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 L/mol \(\cdot\) s. What is the rate law for this reaction?

Step-by-Step Solution

Answer

Rate law: Rate = 0.04 [C2F4]^2.

Step 1: Analyze the graphical data

The graph of \(1/[\mathrm{C}_2\mathrm{F}_4]\) versus time gives a straight line. For a second-order reaction, the plot of \(1/[\text{Reactant}]\) versus time should be linear. Hence, this indicates that the reaction follows second-order kinetics with respect to \(\mathrm{C}_2\mathrm{F}_4\).

Step 2: Identify the rate law format for a second-order reaction

For a second-order reaction of the form \(A \rightarrow \text{products}\), the rate law is given by \(\text{Rate} = k[A]^2\), where \(k\) is the rate constant, and \([A]\) is the concentration of the reactant.

Step 3: Determine the rate constant (k)

The slope of the line on a \(1/[\mathrm{C}_2\mathrm{F}_4]\) versus time plot represents the rate constant \(k\) for a second-order reaction. Given that the slope is +0.04 \(\text{L/mol} \cdot \text{s}\), we have \(k = 0.04 \text{ L/mol} \cdot \text{s}\).

Step 4: Write the rate law

Using the second-order rate law format identified in Step 2 and the calculated rate constant from Step 3, the rate law for the reaction \(\mathrm{C}_2\mathrm{F}_4 \rightarrow \frac{1}{2} \mathrm{C}_4\mathrm{F}_8\) is \(\text{Rate} = 0.04 [\mathrm{C}_2\mathrm{F}_4]^2\).

Key Concepts

Second-Order Reaction

A second-order reaction is one in which the rate of the reaction is proportional to the square of the concentration of one reactant. In simpler terms, if you double the concentration of the reactant, the reaction rate increases by four times. This type of reaction is particularly interesting because it highlights the non-linear dynamics of chemical kinetics. In a second-order reaction, such as the one described in the exercise involving \(\mathrm{C}_{2}\mathrm{F}_{4} \rightarrow \frac{1}{2} \mathrm{C}_{4}\mathrm{F}_{8}\), the plot of the reciprocal of the reactant concentration versus time (\(1/[\mathrm{C}_2\mathrm{F}_4]\) vs. time) is a straight line. This linearity is a hallmark of second-order kinetics. Understanding this concept helps students to analyze how reactions proceed and how concentrations impact the overall rate of reaction.

Rate Law

The rate law is an equation that links the reaction rate to the concentrations of reactants. For a second-order reaction, the rate law is typically expressed in the form \[ \text{Rate} = k[A]^2 \] where \(k\) represents the rate constant and \([A]\) is the concentration of the reactant. The rate law indicates that the rate of reaction depends on the concentration squared, and not just directly proportional as seen in first-order reactions. Key aspects of a rate law:

• Determination from experiments: The form of the rate law is determined through experimental data.
• Specific to reaction conditions: The rate law applies to a specific reaction under given conditions like temperature.

Understanding the rate law is crucial because it allows chemists to predict how changing conditions can influence reaction speed.

Rate Constant

The rate constant, denoted by \(k\), is a crucial part of the rate law equation. It is a proportionality constant that gives insight into the speed of a reaction. While it remains constant for a given reaction at a fixed temperature, it can change with temperature alterations. For the reaction involving \(\mathrm{C}_{2}\mathrm{F}_{4} \rightarrow \frac{1}{2} \mathrm{C}_{4}\mathrm{F}_{8}\), the rate constant is extracted from the slope of the \(1/[\mathrm{C}_2\mathrm{F}_4]\) vs. time graph, which is given as 0.04 L/mol·s. Important Points about the Rate Constant:

• Measurement of reaction speed: A higher rate constant typically indicates a faster reaction under the same conditions.
• Dependence on temperature: Rate constants vary with temperature, following the Arrhenius equation.
• Units: The units of the rate constant vary depending on the order of the reaction. For a second-order reaction, the units are L/mol·s.

The rate constant's value and units are pivotal for calculating reaction rates and understanding reaction dynamics.

Chemical Reaction Analysis

Chemical reaction analysis involves examining and interpreting experimental data to understand how reactions occur and evolve over time. This type of analysis can reveal important details about reaction rates, mechanisms, and kinetics. Given the example reaction, \(\mathrm{C}_{2}\mathrm{F}_{4} \rightarrow \frac{1}{2} \mathrm{C}_{4}\mathrm{F}_{8}\), analyzing a graph of \(1/[\mathrm{C}_2\mathrm{F}_4]\) vs. time was key to determining the nature of the reaction.

• Graphical interpretation: Identifying a straight line in plots like \(1/[\text{Reactant}]\) vs. time is essential for understanding reaction order and deriving constants.
• Predictive power: With known parameters like the rate law and rate constant, chemists can predict how the reaction would behave under different conditions.
• Experimental insights: Combining experimental data and theoretical models provides a comprehensive understanding of chemical reactions.

Overall, chemical reaction analysis gives scientists the tools to rigorously assess and manipulate chemical processes for various applications.