Question

The decomposition reaction of \(\mathrm{N}_{2} \mathrm{O}_{5}\) in carbon tetrachloride is \(2 \mathrm{~N}_{2} \mathrm{O}_{5} \longrightarrow 4 \mathrm{NO}_{2}+\mathrm{O}_{2}\). The rate law is first order in \(\mathrm{N}_{2} \mathrm{O}_{5}\). At \(55^{\circ} \mathrm{C}\) the rate constant is \(4.12 \times 10^{-3} \mathrm{~s}^{-1}\). (a) Write the rate law for the reaction. (b) What is the rate of reaction when \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]=0.050 \mathrm{M} ?(\mathbf{c})\) What happens to the rate when the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is tripled to \(0.150 \mathrm{M}\) ? (d) What happens to the rate when the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is reduced by \(10 \%\) to \(0.045 \mathrm{M}\) ?

Short answer

(a) The rate law for the reaction is: Rate = k [N₂O₅], where k = 4.12 x 10⁻³ s⁻¹ at 55°C. (b) The rate of reaction when [N₂O₅] = 0.050 M is 2.06 x 10⁻⁴ M/s. (c) When the concentration of N₂O₅ is tripled to 0.150 M, the rate of reaction increases to 6.18 x 10⁻⁴ M/s. (d) When the concentration of N₂O₅ is reduced by 10% to 0.045 M, the rate of reaction decreases to 1.85 x 10⁻⁴ M/s.

Step-by-step solution

(a) Write the rate law for the reaction

Since the reaction is first-order in N₂O₅, the rate law for the reaction is: Rate = k [N₂O₅] where Rate is the rate of the reaction, k is the rate constant (4.12 x 10⁻³ s⁻¹ at 55°C), and [N₂O₅] is the concentration of N₂O₅.

(b) Calculate the rate of reaction when [N₂O₅] = 0.050 M

Using the rate law, substitute the rate constant k and the concentration of N₂O₅: Rate = (4.12 x 10⁻³ s⁻¹) (0.050 M) Rate = 2.06 x 10⁻⁴ M/s

(c) What happens to the rate when the concentration of N₂O₅ is tripled to 0.150 M

Since the rate law is first-order in N₂O₅, if the concentration is tripled, the rate of reaction will also triple. Use the rate law to calculate the new rate: Rate = (4.12 x 10⁻³ s⁻¹) (0.150 M) Rate = 6.18 x 10⁻⁴ M/s The rate of reaction increases to 6.18 x 10⁻⁴ M/s.

(d) What happens to the rate when the concentration of N₂O₅ is reduced by 10% to 0.045 M

When the concentration of N₂O₅ is reduced by 10% to 0.045 M, the rate of reaction will be 90% of the rate at 0.050 M concentration. Calculate the new rate using the rate law: Rate = (4.12 x 10⁻³ s⁻¹) (0.045 M) Rate = 1.85 x 10⁻⁴ M/s The rate of reaction decreases to 1.85 x 10⁻⁴ M/s when the concentration is reduced by 10% to 0.045 M.

Key concepts

First Order Reaction

A first-order reaction refers to a chemical reaction where the rate is directly proportional to the concentration of a single reactant. In the context of the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\), the rate is proportional to the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) itself.
This means that if we increase the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\), the rate of reaction will also increase by the same factor. Conversely, when the concentration decreases, the reaction rate decreases proportionally.
Here, the rate law can be expressed as:

• \(\text{Rate} = k[\mathrm{N}_{2} \mathrm{O}_{5}]\), where \(k\) is the rate constant.

Understanding this relationship helps in predicting how changes in reactant concentration affect the reaction speed, which is crucial in both laboratory and industrial settings.

Kinetics

Kinetics is the branch of chemistry that deals with the rates at which chemical reactions occur. It helps us understand how different factors influence the speed of reactions. In our example of \(\mathrm{N}_{2} \mathrm{O}_{5}\) decomposition, kinetics plays a role in determining how quickly the reaction proceeds at a given temperature and concentration of the reactants.
Kinetic studies provide insights into the reaction mechanism and help in optimizing conditions to achieve desired reaction speeds. The analysis of such data can reveal the order of the reaction and the influences of various parameters like temperature, pressure, and concentration.

Rate Constant

The rate constant, denoted as \(k\), is a critical factor in the rate law of chemical reactions. It gives us a measure of how quickly a reaction takes place, independent of reactant concentrations.
For the first-order decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\), \(k\) is given as \(4.12 \times 10^{-3} \mathrm{~s}^{-1}\) at \(55^{\circ} \mathrm{C}\). This constant is unique for any given reaction and depends on factors like temperature. A higher \(k\) value implies a faster reaction.
It's important to note that the units of the rate constant change with the order of reaction; in a first-order reaction, it is \(\mathrm{s}^{-1}\). Understanding the rate constant helps in determining how adjustments in conditions might accelerate or slow down a reaction.

Chemical Decomposition

Chemical decomposition is a process where a single compound breaks down into two or more simpler compounds or elements. In this reaction, \(2\mathrm{~N}_{2} \mathrm{O}_{5}\) decomposes into \(4 \mathrm{NO}_{2}\) and \(\mathrm{O}_{2}\).
This decomposition reaction, like most, is affected by factors such as temperature and concentration, both of which can influence the speed and extent of the reaction.
Such processes are crucial in various applications, including waste management and the chemical industry, where specific compounds are broken down and repurposed in more useful forms. Recognizing the rules governing these reactions, like rate laws, is essential for controlling and optimizing these processes.