Consider the reaction: \(2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \rightarrow\) \(\mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) Initial concentrations and rates for this reaction are given in the table below. $$\begin{array}{|c|c|c|c|} \hline {\text { Experiment }} & {\begin{array}{c} \text { Initial concentration } \\ \text { (mol/L) } \\ \text { [NO] } \end{array}} & \begin{array}{c} \text { Initial rate of } \\ \text { [ } \mathbf{H}_{2} \text { ] } \end{array} & \begin{array}{c} \text { formation of } \mathbf{N}_{2} \\ (\mathbf{m o l} / \mathbf{L} \text { min } \mathbf{)} \end{array} \\ \hline 1 & 0.0060 & 0.0010 & 1.8 \times 10^{-4} \\ 2 & 0.0060 & 0.0020 & 3.6 \times 10^{-4} \\ 3 & 0.0010 & 0.0060 & 0.30 \times 10^{-4} \\ 4 & 0.0020 & 0.0060 & 1.2 \times 10^{-4} \\ \hline \end{array}$$ (a) From the data given, determine the order for each of the reactants, \(\mathrm{NO}\) and \(\mathrm{H}_{2}\), show your reasoning, and write the overall rate law for the reaction. (b) Calculate the value of the rate constant, \(k\), for the reaction. Include units. (c) For experiment 2, calculate the concentration of NO remaining when exactly one-half of the original amount of \(\mathrm{H}_{2}\) has been consumed. (d) The following sequence of elementary steps is a proposed mechanism for the reaction. I. \(\mathrm{NO}+\mathrm{NO} \rightarrow \mathrm{N}_{2} \mathrm{O}_{2}\) II. \(\quad \mathrm{N}_{2} \mathrm{O}_{2}+\mathrm{H}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O}+\mathrm{N}_{2} \mathrm{O}\) III. \(\quad \mathrm{N}_{2} \mathrm{O}+\mathrm{H}_{2} \rightarrow \mathrm{N}_{2}+\mathrm{H}_{2} \mathrm{O}\) Based on the data present, explain why the first step cannot be the rate determining step.

Step by step solution

01

Determine the order of reactants and the rate law

To determine the order of the reaction for the reactants \(\mathrm{NO}\) and \(\mathrm{H}_2\), we compare the initial rates between the experiments. Comparing experiments 1 and 2, the \(\mathrm{NO}\) concentration stays constant, whereas the \(\mathrm{H}_2\) concentration doubles, and the rate doubles as well: $$\frac{3.6 \times 10^{-4}}{1.8 \times 10^{-4}} = 2$$ This indicates that the reaction order for \(\mathrm{H}_2\) is 1. Next, we compare experiments 3 and 4, where \(\mathrm{H}_2\) concentration remains constant, and the \(\mathrm{NO}\) concentration doubles. The rate quadruples: $$\frac{1.2 \times 10^{-4}}{0.3 \times 10^{-4}} = 4$$ Thus, the reaction order for \(\mathrm{NO}\) is 2. Now, we can write the overall rate law for the reaction: $$\text{rate} = k[\mathrm{NO}]^2[\mathrm{H}_2]^1$$

02

Calculate the rate constant, \(k\)

Using the rate law and the data from any of the experiments, we can find the rate constant, \(k\). Let's use experiment 1: $$1.8 \times 10^{-4} = k(0.0060)^2(0.0010)$$ Now, solve for \(k\): $$k = \frac{1.8 \times 10^{-4}}{(0.0060)^2(0.0010)} = 5.0\ \text{L/mol/min}$$

03

Calculate the concentration of NO

For experiment 2, when exactly half of the original amount of \(\mathrm{H}_2\) has been consumed, the remaining concentration of \(\mathrm{H}_2\) will be: $$[\mathrm{H}_2]_{\text{remaining}} = \frac{1}{2}(0.0020\ \text{mol/L}) = 0.0010\ \text{mol/L}$$ Using the rate law, we can relate the rate of formation of \(\mathrm{N}_2\) to the remaining concentrations of the reactants: $$3.6 \times 10^{-4} \ \text{mol/L/min} = k[\mathrm{NO}]^2[\mathrm{H}_2]$$ $$3.6 \times 10^{-4} \ \text{mol/L/min} = 5.0\ \text{L/mol/min} \times [\mathrm{NO}]^2 \times \mathrm{0.0010\ L}$$ Now, we can solve for the remaining concentration of \(\mathrm{NO}\): $$[\mathrm{NO}]^2 = \frac{3.6 \times 10^{-4} \ \text{mol/L/min}}{5.0\ \text{L/mol/min} \times 0.0010\ \text{L}}$$ $$[\mathrm{NO}]^2 = 0.072$$ $$[\mathrm{NO}] = 0.268\ \text{mol/L}$$

