Question

The decomposition of dinitrogen pentoxide has been studied in carbon tetrachloride solvent $\left({\mathrm{CCl}}_4\right)$ at a certain temperature: $$\begin{array}{cc}2{\mathrm{N}}_2{\mathrm{O}}_5& ⟶4{\mathrm{NO}}_2+{\mathrm{O}}_2\\ \left[{\mathrm{N}}_2{\mathrm{O}}_5\right](M)& \text{ Initial Rate }(M/\mathrm{s})\\ 0.92& 0.95\times {10}^{-5}\\ 1.23& 1.20\times {10}^{-5}\\ 1.79& 1.93\times {10}^{-5}\\ 2.00& 2.10\times {10}^{-5}\\ 2.21& 2.26\times {10}^{-5}\end{array}$$ Determine graphically the rate law for the reaction, and calculate the rate constant.

Short answer

The reaction is first-order with rate constant $k\approx 1.03\times {10}^{-5}{s}^{-1}$.

Step-by-step solution

Understanding the Problem

We are given the decomposition of dinitrogen pentoxide (${\mathrm{N}}_2{\mathrm{O}}_5$) in carbon tetrachloride $({\mathrm{CCl}}_4)$ and an experimental data table showing various concentrations $\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$ and their corresponding initial reaction rates. We need to determine the rate law for the reaction and calculate the rate constant based on this data.

Plotting the Data

Plot the initial rate $(M/s)$ against the concentration of ${\mathrm{N}}_2{\mathrm{O}}_5$ $(M)$. This graph helps us visually identify the relationship between the concentration and the rate, which is critical for determining the rate law.

Analyzing the Graph

Once the data is plotted, observe the shape of the graph to determine the order of the reaction with respect to ${\mathrm{N}}_2{\mathrm{O}}_5$. If the graph is a straight line passing through the origin, the relation is linear, suggesting a first-order reaction. If it's a curve, you may need to try different plots, such as rate vs. $[{\mathrm{N}}_2{\mathrm{O}}_5{]}^2$ for a second-order reaction.

Determining the Rate Law

Assume different orders and test which assumption fits the plotted data. Based on the graph plotted earlier, if the plot of rate vs. ${\mathrm{N}}_2{\mathrm{O}}_5$ is linear, determine if it fits a first-order kinetic model. If it fits, the rate law can be expressed as $\text{Rate}=k[{\mathrm{N}}_2{\mathrm{O}}_5]$, where $k$ is the rate constant.

Calculating the Rate Constant (k)

For a first-order reaction, choose one data point from the table. Use the rate law $\text{Rate}=k[{\mathrm{N}}_2{\mathrm{O}}_5]$ to solve for $k$. For example, using the first data point, $\text{Rate}=0.95\times {10}^{-5}M/s$ and $[{\mathrm{N}}_2{\mathrm{O}}_5]=0.92M$, $k=\frac{0.95\times {10}^{-5}}{0.92}$. Calculate $k$ for accuracy.

Verifying the Rate Constant

Check the consistency of the calculated rate constant $k$ using other data points. Ensure that $k$ remains consistent (or sufficiently close) across all data points to confirm the validity of the proposed rate law.

Key concepts

Reaction Rate

In chemical kinetics, the reaction rate measures how quickly a reaction occurs. It is typically expressed as the change in concentration of a reactant or product per unit time. For the decomposition of dinitrogen pentoxide ( 2O5), the reaction rate is determined by observing how the concentration of 2O5 decreases over time.

The initial reaction rate is an important value, especially when examining the early stages of a reaction before any substantial changes occur. Initial rates are often used to deduce important kinetic parameters of a reaction, as they are generally straightforward to measure and better reflect the inherent nature of the process.

Rate Constant

The rate constant, denoted as "k," is a crucial component of the rate law. It reflects how fast a reaction proceeds under given conditions, such as temperature.

The rate constant is derived from experimental data and varies with temperature. It is determined by reorganizing the rate law formula to solve for "k." In the case of the first-order decomposition of 2O5, "k" can be calculated using any of the given concentration and rate data points, ensuring consistent results across multiple trials.

• Temperature dependent: The rate constant changes with temperature.
• Units: The units of "k" depend on the overall order of the reaction.

Reaction Order

Reaction order indicates the power to which the concentration of a reactant is raised in the rate law equation. Understanding the order of a reaction is key to deducing its mechanism.

For the dinitrogen pentoxide decomposition, the reaction order is determined by plotting the initial reaction rates against concentrations. A linear plot suggests a first-order reaction. If the graph were curved, it might indicate a second-order reaction.

• First-order: Rate is directly proportional to the concentration.
• Second-order: Rate depends on the square of the concentration.

Rate Law

The rate law is an equation that connects the reaction rate with the concentration of reactants raised to the power of their respective reaction orders. It is a key expression that is derived experimentally rather than theoretically.

For a first-order reaction like the decomposition of 2O5, the rate law is expressed as $\text{Rate}=k[{\mathrm{N}}_2{\mathrm{O}}_5]$. This equation highlights the direct proportionality between reaction rate and concentration, simplifying the calculation of rate constants.

• Directly established through experiments.
• Linear relationship for first-order reactions.

Dinitrogen Pentoxide Decomposition

The decomposition of dinitrogen pentoxide ( 2O5) is an intriguing reaction that has been studied extensively. It involves the breakdown of 2O5 into nitrogen dioxide (NO2) and oxygen (O2), described by the equation: 2O5 → 4NO2 + O2.

This reaction provides a classic study of first-order kinetics, demonstrable by plotting initial rates against concentrations. The consistency and reproducibility of data make it a reliable model for understanding chemical kinetics principles. Through experiments, the exact analysis of the reaction rate and mechanism becomes accessible to students, reinforcing core kinetic concepts.