Based on their activation energies and energy changes and assuming that all collision factors are the same, rank the following reactions from slowest to fastest.\( \begin{aligned} \text { (a) } E_{a} &=45 \mathrm{kJ} / \mathrm{mol} ; \Delta E=-25 \mathrm{kJ} / \mathrm{mol} \\ \text { (b) } E_{a} &=35 \mathrm{kJ} / \mathrm{mol} ; \Delta E=-10 \mathrm{kJ} / \mathrm{mol} \\ \text { (c) } E_{a} &=55 \mathrm{kJ} / \mathrm{mol} ; \Delta E=10 \mathrm{kJ} / \mathrm{mol} \end{aligned}\)

The Arrhenius equation is a mathematical formula that provides a deep understanding of how the reaction rate is influenced by various factors, including the activation energy and temperature.

It is expressed as \[k = Ae^{\frac{-Ea}{RT}}\] where:

• \(k\) is the rate constant of the reaction,
• \(A\) is the pre-exponential factor, which is a constant for each chemical reaction,
• \(Ea\) is the activation energy,
• \(R\) is the ideal gas constant, and
• \(T\) refers to the temperature in Kelvin.

Simply put, this equation links the rate at which a reaction proceeds to the frequency of effective particle collisions and the energy barrier they must overcome.

In context to the exercise, it explains why a lower activation energy corresponds to a higher reaction rate, assuming all other factors are constant.

Activation energy (\(Ea\)) is the minimum energy barrier that the reacting particles must overcome to result in a successful chemical reaction. It plays a pivotal role in determining the rate at which a reaction occurs.

Higher activation energy implies that the particles need more energy for the reaction to proceed, which can make the reaction slower. In contrast, a lower activation energy typically signifies a faster reaction because fewer particles can surpass the energy barrier. This concept is essential for understanding why certain reactions may progress more rapidly than others.

Referring back to our exercise, it is illustrated that reaction (c), with \(Ea = 55\) kJ/mol, will be the slowest, and reaction (b), with \(Ea = 35\) kJ/mol, will be the fastest due to their respective activation energies.

The rate constant (\(k\)) in the context of chemical reactions is a proportionality constant in the rate equation that is specific to the reaction at a given temperature. It is intimately related to the frequency of successful collisions and the activation energy of a reaction.

The larger the rate constant, the faster the reaction rate since it indicates a greater proportion of effective collisions leading to product formation. As indicated by the Arrhenius equation, the rate constant decreases exponentially with an increase in the activation energy and is influenced by the temperature.

Understanding the relationship between the rate constant and activation energy helps in predicting how changes in conditions, such as temperature, will affect the reaction rate, a concept that underlies the solution provided. When comparing different reactions, those with higher rate constants will invariably be quicker if all other conditions are held equal.