Problem 20 - Chemistry Matter and Change

Question

The rate law for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}] \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) is first order in \(\mathrm{O}_{2}\) and third order overall. What is the rate law for the reaction?

Step-by-Step Solution

Answer

The rate law is \(\text{rate} = k[\mathrm{NO}]^2[\mathrm{O_2}]^1\).

Step 1: Understanding Rate Law

The rate law for a reaction gives the rate of the reaction in terms of the concentration of its reactants. It is generally given as: \[\text{rate} = k[A]^m[B]^n\]where \(\text{rate}\) is the reaction rate, \(k\) is the rate constant, \([A]\) and \([B]\) are concentrations of the reactants, and \(m\) and \(n\) are their respective orders of reaction.

Step 2: Identify the Reaction Orders

According to the problem, the reaction is first order in \(\mathrm{O_2}\). This means the exponent for \([\mathrm{O_2}]\) in the rate law is 1. We also know that the reaction is third order overall. Thus, the sum of the exponents (orders) for all reactants must equal 3.

Step 3: Calculate Order for NO

Let \([\mathrm{NO}]\) have an order of \(x\). The overall reaction order is the sum of the orders of each reactant: \[x + 1 = 3\]Solving for \(x\) gives \(x = 2\). So, the reaction is second order in \([\mathrm{NO}]\).

Step 4: Write the Rate Law

Using the orders determined, the rate law for the reaction is given by: \[\text{rate} = k[\mathrm{NO}]^2[\mathrm{O_2}]^1\]Here, \(k\) is the rate constant, \([\mathrm{NO}]\) has an order of 2, and \([\mathrm{O_2}]\) has an order of 1.

Key Concepts

Understanding Reaction Order

The concept of reaction order helps us understand how changes in concentrations of reactants affect the rate of a chemical reaction. It is expressed as an exponent in the rate law formula. If you've ever wondered why this matters, it's because it tells us how sensitive the reaction rate is to changes in reactant concentrations.
Simply put:

• A zero-order reaction means that changing the concentration of a reactant does not affect the rate.
• In a first-order reaction, the rate changes linearly with the change in concentration of the reactant.
• For second-order reactions, the rate changes quadratically.

In our given reaction, it's stated that the reaction is first order with respect to \(O_2\). This means that if you double the concentration of \(O_2\), the reaction rate will also double.
However, the overall reaction is third order. This hints that the sum of the orders of each reactant equals three, illustrating the combined effect of all reactants on the rate. Identifying individual orders can be vital in understanding the reaction mechanism.

The Role of Rate Constant

The rate constant, symbolized as \(k\), is a crucial factor in the rate law equation. Unlike the reaction orders, it is not influenced by the concentration of reactants. Instead, it depends on other conditions like temperature and the presence of a catalyst.
The rate constant acts as the proportionality factor that links the concentration of reactants to the reaction rate. For example, in the rate law \[\text{rate} = k[\text{NO}]^2[\text{O}_2]^1\], the value of \(k\) will determine how fast the reaction actually proceeds at a given temperature. So even if you have the same concentrations, different values of \(k\) can mean different reaction speeds.
Moreover, conducting experiments at various temperatures allows us to calculate \(k\) using the Arrhenius equation, which relates temperature to the rate constant. Understanding and calculating the rate constant helps chemists in designing reactions with desired speeds, crucial for industries that rely on precise chemical manufacturing conditions.

Chemical Kinetics Demystified

Chemical kinetics is the study of reaction rates and how different conditions impact these rates. Focused on observing and deriving the laws governing the speed of reactions, it provides a deeper insight into reaction mechanisms. This scientific discipline helps us answer 'how fast' and 'through what steps' a reaction takes place.
The study of kinetics involves:

• Determining the rate law that defines the relation between reaction rate and reactant concentration.
• Understanding the influence of temperature and pressure on the reaction speed.
• Identifying catalysts that can speed up reactions without being consumed.

Knowing these aspects of kinetics informs other branches of chemistry, such as thermodynamics, and has practical applications in creating medications, developing sustainable energy, and synthesizing new materials.
For students trying to master chemical kinetics, it's essential to focus on experimental data and mathematical models provided in exercises, like studying reaction order and rate laws, to develop a concrete understanding of this dynamic field.