Problem 38 - Chemistry Matter and Change

Question

Relate the size of the activation energy of an elementary step in a complex reaction to the rate of that step.

Step-by-Step Solution

Answer

The reaction rate of an elementary step in a complex reaction is inversely related to its activation energy. According to the Arrhenius equation, \(k = Ae^{\frac{-Ea}{RT}}\), a higher activation energy (Ea) corresponds to a slower reaction rate, while a lower activation energy corresponds to a faster reaction rate. This is because a lower activation energy requires a smaller energy barrier for molecules to overcome, allowing more molecules to participate in the reaction, and vice versa for a higher activation energy.

Step 1: Define activation energy and reaction rate

Activation energy (Ea) is the minimum amount of energy required for a chemical reaction to occur. It is the energy barrier that molecules need to overcome for the reaction to proceed. Reaction rate, on the other hand, is a measure of how fast a chemical reaction occurs. It is expressed as the change in concentration of reactants or products per unit time. The higher the reaction rate, the faster the reaction proceeds.

Step 2: Introduce the Arrhenius equation

The Arrhenius equation is a mathematical expression that relates the rate constant (k) of a chemical reaction to its activation energy (Ea), and the temperature (T) at which the reaction occurs. The equation is given by: \[k = Ae^{\frac{-Ea}{RT}}\] where: - k is the rate constant - A is the pre-exponential factor, also called the frequency factor - Ea is the activation energy of the reaction - R is the universal gas constant, approximately equal to 8.314 J/mol K - T is the temperature in Kelvin

Step 3: Relate the activation energy to the reaction rate

From the Arrhenius equation, we can see that the rate constant (k) is directly proportional to the exponential dependence of activation energy (Ea) and temperature (T). Thus, the rate of the reaction is dependent on the activation energy of the elementary step in the complex reaction. If the activation energy of an elementary step is low, then the reaction rate will be higher. This is because the energy barrier that molecules need to overcome will be smaller, allowing more molecules to have enough energy to participate in the reaction. On the other hand, if the activation energy is high, then the reaction rate will be lower. This is because fewer molecules will have the required energy to surpass the energy barrier and participate in the reaction. In summary, there is an inverse relationship between activation energy and reaction rate. A higher activation energy results in a slower reaction rate, while a lower activation energy results in a faster reaction rate.

Key Concepts

reaction rate

When we talk about reaction rate, we're discussing how quickly a chemical reaction occurs. It's a measure of the speed at which reactants turn into products. This speed is expressed as the change in concentration of reactants or products over time. Imagine a busy highway full of cars—each car representing a molecule in a reaction. The more cars (or molecules) that can move through a specific point in a certain time, the higher the reaction rate.

Reaction rates can vary based on several factors. Some of these include:

• Temperature: Usually, higher temperatures increase reaction rates.
• Concentration: More reactant molecules typically lead to faster reactions.
• Physical state: Reactions tend faster in liquid or gas phases compared to solids.
• Catalysts: These often speed up reactions without being consumed.

Understanding reaction rates is crucial in fields like chemistry and manufacturing, where controlling how fast reactions occur can be vital.

Arrhenius equation

The Arrhenius equation is a handy tool for chemists to predict how the reaction rate changes with temperature and activation energy. This equation is a mathematical representation that shows the rate constant \( k \) as a function of temperature \( T \) and activation energy \( Ea \). Here's the equation:\[ k = Ae^{\frac{-Ea}{RT}} \]

• **A** is the pre-exponential or frequency factor, symbolizing how often molecules collide in the right orientation.
• **Ea** stands for activation energy, representing the energy molecules need to start a reaction.
• **R** is the universal gas constant, approximately 8.314 J/mol K.
• **T** is the absolute temperature measured in Kelvin.

From this equation, it's easy to see that the reaction rate is highly sensitive to the temperature. As the temperature increases, the exponential factor in the equation becomes larger, leading to a greater rate constant \( k \). Chemists use the Arrhenius equation to understand and predict how changes in temperature and activation energy will affect the speed of a reaction.

elementary step in reaction

In a chemical reaction, an elementary step is a single reaction event that represents the most basic underlying processes. Think of it as a building block of a more complex reaction.

Elementary steps involve the direct breaking or forming of chemical bonds and can't be broken down into simpler steps. Each step is characterized by its own reaction rate and activation energy. Understanding these individual steps is crucial in learning how an overall chemical reaction proceeds.

Characteristics of elementary steps include:

• Each step has its own specific rate constant, which can be used alongside the Arrhenius equation to determine how temperature and activation energy influence it.
• The rate of an elementary step is usually represented in a rate law, derived from its molecularity (the number of molecules involved).
• Elementary steps can be either unimolecular (involving one molecule), bimolecular (two molecules), or termolecular (three molecules, which are rare).

By analyzing these steps, chemists can gain insight into the mechanisms of complex reactions and potentially find ways to control or improve them.