Problem 65 - Chemistry Matter and Change

Question

Suppose that a generic chemical reaction has the rate law of rate \(=[A]^{2}[B]^{3}\) and that the reaction rate under a given set of conditions is \(4.5 \times 10^{-4} \mathrm{mol} /(\mathrm{L} \cdot \min ) .\) If the concentrations of both \(\mathrm{A}\) and \(\mathrm{B}\) are doubled and all other reaction conditions remain constant, how will the reaction rate change?

Step-by-Step Solution

Answer

The reaction rate will increase 32 times.

Step 1: Understand the Rate Law

The rate law given is \( \text{rate} = k [A]^2 [B]^3 \), where \( k \) is the rate constant. The rate depends on the concentrations of \( A \) and \( B \). Initially, the rate is \(4.5 \times 10^{-4} \text{ mol/L} \cdot \text{min} \).

Step 2: Express Initial Rate with Concentrations

Let's express the initial rate in terms of the concentrations: \( \text{rate}_{\text{initial}} = k [A]_{\text{initial}}^2 [B]_{\text{initial}}^3 = 4.5 \times 10^{-4} \text{ mol/L} \cdot \text{min} \).

Step 3: Calculate New Concentrations

If the concentrations of both \( A \) and \( B \) are doubled, the new concentrations are \([A]_{\text{new}} = 2[A]_{\text{initial}}\) and \([B]_{\text{new}} = 2[B]_{\text{initial}}\).

Step 4: Determine New Rate Expression

Substitute the new concentrations into the rate law: \( \text{rate}_{\text{new}} = k (2[A])^2 (2[B])^3 = k \cdot 4[A]^2 \cdot 8[B]^3 \).

Step 5: Simplify and Compare

Simplify the expression: \( \text{rate}_{\text{new}} = k \cdot 32 [A]^2 [B]^3 \). The new rate is 32 times the original rate: \( \text{rate}_{\text{new}} = 32 \times \text{rate}_{\text{initial}} = 32 \times 4.5 \times 10^{-4} \text{ mol/L} \cdot \text{min} \).

Step 6: Final Calculation

Calculate the new rate: \( \text{rate}_{\text{new}} = 32 \times 4.5 \times 10^{-4} = 1.44 \times 10^{-2} \text{ mol/L} \cdot \text{min} \).

Key Concepts

Rate Law

A rate law expresses the speed of a chemical reaction in terms of the concentrations of the reactants. The general form is \( \text{rate} = k [A]^m [B]^n \), where \( k \) is the rate constant, and \( m \) and \( n \) are the reaction orders. In the given exercise, the rate law is \( \text{rate} = [A]^2[B]^3 \), showing that the reaction rate is directly proportional to the squared concentration of \( A \) and the cubed concentration of \( B \). This means that any change in these concentrations will significantly affect the reaction speed. Understanding how to interpret and manipulate the rate law is essential for predicting how changes in conditions will influence overall reaction rates.

Concentration Effect

The concentration effect in chemical kinetics refers to how changes in the concentration of reactants influence the rate of a reaction. If you increase the concentration of a reactant, typically, the reaction rate increases because more reactant molecules are available to collide and react.

In the provided problem, doubling the concentrations of both substances \( A \) and \( B \) leads to a notable change in the rate. The rate law indicates the dependency: since \( A \) has an exponent of 2 and \( B \) of 3 in the rate expression, doubling yields \( (2)^2 = 4 \) for \( A \) and \( (2)^3 = 8 \) for \( B \). Consequently, the overall effect is a 32-fold increase in the reaction rate, calculated as \( 4 \times 8 = 32 \). It illustrates how concentration changes can exponentially affect reaction dynamics.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions and the factors affecting them. It's about understanding how and why reactions proceed over time.

Key factors in kinetics include:

• Temperature
• Concentration of reactants
• Presence of catalysts
• Nature of the reactants

In our case, kinetics explains how varying concentrations impact the rate. It provides insights to engineers and scientists looking to optimize conditions for desired reaction rates. By performing calculations based on the rate law, scientists can predict changes in rate with changing conditions, just as exhibited when the concentrations doubled in our exercise.

Reaction Mechanism

A reaction mechanism is a step-by-step sequence of elementary reactions by which an overall chemical change occurs. A mechanism gives insight into how reactants transform into products at the molecular level.

Understanding a mechanism involves identifying:

• Intermediate species
• The sequence of steps
• The rate-determining step (slowest step)

In the rate law provided, the exponents in the expression \( [A]^2[B]^3 \) suggest how the reactants engage in the mechanism. A higher power, like the cube on \( B \), might suggest that three times as much \( B \) is involved in the rate-determining step, hinting at the complexity of that step in the mechanism. Precise understanding of mechanisms assists in designing reactions, particularly in industrial and synthetic chemistry.