Question

Write the Arrhenius equation, and define all terms.

Short answer

The Arrhenius equation is $k=A{e}^{-\frac{E_a}{RT}}$, where $k$ is the rate constant, $A$ is the pre-exponential factor, $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the temperature in Kelvin.

Step-by-step solution

Identify the Equation

The Arrhenius equation is used to express the temperature dependence of reaction rates. The equation is:$$k=A{e}^{-\frac{E_a}{RT}}$$

Define Each Term

In the Arrhenius equation:- $k$ stands for the rate constant of the reaction.- $A$ represents the pre-exponential factor or frequency factor.- $e$ is the base of the natural logarithm.- $E_a$ is the activation energy required for the reaction to occur.- $R$ is the universal gas constant, approximately equal to 8.314 J/(mol·K).- $T$ is the temperature in Kelvin.

Understand the Equation Relationship

The Arrhenius equation suggests that the rate constant $k$ increases exponentially with an increase in temperature $T$ because the exponential term $e^{-\frac{E_a}{RT}}$ decreases as $T$ increases. Conversely, an increase in the activation energy $E_a$ decreases the rate constant for a given temperature.

Key concepts

Temperature Dependence

Temperature plays a critical role in the speed of chemical reactions. According to the Arrhenius equation, as temperature increases, the reaction rate typically increases as well. This is because higher temperatures provide molecules with more energy, which makes them move faster and collide more often. More importantly, these collisions are more likely to have sufficient energy to overcome the activation energy barrier, leading to successful reactions.
In practical terms, this means that for most reactions, a small increase in temperature can lead to a significant increase in the reaction rate. This is why reactions tend to occur faster at higher temperatures. However, the exact relationship depends on the specific reaction and its unique parameters, such as the activation energy and pre-exponential factor.

Reaction Rate

The reaction rate is a measure of how quickly reactants are converted into products in a chemical reaction. It is directly related to the rate constant, denoted as $k$ in the Arrhenius equation, which describes the speed at which a reaction proceeds at a given temperature.
Molecular interactions and the frequency with which they occur impact the reaction rate. An increase in temperature generally leads to a higher reaction rate, as molecules move more energetically and collide more frequently.
Understanding reaction rates is essential for both industrial applications and everyday chemical reactions. It helps scientists and engineers control how fast or slow a reaction occurs, making it crucial for processes in fields such as pharmaceuticals, manufacturing, and environmental science.

Activation Energy

Activation energy is a critical component in determining a chemical reaction's rate. It represents the minimum energy that colliding reactant molecules must have for a reaction to occur. In the Arrhenius equation, it is denoted by $E_a$.
A higher activation energy means that fewer molecules will have sufficient energy to undergo the reaction, resulting in a slower reaction rate. Conversely, a lower activation energy allows more molecules to participate in the reaction, increasing the reaction rate.
By understanding activation energy, chemists can predict how changes in conditions, such as temperature or the use of a catalyst, will affect the rate at which a reaction proceeds. Catalysts, for example, work by providing an alternative pathway for the reaction with a lower activation energy, thus enhancing the reaction rate without being consumed by the reaction.

Universal Gas Constant

The universal gas constant, represented by $R$, is a constant used in various equations, including the Arrhenius equation. It provides a link between energy, temperature, and the amount of substance involved in a reaction. The value of $R$ is approximately 8.314 J/(mol·K).
In the context of the Arrhenius equation, $R$ ties together the temperature and the energy aspects of the rate constant calculation. It ensures the units for energy and temperature are consistent across the equation, allowing accurate calculation of the rate constant $k$.
Understanding the role of the universal gas constant helps in various chemical calculations, ensuring that reaction rates and energies can be correctly compared and evaluated across different scenarios and temperature conditions.