Problem 23 - Introductory Chemistry Atoms First

Question

Use the given kinetics data to write the rate law for the reaction $$2 \mathrm{NO}+\mathrm{O}_{2} \rightarrow 2 \mathrm{NO}_{2}$$ $$\begin{array}{cccc} \text { Experiment } & \text { Initial [NO] } & \text { Initial }\left[\mathrm{O}_{2}\right] & \text { Rate of } \mathrm{NO}_{2} \text { formation (M/s) } \\ \hline 1 & 0.015 \mathrm{M} & 0.015 \mathrm{M} & 0.048 \\ 2 & 0.030 \mathrm{M} & 0.015 \mathrm{M} & 0.192 \\ 3 & 0.015 \mathrm{M} & 0.030 \mathrm{M} & 0.096 \\ 4 & 0.030 \mathrm{M} & 0.030 \mathrm{M} & 0.384 \end{array}$$

Step-by-Step Solution

Answer

The rate law for the reaction \(\mathrm{2NO + O_2 \rightarrow 2NO_2}\) is given by \[Rate = k[\mathrm{NO}]^{2}[\mathrm{O}_{2}]^{1}\], where \(k\) is the rate constant, \([\mathrm{NO}]^2\) represents the squared concentration of NO, and \([\mathrm{O}_2]^1\) represents the concentration of O2. This was determined by analyzing the provided kinetics data and deducing the reaction orders for both NO and O2.

Step 1: Determine how the rate of the reaction changes with reactant concentrations

Compare the rates of NO2 formation between different experiments. When the initial concentration of NO doubles (from Experiments 1 to 2 and from Experiments 3 to 4), the rate of NO2 formation quadruples. This indicates that the reaction order with respect to NO is 2. Similarly, when the initial concentration of O2 doubles (from Experiments 1 to 3), the rate of NO2 formation doubles, indicating that the reaction order with respect to O2 is 1.

Step 2: Write the rate law equation based on the determined reaction orders

Now that we have the reaction orders for both NO and O2, we can write the rate law equation. The general form of a rate law equation is: \[ Rate = k[A]^{m}[B]^{n} \] Where k is the rate constant, [A] and [B] are the reactant concentrations, and m and n are their respective reaction orders. Based on our analysis, the reaction order for NO (A) is 2 and for O2 (B) is 1. Therefore, the rate law equation for the given reaction is: \[ Rate = k[\mathrm{NO}]^{2}[\mathrm{O}_{2}]^{1} \]

Key Concepts

Reaction Order

The reaction order is a crucial aspect in the study of chemical kinetics. It tells us how the concentration of a reactant affects the rate of a reaction. In the context of the problem provided, the reaction order with respect to a specific reactant, like NO or O₂, is determined by observing how changes in concentration affect the reaction rate.
For instance, if the concentration of NO is doubled and the rate of formation of NO₂ increases fourfold, we conclude that NO affects the reaction rate with a second-order dependency. Similarly, if doubling O₂ concentration doubles the reaction rate, O₂ has a first-order effect.
These observations can be succinctly expressed in the rate law equation through the exponents corresponding to each reactant, helping us understand the relationship between concentration and rate dynamics.

Rate Constant

The rate constant, denoted as 'k' in the rate law equation, is pivotal in describing the speed of the reaction at a specific temperature. It serves as a proportionality constant that links the rate of reaction to the concentrations of the reactants raised to their respective powers (which are the reaction orders).
The rate constant has unique units depending on the overall reaction order. Its value is determined experimentally, usually by calculating it from data obtained from experiments like those provided in the exercise, where concentration and rate information is available.
For example, by using the rate law equation, one can calculate 'k' by plugging in known values for reactant concentrations and the measured reaction rate, further enhancing the quantitative understanding of the reaction kinetics.

Kinetics Data

Kinetics data, often presented in tables, is fundamental for analyzing and understanding chemical reaction rates. This data typically includes various experimental conditions, such as initial concentrations of reactants and corresponding reaction rates.
Analyzing this data allows chemists to determine the reaction order by observing how changes in reactant concentrations affect the reaction rate, as seen in the provided experiments.
For example, by comparing different experiments where varying concentrations of NO and O₂ are used, one can deduce the order with respect to each reactant. This data-driven approach enhances our understanding of the kinetics, guiding us in constructing accurate models like the rate law equation.

Concentration Effect

The concentration effect is a primary factor influencing the rate of chemical reactions. It refers to how changes in the concentration of reactants affect the overall reaction rate.
When a reactant's concentration increases, it usually results in more frequent collisions between particles, thus potentially increasing the rate of reaction.
In our exercise, we see this effect with NO and O₂: doubling NO concentration quadruples the rate, indicating a second-order effect, while doubling O₂ concentration doubles the rate, illustrating a first-order effect.
Understanding concentration effects is crucial for deriving the reaction order and eventually constructing an accurate rate law equation which predicts how the reaction proceeds under different conditions.