life values, calculate k. For example, when [NH3]0 = 0.0031 M and t1/2 = 7.6 minutes, \( k = \frac{1}{{7.6 \times 0.0031}} \approx 43.0 \, \text{M}^{-1}\text{min}^{-1} \). The unit of k for second order reactions is M-1min-1. Repeating calculation for all given data points can cross-check the rate constant value and its consistency.

Key Concepts

Reaction Order

The order of a chemical reaction is a crucial concept in chemical kinetics. It defines how the rate of reaction is related to the concentration of reactants. In the exercise, ammonia decomposes over a hot tungsten wire, and the goal is to determine the reaction order by examining how the half-lives change with different initial concentrations.

Unlike first-order reactions, where half-life remains constant regardless of concentration, second-order reactions exhibit a change in half-life depending on concentration. Specifically, for second-order reactions, the half-life increases as the initial concentration decreases. Looking at the given data, we observe that as the concentration of NH3 decreases from 0.0031 M to 0.00068 M, the half-life decreases from 7.6 to 1.7 minutes. This correlation indicates that the decomposition reaction follows second-order kinetics.

Understanding reaction order helps in predicting how changes in concentration will affect the reaction rate, which is essential for applications ranging from industrial synthesis to everyday chemical reactions.

Rate Constant

The rate constant, denoted as \( k \), is a proportionality factor in the rate equation that is specific to a particular reaction at a given temperature. For a second-order reaction, the rate constant can be determined using the relationship between half-life, initial concentration, and \( k \).

From the formula for a second-order reaction:

• Half-life, \( t_{1/2} = \frac{1}{k[N]_0} \)

This can be rearranged to solve for \( k \) as follows:

• \( k = \frac{1}{t_{1/2}[N]_0} \)

By substituting the values from our experiment:

• For \([\text{NH}_3]_0 = 0.0031\, \text{M}\) and \( t_{1/2} = 7.6\, \text{min} \), \( k \approx 43.0\, \text{M}^{-1} \text{min}^{-1} \).

The consistent value of \( k \) across all calculations confirms the reaction is second-order and provides insight into how rapidly the reaction proceeds under given conditions. The consistency of the rate constant across different data points is critical for validating the order of reaction and reliability of experimental data.

Half-Life

Half-life in the context of chemical reactions is the time required for half of the reactant to be consumed. This concept is particularly important as it helps define the kinetics of a chemical process. Notably, the behavior of half-life can differ significantly between reaction orders.

In the given exercise, the half-lives change with varying initial concentrations, key evidence that the reaction is not first-order. For second-order reactions, the formula

• \( t_{1/2} = \frac{1}{k[N]_0} \)

indicates that the half-life is inversely proportional to the initial concentration. As a result, a higher concentration results in a shorter half-life. This inverse relationship is what distinguishes second-order reactions in terms of half-life behavior.

Understanding the half-life allows chemists not only to predict how long a reaction will proceed before half of the reactant is used up but also provides insights into the speed and mechanism of the reaction over time.

Second-Order Reaction

Second-order reactions involve a rate that is directly proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. The decomposition of ammonia on the tungsten wire is an example of a second-order reaction, evidenced by the observed changes in half-life with varying initial concentrations.

For a second-order reaction, the rate equation can be written as:

• \( \text{Rate} = k[N]^2 \)

This highlights that the reaction rate is sensitive to changes in concentration, making it distinctly different from first-order reactions (which depend linearly on concentration) or zero-order reactions (independent of concentration changes).

The kinetic study of second-order reactions allows chemists to develop a deeper understanding of reaction mechanism and dynamics, optimize conditions for industrial processes, and accurately predict how a reactant's concentration will evolve over time. Consequently, mastering this concept is essential for anyone studying or working in the field of chemistry.