Problem 105 - Chemistry: An Atoms-Focused Approach

Question

In the following mechanism for NO formation, oxygen atoms are produced by breaking \(\mathrm{O}=\mathrm{O}\) bonds at high temperature in a fast reversible reaction. If \(\Delta[\mathrm{NO}] / \Delta t=k\left[\mathrm{N}_{2}\right]\left[\mathrm{O}_{2}\right]^{1 / 2},\) which step in the mechanism is the rate-determining step? $$\begin{aligned} (1)\quad\quad\quad\quad\quad\mathrm{O}_{2}(g) & \rightleftharpoons 2 \mathrm{O}(g) \\ (2)\quad\quad\mathrm{O}(g)+\mathrm{N}_{2}(g) & \rightarrow \mathrm{NO}(g)+\mathrm{N}(g) \\ (3)\quad\quad\mathrm{N}(g)+\mathrm{O}(g) & \rightarrow \mathrm{NO}(g) \\ overall \quad \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) & \rightarrow 2 \mathrm{NO}(g) \end{aligned}$$

Step-by-Step Solution

Answer

The rate-determining step in the reaction mechanism for the formation of NO is the second step, where O and N2 react to form NO and N. This is because the rate law for this step matches the given rate law for the overall reaction, both having the same dependence on the concentrations of N2 and O2.

Step 1: Examine the rate law

We are given the rate law for the overall reaction: \(\frac{\Delta[\mathrm{NO}]}{\Delta t} = k\left[\mathrm{N}_{2}\right]\left[\mathrm{O}_{2}\right]^{1 / 2}\). The rate law expresses the rate of NO formation in terms of reactant concentrations and the reaction rate constant.

Step 2: Consider the first step in the mechanism

First, look at the equilibrium step: \(\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}(g)\). Since O2 is consumed, the rate law for this step can be written as follows: \(\frac{\Delta[\mathrm{O}_{2}]}{\Delta t} = k_{1}[\mathrm{O}_{2}] - k_{-1}[\mathrm{O}]^{2}\). As the reaction reaches equilibrium, the forward and reverse rates become equal, so the rate expression becomes: \(k_{1}[\mathrm{O}_{2}] = k_{-1}[\mathrm{O}]^{2}\). We can now solve for the concentration of O: \([\mathrm{O}] = \sqrt{\frac{k_{1}}{k_{-1}}[\mathrm{O}_{2}]}\).

Step 3: Examine the second step in the mechanism

Now, let's look at the second step, where O and N2 react to form NO and N: \(\mathrm{O}(g)+\mathrm{N}_{2}(g) \rightarrow \mathrm{NO}(g)+\mathrm{N}(g)\). Since this reaction consumes O and N2, the rate law for this step will be: \(\frac{\Delta[\mathrm{NO}]}{\Delta t} = k_{2}[\mathrm{O}][\mathrm{N}_{2}]\).

Step 4: Replace the concentration of O in step 3

Now, we substitute the expression for the concentration of O obtained in step 2 into the rate law for step 2: \(\frac{\Delta[\mathrm{NO}]}{\Delta t} = k_{2}\sqrt{\frac{k_{1}}{k_{-1}}[\mathrm{O}_{2}]}[\mathrm{N}_{2}]\).

Step 5: Compare the rate laws

Finally, we compare the given rate law for the overall reaction with the rate law derived for step 2: $$\frac{\Delta[\mathrm{NO}]}{\Delta t} = k_{eff}\left[\mathrm{N}_{2}\right]\left[\mathrm{O}_{2}\right]^{1 / 2}$$ $$\frac{\Delta[\mathrm{NO}]}{\Delta t} = k_{2}\sqrt{\frac{k_{1}}{k_{-1}}[\mathrm{O}_{2}]}[\mathrm{N}_{2}]$$ By comparing both expressions, we can see that they have the same dependence on the concentrations of N2 and O2. Therefore, the second step of the mechanism is the rate-determining step.

Key Concepts

Rate laws

Rate laws are mathematical expressions that predict the rate of a chemical reaction based on the concentration of its reactants and a rate constant. In this context, for the NO formation mechanism given, the rate law is expressed as \(\frac{\Delta[\mathrm{NO}]}{\Delta t} = k[\mathrm{N}_{2}][\mathrm{O}_{2}]^{1/2}\). This tells us that the rate of formation of NO depends on the concentration of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\).

In general, a rate law involves the rate constant (k) and the concentrations of the reactants raised to specific powers, known as orders of reaction. These powers indicate how changes in concentration affect the reaction rate. They are normally determined experimentally and can be different from the stoichiometric coefficients seen in the balanced chemical equation.

For the NO formation, the given rate law suggests the reaction is first-order with respect to \(\mathrm{N}_{2}\) and one-half order with respect to \(\mathrm{O}_{2}\). This implies a more complex mechanism where the initial bond breakage of \(\mathrm{O}_{2}\) influences how \(\mathrm{O}_{2}\) behaves within the reaction.

Rate-determining step

The rate-determining step is a fundamental concept in reaction mechanisms, indicating the slowest step in the reaction sequence. It essentially limits the speed of the overall reaction, acting as a bottleneck in the pathway.

For the mechanism discussed for NO formation, the task was to identify which step aligned with the overall rate law given: \(\frac{\Delta[\mathrm{NO}]}{\Delta t} = k[\mathrm{N}_{2}][\mathrm{O}_{2}]^{1/2}\).
After evaluating each step, it was determined that the second step is the rate-determining step. This is because its derived rate expression matches the given overall rate law, specifically in terms of how it depends on \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\).

This means that although Step 1 reaches equilibrium quickly and Step 3 follows quickly after the second, it's the reaction between \(\mathrm{O}\) + \(\mathrm{N}_{2}\) in Step 2 that governs the rate at which \(\mathrm{NO}\) is produced.

Chemical kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the rate and pathway of chemical reactions. It helps chemists determine how a reaction proceeds from reactants to products and to design their conditions accordingly.

In the NO formation reaction, chemical kinetics involves studying the three steps detailed in the mechanism to identify how they contribute to the overall process. Importantly, it also involves looking at how intermediate species, like \(\mathrm{O}(g)\) or \(\mathrm{N}(g)\), play roles in different steps until the final product, \(\mathrm{NO}(g)\), is achieved.

By understanding these kinetic principles, including aspects like activation energy, collision frequency, and molecular orientation, chemists can predict and control reaction rates. For industrial processes, this means optimizing conditions to maximize yield and efficiency while minimizing time and cost.