Problem 33 - General Chemistry: Principles and Modern Applications

Question

The reaction \(A+B \longrightarrow C+D\) is second order in \(A\) and zero order in B. The value of \(k\) is \(0.0103 \mathrm{M}^{-1} \mathrm{min}^{-1}.\) What is the rate of this reaction when \([\mathrm{A}]=0.116 \mathrm{M}\) and \([\mathrm{B}]=3.83 \mathrm{M} ?\)

Step-by-Step Solution

Answer

The rate of reaction when \(\displaystyle [A]=0.116 M\) and \(\displaystyle [B]=3.83 M\) is \(\displaystyle 0.139 M min^{-1}\)

Step 1: Write out the rate law for the reaction

Using the information about the order of the reaction with respect to each reactant, we can write out the rate law for the reaction. For a reaction \(\displaystyle jA+kB \longrightarrow\), the rate law is usually \(\displaystyle Rate=k[A]^{j}[B]^{k}\) where \(\displaystyle k\) is the rate constant, and the exponents reflect the order of the reaction with respect to each reactant. Here, since the reaction is second order in \(\displaystyle A\) and zero order in \(\displaystyle B\), the rate law becomes \(\displaystyle Rate = k[A]^{2}[B]^{0}\)

Step 2: Simplify the rate law expression

Since any number or quantity raised to the power of zero equals one, we can simplify the expression given above by getting rid of \(\displaystyle [B]^{0}\). This leaves us with \(\displaystyle Rate = k[A]^{2}\)

Step 3: Substitute given values into the rate law

Substitute the given values for \(\displaystyle k\), \(\displaystyle [A]\), and \(\displaystyle [B]\) into the rate law. The rate constant \(\displaystyle k\) given is \(\displaystyle 0.0103 \mathrm{M}^{-1} \mathrm{min}^{-1}\) and the concentrations of \(\displaystyle A\) and \(\displaystyle B\) are \(\displaystyle 0.116 \mathrm{M}\) and \(\displaystyle 3.83 \mathrm{M}\), respectively. Remember that since we found the rate law for this reaction is \(\displaystyle Rate = k[A]^{2}\), we no longer need the concentration of \(\displaystyle B\). Therefore, substituting the given values gives us \(\displaystyle Rate = 0.0103 M^{-1} min^{-1} * (0.116 M)^{2}\)

Step 4: Solve for the Rate

To get \(\displaystyle Rate\), we proceed to simplify \(\displaystyle 0.0103 M^{-1} min^{-1} * (0.116 M)^{2}\). This results in \(\displaystyle Rate = 0.139 M min^{-1}\). So, the rate of reaction when \(\displaystyle [A]=0.116 M\) is \(\displaystyle 0.139 M min^{-1}\)

Key Concepts

Understanding Rate Law

The rate law of a reaction is crucial in understanding how the rate depends on the concentration of the reactants. In simple terms, the rate law expresses the rate of a chemical reaction as a mathematical equation that incorporates the concentrations of reactants and a rate constant.
For a given reaction, the rate law is determined by the reaction mechanism and is not directly deducible from the balanced chemical equation.
For the reaction involving substances A and B, like in our example, the rate law is written as:

• \(\text{Rate} = k [A]^m [B]^n\)

where \( k \) is the rate constant, and \( m \) and \( n \) are the orders of reaction with respect to reactants \( A \) and \( B \), respectively.
The powers to which the concentrations are raised give insights into how changes in concentration affect the reaction rate. In a second-order reaction with respect to \( A \), as in our example, even a small change in \( [A] \) significantly impacts the reaction rate, whereas a zero order in \( B \) indicates the rate is independent of \( [B] \).

Calculating Reaction Rate

Calculating the reaction rate is about determining how fast a reaction is proceeding at a given moment. The reaction rate is determined using the rate law equation. For our specific example, the reaction \( A+B \to C+D \) is second order in \( A \) with no dependence on \( B \). Hence, our simplified rate law is:

• \( \text{Rate} = k [A]^2\)

With given values for the concentration of \( A \) and the rate constant \( k \), you can calculate the rate directly by substituting them into the rate law equation.
Let's break it down:

• Given: \( [A] = 0.116 \text{ M} \)
• Rate constant \( k = 0.0103 \text{ M}^{-1} \text{ min}^{-1} \)
• Substitute into \( \text{Rate} = k [A]^2 \):

\(\text{Rate} = 0.0103 \text{ M}^{-1} \text{ min}^{-1} \times (0.116 \text{ M})^2 = 0.139 \text{ M min}^{-1}\)
This computation concludes that the reaction's rate is \( 0.139 \text{ M min}^{-1} \), highlighting the importance of the concentration in determining how quickly products are formed.

Introduction to Chemical Kinetics

Chemical kinetics is a branch of chemistry that deals with measuring and studying how fast reactions occur. It encompasses the analysis of speeds at which reactants transform into products and the factors affecting these speeds.
Central concepts in chemical kinetics include:

• The **reaction rate**, which tells us how concentration changes over time.
• The **rate constant \( k \)**, a crucial factor in kinetics, which may change with temperature, among other conditions.
• **Reaction order**, specifying whether concentration changes impact rates linearly or otherwise.

In our example, you have seen how the reaction rate depends heavily on the concentration of \( A \), since the reaction is second order in \( A \).
Moreover, kinetics allows chemists to decipher the step-by-step pathway or mechanism of a reaction. By exploring these mechanisms, they can influence or control the reaction conditions to optimize rates and yields, essential in industrial chemical production or everyday processes in lab settings.