Question

The decomposition of nitrosyl chloride $$ \mathrm{NOCl}(g) \longrightarrow \mathrm{NO}(g)+\frac{1}{2} \mathrm{Cl}_{2}(g) $$ is a second-order reaction. If it takes 0.20 min to decompose \(15 \%\) of a \(0.300 \mathrm{M}\) solution of nitrosyl chloride, what is \(k\) for the reaction?

Short answer

Answer: The rate constant for the decomposition of nitrosyl chloride is \(k = 0.00350 \, \mathrm{M^{-1}s^{-1}}\).

Step-by-step solution

Calculate the 15% decomposition quantity

The exercise provides that after 0.20 minutes, 15% of the initial concentration has decomposed. Thus, decode the percentage to a decimal value and multiply by the initial concentration to obtain the quantity of nitrosyl chloride that has decomposed: \(0.15 * 0.300 M = 0.045 M\)

Calculate the concentration of nitrosyl chloride at 0.20 minutes

Subtract the decomposition quantity from the initial concentration to find the concentration of nitrosyl chloride at the given time: \([\mathrm{A}]_{t} = [\mathrm{A}]_{0} - 0.045 M = 0.300 M - 0.045 M = 0.255 M\)

Convert the given time in minutes to seconds

Since we were given the time in minutes, we need to convert it into seconds for the formula calculation: \(t = 0.20 \, \text{min} \times 60 \, \frac{\text{s}}{\text{min}} = 12 \, \text{s}\)

Use the second-order reaction formula to find the rate constant

Now we have all the values needed to find \(k\) using the given formula: $$ \frac{1}{[\mathrm{A}]_{t}}-\frac{1}{[\mathrm{A}]_{0}}=kt $$ Plug in the known values: $$ \frac{1}{0.255}-\frac{1}{0.300}=k(12) $$ Calculate the value of \(k\): $$ k=\frac{\frac{1}{0.255}-\frac{1}{0.300}}{12} $$ $$ k = 0.00350 \, \mathrm{M^{-1}s^{-1}} $$ The rate constant for the decomposition of nitrosyl chloride is \(k = 0.00350 \, \mathrm{M^{-1}s^{-1}}\).

Key concepts

Nitrosyl Chloride Decomposition

Understanding the decomposition of nitrosyl chloride (NOCl) is essential in the field of chemical kinetics. Nitrosyl chloride is a gas that decomposes into nitrogen monoxide (NO) and chlorine (Cl₂) gas. The reaction can be written as:
\[\begin{equation} \mathrm{NOCl}(g) \longrightarrow \mathrm{NO}(g) + \frac{1}{2} \mathrm{Cl}_{2}(g) \end{equation}\]
This process is an example of a unimolecular reaction, where a single molecule of nitrosyl chloride splits into two products. These reactions are often studied since they provide insights into reaction mechanisms and conditions that influence the rate of reaction. Mastery of these reactions is not just important for academic purposes but is also critical in various industrial applications such as the synthesis of fine chemicals and the manufacturing of pharmaceuticals.

Reaction Kinetics

Reaction kinetics is the study of the rates of chemical processes and the factors that affect these rates. It is a central concept in chemistry because it enables us to understand how different conditions such as temperature, pressure, and concentration influence reaction speeds.
For a second-order reaction, the rate of reaction depends on the concentration of one reactant raised to the second power, or on the concentrations of two reactants each raised to the first power. The rate law for a second-order reaction where the concentration of a single reactant is squared can be expressed as:
\[\begin{equation} rate = k[\mathrm{A}]^{2} \end{equation}\]
where k is the rate constant, and [A] is the concentration of the reactant. The units of k in this case are typically M^-1s^-1. Knowing the rate constant allows chemists to predict how fast the reaction will proceed under given conditions, which is important for control and optimization of industrial chemical reactions.

Chemistry Problem Solving

Solving problems in chemistry, particularly those involving reaction kinetics, requires a systematic approach that often includes translating word problems into mathematical equations. A key aspect in problem-solving is understanding the units and converting them when necessary. For instance, when working with reaction rates, you may need to convert time measurements from minutes to seconds to use the appropriate units in rate equations.
Furthermore, it's vital to correctly interpret the given data. In our example with nitrosyl chloride, the 15% decomposition is not simply a percentage; it represents a change in concentration, a value that we need for calculations.
Students honing their problem-solving skills in chemistry learn to analyze the problem, identify what is known and what needs to be found, perform the necessary calculations, and then interpret the results. Exercises such as calculating the rate constant for the decomposition of nitrosyl chloride provide real-life contexts, practice in critical thinking, and application of mathematical skills, all of which are indispensable for both academic success and professional careers in science.