Problem 4 - Chemistry in 30 days

Question

For the reaction $$ \mathrm{Cl}_{2}(\mathrm{~g})+2 \mathrm{NO}(\mathrm{g}) \rightarrow 2 \mathrm{NOCl}(\mathrm{g}) $$ It is found that doubling the concentrations of both the reactants increases the rate by a factor of 8 . If only the concentration of \(\mathrm{Cl}_{2}^{\prime}\) is doubled, the rate increases by a factor of 2 . The order of the reaction with respect to NO is (a) \(\underline{1}\) (b) \(\overline{0}\) (c) 2 (d) \(1 / 2\)

Step-by-Step Solution

Answer

The order of the reaction with respect to NO is 2 (option c).

Step 1: Understanding the Rate Law

The rate of a reaction is generally expressed as \( \text{Rate} = k [\text{Cl}_2]^x [\text{NO}]^y \), where \( k \) is the rate constant, and \( x \), \( y \) are the orders of the reaction with respect to \( \text{Cl}_2 \) and \( \text{NO} \), respectively.

Step 2: Effect of Doubling Both Reactants

Doubling the concentrations of both \( \text{Cl}_2 \) and \( \text{NO} \) increases the rate by a factor of 8. This means: \[ (2^x)(2^y) = 2^{x+y} = 8 \] So, \( x+y = 3 \).

Step 3: Effect of Doubling Only Cl₂

Doubling only the concentration of \( \text{Cl}_2 \) leads to a doubled rate, meaning: \[ 2^x = 2 \] Thus, \( x = 1 \).

Step 4: Determine Order with Respect to NO

With \( x = 1 \), substitute into \( x+y = 3 \) to find \( y \): \[ 1 + y = 3 \] So, \( y = 2 \).

Step 5: Conclusion

The order of the reaction with respect to \( \text{NO} \) is 2. Therefore, answer choice (c) is correct.

Key Concepts

Rate law

Understanding the concept of a rate law is key in chemical kinetics. The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants.
It is represented by the equation:

• \( \text{Rate} = k[\text{A}]^m[\text{B}]^n \)

Here, \( k \) is the rate constant, and \( m \), \( n \) indicate the reaction orders of the reactants \( \text{A} \) and \( \text{B} \), respectively. Each order provides insight into how changes in concentration affect the reaction rate. Calculating and understanding orders helps determine reaction mechanisms. It is crucial for predicting how a reaction behaves under different conditions.
The exponents in the rate law represent the order:

• \( m = 0 \) indicates zero influence of \( \text{A} \)'s concentration on the rate.
• \( m = 1 \) depicts a direct proportional effect, where doubling \( [\text{A}] \) doubles the rate.
• Higher orders, like 2, amplify the effect. Doubling \( [\text{A}] \) quadruples the rate. 

By solving equations based on experimental data, you determine specific values of \( m \) and \( n \). This outlines how each reactant contributes to the overall reaction rate.

Chemical kinetics

Chemical kinetics is the study of reaction rates and the mechanisms by which reactions occur. It provides insight into:

• The speed or rate of chemical reactions.
• How various factors like temperature, pressure, and concentration influence these rates. 

Understanding the principles of chemical kinetics allows chemists to predict the progress of reactions over time. They can also manipulate conditions to either speed up or slow down chemical changes to suit industrial or laboratory needs. Key aspects include understanding activation energy, frequency of collisions, and the effectiveness of these collisions in forming products.
Pivotal concepts in chemical kinetics:

• Rate laws describe the quantitative relation between concentration and rate.
• Order of reaction helps determine the step-wise reaction mechanism.
• Factors affecting rate like temperature lead to an exponential change in rate often depicted by the Arrhenius equation.

Chemical kinetics intricately links molecular behavior and macroscopic reactions, paving the way for advances in fields like catalysis and energy production.

Reaction rate

The reaction rate describes how fast or slow a reaction proceeds. It is a measure of the change in concentration of reactants or products over time.
Reaction rates are central in chemical kinetics and are influenced by multiple elements.
Some critical factors affecting reaction rates include:

• Concentration: Higher concentration typically increases the likelihood of collisions, thereby accelerating the rate.
• Temperature: Higher temperatures generally increase reaction rates by providing more energy to the molecules, resulting in more frequent and energetic collisions.
• Surface Area: For reactions involving solids, greater surface area allows more collisions to occur.
• Catalysts: Substances that increase the rate without being consumed. They lower the activation energy needed. 

Incorporating reaction rate knowledge helps industries optimize processes to maximize yield and minimize time.
Reaction rate studies are not just theoretical but have practical applications in everyday life, including cooking, pharmacology, and environmental science.

Concentration effect

The concentration effect relates to how the molarity of reactants influences the reaction rate.
When concentrations change, it directly affects how often molecules collide.

• Doubling Concentration: Doubling the concentration of a reactant can significantly increase the rate, often by a power dependent on the order for that reactant in the rate law.
• Order Specific Influence: If a reaction is first-order with respect to a reactant, doubling the concentration doubles the rate. With a second order, the impact quadruples. This was evident in the example problem, where doubling both reactant concentrations increased rate eightfold due to the second-order nature regarding \( \text{NO} \).
• Partial Orders: Not all reactants have the same order. Some may barely affect the rate, adding layer complexity to understanding dynamics.
  

The concept underscores why precise measurement and control of reactants are crucial in experiments and industrial applications. By managing concentration levels, reactions can be directed towards desired outcomes more efficiently.