Problem 33 - Chemistry and Chemical Reactivity

Question

Gaseous \(\mathrm{NO}_{2}\) decomposes at \(573 \mathrm{K}\) $$ \mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g}) $$ The concentration of \(\mathrm{NO}_{2}\) was measured as a function of time. A graph of \(1 /\left|\mathrm{NO}_{2}\right|\) versus time gives a straight line with a slope of 1.1 L/mol \(\cdot\) s. What is the rate law for this reaction? What is the rate constant?

Step-by-Step Solution

Answer

The rate law is Rate = 1.1 [NO2]^2 with rate constant 1.1 L/mol⋅s.

Step 1: Identify the Reaction Order

The graph of \(\frac{1}{[\text{NO}_2]}\) versus time shows a straight line, indicating that the reaction follows second-order kinetics. For a second-order reaction, a plot of \(\frac{1}{[A]}\) versus time is linear.

Step 2: Write the Integrated Rate Law

For a second-order reaction, the integrated rate law is given by:\[\frac{1}{[A]} = kt + \frac{1}{[A]_0}\]where \([A]\) is the concentration at time \(t\), \(k\) is the rate constant, and \([A]_0\) is the initial concentration.

Step 3: Identify the Slope of the Graph

In the graph \(\frac{1}{[\text{NO}_2]}\) versus time, the slope represents the rate constant \(k\). Given that the slope is 1.1 L/mol \(\cdot\) s, the rate constant \(k\) is thus 1.1 L/mol \(\cdot\) s.

Step 4: State the Rate Law

The rate law for a second-order reaction is expressed as:\[\text{Rate} = k [\text{NO}_2]^2\]Using the rate constant from the graph, this becomes:\[\text{Rate} = 1.1\, [\text{NO}_2]^2\]

Key Concepts

Second-Order Reactions

Chemical kinetics focuses on understanding reaction rates and determining reaction orders. Second-order reactions are a specific type where the sum of the powers of the concentration terms in the rate equation is two. This is quite different from first-order reactions, where the rate depends on the concentration of a single reactant to the first power. In a second-order reaction, the rate can depend on either the square of the concentration of a single reactant or on the product of the concentrations of two different reactants.

For reactions like the decomposition of \( ext{NO}_2\) into \( ext{NO}\) and half \( ext{O}_2\), plotting the reciprocal concentration \[ \frac{1}{[ ext{NO}_2]} \] against time produces a straight line. The linear relationship here confirms the reaction is second-order. These reactions can vary significantly in their characteristics from zero or first-order reactions, making understanding their unique traits crucial.

Overall, identifying the reaction order is pivotal because it dictates the form of the rate law and aids in predicting how the concentration of reactants will change over time.

Rate Law

A rate law expresses the relationship between the reactant concentrations and the rate of a chemical reaction. For second-order reactions, the rate law includes the concentration of the reactant raised to the power of two. The general form of the rate law for a reaction like \( ext{NO}_2 ightarrow ext{NO} + rac{1}{2} ext{O}_2\) is: \[ ext{Rate} = k [ ext{NO}_2]^2 \] where \(k\) is the rate constant.

To derive this, we write the differential rate law which describes how the reaction rate changes with concentration. For the decomposition of \(\text{NO}_2\), it is expressed as: \[ rac{d[ ext{NO}_2]}{dt} = -k[ ext{NO}_2]^2 \] Integrating this equation gives the integrated rate law, useful for determining concentrations at any given time.

Understanding the rate law is essential for predicting reaction behavior and for providing insights into the mechanism. This informs how changes in conditions, like concentration and temperature, could impact the rate of the reaction.

Rate Constant Determination

The rate constant, symbolized as \(k\), is a proportionality factor in the rate law of a chemical reaction. Its value gives insight into the speed of the reaction. In the context of second-order reactions, determining the rate constant involves observing the slope of a plot of \[ rac{1}{[ ext{A}]} \] versus time, where \(A\) represents the reactant.

For our reaction with \(\text{NO}_2\), the given slope of the graph is 1.1 L/mol \(\cdot\) s. This value corresponds exactly to the rate constant for the reaction. It tells us how rapidly the \(\text{NO}_2\) concentration decreases over time relative to its initial concentration.

A crucial aspect of interpreting the rate constant is understanding its units, which change with reaction order. For second-order reactions, the units are typically L/mol \(\cdot\) s, reflective of the reaction's dependence on two concentration factors. Accurate determination of \(k\) allows chemists to model how fast a reaction proceeds, which is vital for both industrial applications and theoretical insights.