Problem 35 - Chemistry and Chemical Reactivity

Question

For the reaction \(\mathrm{C}_{2} \mathrm{F}_{4} \rightarrow^{1 / 2} \mathrm{C}_{4} \mathrm{F}_{8,}\) a graph of \(1 /\left[\mathrm{C}_{2} \mathrm{F}_{4}\right]\) versus time gives a straight line with a slope of \(+0.04 \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} .\) What is the rate law for this reaction?

Step-by-Step Solution

Answer

Rate law: \( ext{rate} = 0.04 \, {L/mol} \cdot {s} \times [ {C}_2 {F}_4]^2 \).

Step 1: Identify Reaction Order

The straight line in the graph of \( \frac{1}{[{C}_2 {F}_4]} \) versus time indicates that the reaction follows a second-order rate law.

Step 2: Write the General Rate Law for Second-Order Reactions

The general rate law for a second-order reaction is \( ext{rate} = k [{C}_2 {F}_4]^2 \) where \( k \) is the rate constant.

Step 3: Determine Rate Constant from the Graph Slope

For a second-order reaction, the slope of the graph \( \frac{1}{[{A}]} \) versus time is equal to \( k \). Given the slope is \(+0.04 \, {L/mol} \cdot{s} \), the rate constant \( k \) is \( 0.04 \, {L/mol} \cdot {s} \).

Step 4: Write the Specific Rate Law with the Known Constant

Now, substituting the rate constant into the general rate law, the specific rate law for this reaction becomes \( ext{rate} = 0.04 \, {L/mol} \cdot {s} \times [{C}_2 {F}_4]^2 \).

Key Concepts

Reaction Order

The reaction order is a crucial concept in chemical kinetics, determining how the concentration of reactants affects the rate of reaction. For a reaction involving compound \( \mathrm{C}_2 \mathrm{F}_4 \rightarrow^{1/2} \mathrm{C}_4 \mathrm{F}_8 \), the reaction order can be identified by analyzing how the concentration changes with time.
In practical terms, looking at the graph of \( \frac{1}{[\mathrm{C}_2 \mathrm{F}_4]} \) versus time can reveal the order. If this results in a straight line, it confirms a second-order reaction. This is because a second-order reaction has the form \( \mathrm{rate} = k [A]^2 \).

• Zero-order reactions: Concentration does not affect the rate.
• First-order reactions: Rate is directly proportional to concentration.
• Second-order reactions: Rate is proportional to the square of the concentration.

Understanding the reaction order helps predict how changes in concentration affect the reaction rate. It aids in determining the rate law and further kinetic analysis.

Rate Law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. In simple terms, it tells us how the speed of the reaction varies with the concentration of certain compounds involved.
For the reaction \( \mathrm{C}_2 \mathrm{F}_4 \rightarrow^{1/2} \mathrm{C}_4 \mathrm{F}_8 \), the rate law is important to determine because it uses the reaction order as a component. For a second-order reaction like this, the general rate law is \( \mathrm{rate} = k [\mathrm{C}_2 \mathrm{F}_4]^2 \).

• The rate law is derived from experimental data rather than the stoichiometry of the reaction.
• It includes the rate constant \( k \), providing crucial data about the reaction's velocity at a certain temperature.

The specific rate law formulated through the reaction order and use of the rate constant enhances our ability to predict how the reaction progresses under various conditions.

Rate Constant

The rate constant, denoted as \( k \), is a vital component in the rate law, responsible for linking the reaction rate with the concentrations of reactants. It's unique to every reaction and remains constant only at a specific temperature.
For the reaction \( \mathrm{C}_2 \mathrm{F}_4 \rightarrow^{1/2} \mathrm{C}_4 \mathrm{F}_8 \), the rate constant is determined using the slope of the graph \( \frac{1}{[\mathrm{C}_2 \mathrm{F}_4]} \) versus time, which is consistent with its second-order nature.

• For a second-order reaction, the slope of this graph equates to the rate constant \( k \).
• Given the slope is \( 0.04 \, \mathrm{L}/\mathrm{mol} \cdot \mathrm{s} \), this value serves as the rate constant for the specific reaction.

Understanding and determining the rate constant allows chemists to predict not just the speed but also how sensitive the rate is to concentration changes, thereby providing deeper insights into the reaction dynamics.