Question

Two mechanisms are proposed for the reaction $$\begin{array}{cl}2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g) & \\ \text { Mechanism 1: } \mathrm{NO}+\mathrm{O}_{2} \rightleftharpoons \mathrm{NO}_{3} & \text { (fast) } \\ \mathrm{NO}_{3}+\mathrm{NO} \longrightarrow 2 \mathrm{NO}_{2} & \text { (slow) } \\ \text { Mechanism 2: } \mathrm{NO}+\mathrm{NO} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2} & \text { (fast) } \\ \mathrm{N}_{2} \mathrm{O}_{2}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{NO}_{2} & \text { (slow) } \end{array}$$ Show that each of these mechanisms is consistent with the observed rate law: rate \(=k[\mathrm{NO}]^{2} \times\left[\mathrm{O}_{2}\right]\).

Short answer

Answer: Yes, both Mechanism 1 and Mechanism 2 are consistent with the observed rate law.

Step-by-step solution

Identify the slow (rate-determining) step in Mechanism 1

The slow step for Mechanism 1 is given as NO3 + NO → 2NO2.

Derive the rate law for the slow step

To derive the rate law, we simply need to express the rate of the reaction in terms of the concentration of the reactants involved in the slow step (NO3 and NO). We'd get: rate = k1'[NO3][NO] Here, k1' represents the rate constant for the slow step of Mechanism 1.

Calculate the concentration of the intermediate NO3

NO3 is an intermediate, and we need to express its concentration in terms of the reactants present in the overall reaction (i.e. NO and O2). We'll use the first (fast) step of Mechanism 1 for this: NO + O2 <=> NO3 The equilibrium constant (K1) for this reaction is given by: K1 = {[NO3]} / {[NO][O2]} Solving for the concentration of NO3, we get: [NO3] = K1[NO][O2]

Substitute the calculated intermediate concentration back into the rate law

Now that we have [NO3] in terms of [NO] and [O2], we can substitute this expression back into the rate law derived in Step 2: rate = k1'K1[NO][O2][NO] Which simplifies to: rate = k[NO]²[O2] Since this derived rate law matches the observed rate law, Mechanism 1 is consistent with the observed rate law. #For Mechanism 2#

Identify the slow (rate-determining) step in Mechanism 2

The slow step for Mechanism 2 is given as N2O2 + O2 → 2NO2.

Derive the rate law for the slow step

To derive the rate law, we express the rate of the reaction in terms of the concentration of the reactants involved in the slow step (N2O2 and O2). We'd get: rate = k2'[N2O2][O2] Here, k2' represents the rate constant for the slow step of Mechanism 2.

Calculate the concentration of the intermediate N2O2

N2O2 is an intermediate, and we need to express its concentration in terms of the reactants present in the overall reaction (i.e. NO and O2). We'll use the first (fast) step of Mechanism 2 for this: NO + NO <=> N2O2 The equilibrium constant (K2) for this reaction is given by: K2 = {[N2O2]} / {[NO]²} Solving for the concentration of N2O2, we get: [N2O2] = K2[NO]²

Substitute the calculated intermediate concentration back into the rate law

Now that we have [N2O2] in terms of [NO], we can substitute this expression back into the rate law derived in Step 2: rate = k2'K2[NO]²[O2] Which simplifies to: rate = k[NO]²[O2] Since this derived rate law matches the observed rate law, Mechanism 2 is also consistent with the observed rate law. Both mechanisms are consistent as requested.

Key concepts

Rate-Determining Step

Understanding the rate-determining step in a chemical reaction is essential for grasping how reactions occur at the molecular level. The rate-determining step is the slowest step in a reaction mechanism and, as such, dictates the overall reaction rate, much like the slowest runner in a relay race determines the team’s overall time. It allows us to predict the reaction kinetics by focusing on this single step rather than the entire complex reaction sequence.

In our exercise, the rate-determining steps for Mechanism 1 and Mechanism 2 are the creation of NO₂ from NO₃ and NO, and from N₂O₂ and O₂, respectively. These slow steps are crucial because they require the highest activation energy to proceed and occur less frequently compared to other, faster steps. By examining the rate law, which in this exercise is given as rate = k[NO]²[O₂], we can confirm that the proposed mechanisms align with the experimentally determined rate law. In daily life, understanding the rate-determining step helps chemists design reactions that are faster and more efficient, such as in the synthesis of pharmaceuticals or the breakdown of pollutants.

Reaction Intermediates

Reaction intermediates are species often radicals or ions, that are formed during the middle of a reaction mechanism but do not appear in the overall balanced equation because they are not final products or initial reactants. They are crucial puzzle pieces in the complete picture of how a chemical reaction proceeds.

For instance, NO₃ in Mechanism 1 and N₂O₂ in Mechanism 2 are intermediates. Engineers use their understanding of intermediates to develop catalysts that stabilize these fleeting species, leading to faster and more controllable reactions. In the environment, such intermediates may be responsible for phenomena like ozone layer depletion, making their study vital for environmental chemistry as well.

Equilibrium Constant

The equilibrium constant, denoted as K, is a number that expresses the ratio of the concentration of the products to the reactants at equilibrium for a reversible reaction. A large equilibrium constant indicates a high concentration of products at equilibrium, while a small value signifies mostly reactants. It's a critical factor when predicting the extent and direction of a chemical reaction.

In the aforementioned mechanisms, K1 and K2 represent the equilibrium constants for the formation of intermediates NO₃ and N₂O₂. They are used to calculate the concentrations of these intermediates, which are then used in the rate laws. In everyday life, equilibrium constants explain why certain reactions, like the synthesis of ammonia by the Haber process, occur under specific temperature and pressure conditions to optimize yield.