Question

If the reaction $$2 \mathrm{NO}_{2}(g)+\mathrm{F}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2} \mathrm{~F}(g)$$ occurred by a one-step process, what would be the expected rate law for the reaction? The actual rate law is rate \(=k\left[\mathrm{NO}_{2}\right]\left[\mathrm{F}_{2}\right]\), why is this a better rate law?

Short answer

The stoichiometrically expected rate law is \(\text{rate} = k[\text{NO}_{2}]^2[\text{F}_{2}]\), but the experimentally determined actual rate law is \(\text{rate} = k[\text{NO}_{2}][\text{F}_{2}]\). The actual rate law accounts for the reaction mechanism and is found to be more accurate.

Step-by-step solution

Determine the expected rate law from the stoichiometry

Based on the stoichiometry of the given chemical equation, the expected rate law assumes that the rate of the reaction is directly proportional to the concentration of the reactants, each raised to a power equal to its stoichiometric coefficient. Thus, for the equation \(2 \text{NO}_{2}(g) + \text{F}_{2}(g) \rightarrow 2 \text{NO}_{2}\text{F}(g)\), the expected rate law would be \(\text{rate} = k[\text{NO}_{2}]^2[\text{F}_{2}]\).

Compare the expected rate law with the actual rate law

Comparing the expected rate law \(\text{rate} = k[\text{NO}_{2}]^2[\text{F}_{2}]\) with the actual rate law \(\text{rate} = k[\text{NO}_{2}][\text{F}_{2}]\), we observe that the actual rate law indicates a first-order dependency on both \(\text{NO}_{2}\) and \(\text{F}_{2}\), rather than second-order on \(\text{NO}_{2}\) and first-order on \(\text{F}_{2}\) as would be expected from the stoichiometry.

Explain why the actual rate law is more suitable

The actual rate law \(\text{rate} = k[\text{NO}_{2}][\text{F}_{2}]\) is determined experimentally and takes into account the reaction mechanism, which involves the actual steps and intermediates in the reaction. This could indicate that the reaction does not occur in a single step as the stoichiometry would suggest, but through a mechanism with rate-determining step(s) that define the observed rate law. It reflects the real process better than the expected law based only on stoichiometry.

Key concepts

Reaction Rate Law

Understanding the reaction rate law is crucial for predicting how quickly a chemical reaction will proceed. It relates the speed of a reaction to the concentration of the reactants. The rate law expression has the general form \(\text{rate} = k[\text{A}]^{m}[\text{B}]^{n}...\), where \(k\) is the rate constant, and \(m\) and \(n\) represent the orders of the reaction with respect to reactants \(A\) and \(B\), respectively. These orders are often determined experimentally, not merely deduced from the coefficients in the balanced chemical equation.

Why is this distinction important? Because the reaction rate law gives us insight into the chemical kinetics of a system. It helps us understand how the reaction rate will change if we alter the concentrations of the reactants. For instance, if the rate law is first-order in \(\text{NO}_{2}\), doubling the concentration of \(\text{NO}_{2}\) will double the reaction rate, which is crucial information for both theoretical studies and practical applications, such as industrial chemical manufacturing.

Reaction Stoichiometry

Reaction stoichiometry deals with the quantitative relationships between the amounts of reactants and products in a chemical reaction. It's based on the principle of the conservation of mass, where the total amount of each element must be the same in the reactants and products.

When writing a balanced chemical equation, the coefficients imply the relative amounts of substances involved. For example, in the reaction \(2 \text{NO}_{2}(g) + \text{F}_{2}(g) \rightarrow 2 \text{NO}_{2}\text{F}(g)\), the coefficients suggest for every two moles of \(\text{NO}_{2}\) reacting, one mole of \(\text{F}_{2}\) is required to produce two moles of \(\text{NO}_{2}\text{F}\). While this stoichiometry is invaluable for preparing and scaling up reactions, it doesn't always reflect the reaction's kinetics. That's where the rate law and reaction mechanism come into play, providing a more detailed glimpse into how these reactions occur on a molecular level.

Chemical Reaction Mechanism

The chemical reaction mechanism is essentially the 'story' of how reactants transform into products at the molecular level. It details the sequence of steps, including how bonds are broken and formed, and the intermediates that appear throughout the process.

When the rate law doesn't align with the stoichiometry, it often means the reaction proceeds through several steps, each with its own rate, and not as a one-step process as the balanced equation might suggest. For our \(\text{NO}_{2}\) and \(\text{F}_{2}\) example, the actual rate law implies that the formation of the product \(\text{NO}_{2}\text{F}\) is controlled by a rate-determining step that involves equal contributions (in terms of reaction order) from both \(\text{NO}_{2}\) and \(\text{F}_{2}\). This step is slower than the others and thus dictates the overall rate of the reaction.

Order of Reaction

The order of reaction is a term used in chemical kinetics to describe how the rate is affected by the concentration of one or more reactants. For example, a reaction is first order in \(\text{NO}_{2}\) if the rate doubles when the concentration of \(\text{NO}_{2}\) doubles. A reaction can be zero, first, second, or even fractional order for a given reactant.

What's essential to realize is that the overall reaction order is the sum of the orders with respect to each reactant involved in the rate-determining step. This does not necessarily correlate with stoichiometric coefficients. Therefore, experimental determination is key to establishing the correct rate law. In our exercise, while stoichiometry suggested a second-order dependence on \(\text{NO}_{2}\) (due to its coefficient of 2), the experimental rate law showed that the reaction is first-order in both \(\text{NO}_{2}\) and \(\text{F}_{2}\), indicating a one-to-one relationship with the rate of reaction and the concentration of each reactant.