04

Analyze the proposed mechanism

The first step of the proposed mechanism is: $$\mathrm{NO}+\mathrm{NO} \rightarrow \mathrm{N}_{2} \mathrm{O}_{2}$$ If this step were the rate-determining step, the rate law would depend solely on the concentration of \(\mathrm{NO}^2\). However, from the data given, we have determined that the rate law depends on both \(\mathrm{NO}^2\) and \(\mathrm{H}_2^1\). Hence, the first step cannot be the rate-determining step of the reaction.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Law Determination

Understanding how the speed of a chemical reaction is affected by the concentration of the reactants is paramount in the field of chemical kinetics. Rate law determination is a critical step in analyzing how these concentrations affect the reaction rate. It involves conducting a series of experiments, where the reactant concentrations are varied while monitoring the rate at which products are formed.

Using the data from the experiments, we can decipher the relationship between the concentration of each reactant and the reaction rate, commonly represented by the rate law equation:
\[ \text{rate} = k[\text{Reactant}_1]^{n_1}[\text{Reactant}_2]^{n_2}\dots \] where \( k \) is the rate constant, and \( n_1, n_2, \dots \) are the orders of the reaction with respect to each reactant.

Experimental Data Analysis

In the presented problem, the comparison of experimental data allowed us to determine the reaction order for each reactant: \( \text{NO} \) and \( \text{H}_2 \). By holding one reactant concentration constant and varying the other, we can infer their individual impact on the reaction rate. This method reveals the direct proportionality signifying a first-order dependence, or a squared relationship indicating a second-order reaction, for the respective reactants.

Rate Constant Calculation

Once the rate law is established, the next step is determining the rate constant, symbolized by \( k \). This constant provides a measure of the intrinsic reactivity of the reaction, unaffected by reactant concentrations. Calculating the rate constant involves rearranging the rate law equation and solving for \( k \) using experimental data.

This process requires the selection of one experiment's data with known concentrations and reaction rate. Upon substitution, the algebraic solution will yield the rate constant as shown in the example:
\[ k = \frac{\text{rate}}{[\text{NO}]^2[\text{H}_2]} \] It is crucial to include appropriate units for the rate constant, reflecting the order of the reaction, which helps ensure clarity in its representation and usage in subsequent calculations.

Significance of the Rate Constant

The magnitude of the rate constant gives insight into the speed of the reaction under standard conditions. Higher values indicate faster reactions, and they can significantly vary depending on the nature of the reaction and temperature.

Reaction Order

The reaction order is intrinsic to the rate law and reveals how the rate is affected by the concentration of each reactant. It is determined empirically from experimental data, as the order cannot generally be inferred from the stoichiometry of the reaction alone.

The combined orders of all reactants give us the overall order of the reaction. In our example, the reaction order for \( \text{NO} \) is 2, and for \( \text{H}_2 \) is 1, making the overall reaction order 3. The reaction order provides deeper insight into the kinetics of the reaction, influencing how we understand and predict the reaction's behavior under different conditions.

Implications of Different Orders

A zero-order reaction implies that the rate is independent of the concentration of the reactant, a first-order suggests a direct proportional relationship, and a second-order indicates the rate is proportional to the square of the reactant concentration. Reaction orders can also be fractional or negative, revealing even more about the complex interactions within the reaction mechanism.

Reaction Mechanism Analysis

Delving into reaction mechanism analysis allows chemists to theorize the step-by-step sequence that leads to the formation of products. It's an intricate puzzle involving the identification of intermediate species and determining which step controls the overall speed of the reaction – the rate-determining step.

By comparing the derived rate law from experimental data with the proposed mechanism steps, we can evaluate the consistency between the two. The exercises sometimes reveal that the rate-determining step cannot solely involve the formation of an intermediate if the rate law includes reactants that interact with that intermediate.

Validity of Proposed Mechanisms

In our example, even though one might first consider the direct interaction between \( \text{NO} \) molecules as the initial step, the reaction order with respect to \( \text{H}_2 \) indicates that \( \text{H}_2 \) also plays a significant role, which conflicts with making the first step rate-determining. The true reaction mechanism often requires a substantial evidential basis to confirm and may require further experimentation beyond initial rate studies.