Question

The reaction \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) has the rate constant \(k=0.63 M^{-1} \mathrm{s}^{-1}\) . (a) Based on the units for \(k,\) is the reaction first or second order in \(\mathrm{NO}_{2} ?\) ? (b) If the initial concentration of \(\mathrm{NO}_{2}\) is \(0.100 \mathrm{M},\) how would you determine how long it would take for the concentration to decrease to 0.025 \(\mathrm{M}\) ?

Short answer

The reaction is second-order with respect to \(NO_2\) since the rate constant has units of \(M^{-1}s^{-1}\). To find the time for the concentration of \(NO_2\) to decrease to 0.025 M, we can use the integrated rate law equation for a second-order reaction: \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\). Plugging in the initial concentration (\([A]_0 = 0.100\,M\)), the given rate constant (\(k=0.63\, M^{-1}s^{-1}\)), and the final concentration (\([A]_t = 0.025\,M\)), we can solve for t to find it takes approximately 92.06 seconds for the concentration of \(NO_2\) to decrease to 0.025 M.

Step-by-step solution

Determine the order of the reaction with respect to \(NO_2\)

To determine the order of the reaction with respect to \(NO_2\), we need to analyze the units of the rate constant (k). The general rate law equation for a reaction can be represented as follows: \(rate = k[A]^m[B]^n\) Since we are considering the reaction \(2NO_2 \rightarrow 2NO + O_2\), the general rate law equation for this reaction can be represented as: \(rate = k[NO_2]^n\), where n determines the order of the reaction with respect to \(NO_2\). From the exercise, we have the rate constant \(k=0.63 M^{-1}s^{-1}\). Now, to find the order of the reaction, n, compare the units of k with the given units: \(M^{1-n}s^{-1} = M^{-1}s^{-1}\) Comparing the units, we can conclude that n = 2, which indicates that the reaction is second-order with respect to \(NO_2\).

Determine the time for the concentration of \(NO_2\) to decrease to 0.025 M

Now that we know the reaction is second-order in \(NO_2\), we can use the integrated rate law equation for a second-order reaction: \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\) Where: - \([A]_t\) is the concentration of \(NO_2\) at time t - \([A]_0\) is the initial concentration of \(NO_2\) - k is the rate constant - t is the time We are given the initial concentration of \(NO_2\) as \([A]_0 = 0.100\, M\), and we need to find the time (t) it would take for the concentration to decrease to \([A]_t = 0.025\, M\). We will plug these values into the equation along with the rate constant: \(\frac{1}{0.025} - \frac{1}{0.100} = 0.63t\) Now, we simply need to solve for t: \(t = \frac{\frac{1}{0.025} - \frac{1}{0.100}}{0.63}\) Calculating the value, we get: \(t \approx 92.06\,s\) Therefore, it will take approximately 92.06 seconds for the concentration of \(NO_2\) to decrease to 0.025 M.

Key concepts

Rate Constant

The rate constant, denoted by the symbol k, is a crucial factor in the study of reaction kinetics. It provides valuable information about the speed at which a reaction proceeds. Specifically, k represents the proportionality constant that connects the rate of a chemical reaction to the concentrations of the reactants raised to a power determined by the reaction's order.

For a second-order reaction such as the decomposition of NO₂, the rate constant has units of M^-1s^-1. In this example, the given rate constant k is 0.63 M^-1s^-1. These units tell us that as the concentration of the reactant, in this case NO₂, changes, the rate of the reaction will be impacted accordingly. It is also indicative that the reaction is second-order with respect to NO₂, as inferred from the exercise.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur and the factors that affect these rates. When examining reaction kinetics, we not only look at the speed of a reaction but also how different conditions, such as concentration, temperature, and catalysts, can influence this speed.

The significance of understanding reaction kinetics cannot be overstated as it allows chemists to predict how fast a reaction will occur under various circumstances, which is key for industrial applications, pharmaceuticals production, and even in environmental monitoring. In classroom terms, studying the kinetics of a reaction, like the one between NO₂ molecules forming NO and O₂, allows us to calculate reaction rates and further gain insight into the mechanism of the reaction, which can be crucial for developing new reactions or improving existing ones.

Integrated Rate Law

The integrated rate law is a mathematical expression that allows us to relate the concentrations of reactants or products over time for a given reaction. Unlike the differential rate law that describes the rate at a specific instant, the integrated rate law gives an overall picture from the start of the reaction to a certain time t.

For a second-order reaction, the integrated rate law equation is typically presented as:\begin{center}\(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\)t, \([A]_0\) is the initial concentration, k is the rate constant, and t is time. By rearranging this equation, we can calculate the time it takes for a reactant to reach a certain concentration from its initial value, as demonstrated in the solution provided for the NO₂ reaction.