Question

In Arrhenius equation: \(\mathrm{K}=\mathrm{Ae}^{-\mathrm{Ea} \mathrm{KT}}\) a. The pre-exponential factor has the units of rate constant of the reaction b. The exponential factor is a dimensionless quantity c. The exponential factor has the units of reciprocal of temperatures d. The pre-exponential factor has the units of rate of the reaction

Short answer

Options (a) and (b) are correct.

Step-by-step solution

Understanding the Arrhenius Equation

The Arrhenius equation is given by \( K = Ae^{-\frac{E_a}{RT}} \), 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. This equation is used to describe how the rate constant changes with temperature.

Analyzing the Pre-exponential Factor

The pre-exponential factor \( A \) in the Arrhenius equation is a constant that has the same units as the rate constant \( K \). For a reaction with rate constant \( K \, (\text{s}^{-1}) \), \( A \) will also have units \( \text{s}^{-1} \). Option (a) correctly states this property.

Analyzing the Exponential Factor

The exponential factor \( e^{-\frac{E_a}{RT}} \) is dimensionless. This is because the exponent \( -\frac{E_a}{RT} \) is dimensionless, with \( E_a \) and \( RT \) having the same units (energy per mole). Thus, option (b) is correct. Options (c) and (d) are incorrect as they imply incorrect interpretations of the units in the equation.

Key concepts

Rate Constant

The rate constant, often denoted by the symbol \( K \), is a fundamental concept in chemical kinetics. It refers to a proportionality factor that relates the reaction rate to the concentration of reactants. In simpler terms, it tells us how fast a reaction proceeds under a given set of conditions. The units of the rate constant depend on the order of the reaction. For example, a first-order reaction will have a rate constant with units of \( \text{s}^{-1} \), while a second-order reaction's rate constant might have units of \( \text{M}^{-1}\text{s}^{-1} \).
Understanding the rate constant is crucial as it

• determines the speed of a reaction
• is temperature-dependent, as described by the Arrhenius equation
• can be influenced by factors such as catalysts and pressure

Hence, learning about rate constants gives insight into the dynamics of chemical reactions and their efficiency under varying conditions.

Activation Energy

Activation energy, represented by \( E_a \), is the minimum energy that reactant molecules must possess for a reaction to occur. Think of it as an energy barrier. Reactants must overcome this barrier for the transformation into products. In the Arrhenius equation, activation energy plays a key role in determining the rate at which a reaction proceeds.
- High activation energy implies a slower reaction, as fewer molecules have the necessary energy to react.- Low activation energy results in a faster reaction, as more molecules can overcome the energy hurdle.
Activation energy is influenced by several factors:

• Catalysts: These substances lower the activation energy, thus speeding up the reaction.
• Temperature: Higher temperatures provide reactant molecules with more kinetic energy, enhancing their ability to overcome the activation energy.

Understanding activation energy helps predict how reactions will change under different conditions, making it a vital concept in both academic and industrial chemistry.

Temperature Dependence

Temperature has a pronounced effect on reaction rates, as described by the Arrhenius equation. When we talk about temperature dependence, we're referring to how the rate of a reaction changes with varying temperatures. According to the Arrhenius equation, the rate constant:\[ K = Ae^{-\frac{Ea}{RT}} \]will increase with rising temperatures. This is because increasing temperature provides more energy to reactant molecules. So, the fraction of molecules with energies exceeding the activation energy increases.
Key aspects of temperature dependence include:

• Higher temperatures typically speed up reactions by increasing molecular collisions and energy levels.
• The Arrhenius plot, a graph of \( \ln K \) versus \( \frac{1}{T} \), is linear and can be used to determine the activation energy.
• Experiments must often be carried out at controlled temperatures to observe predictable reaction rates.

Temperature dependence is a cornerstone of kinetic studies, giving insight into ideal reaction conditions for desired reaction velocities.