Question

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.$\begin{array}{rl}\text{ (a) }{E}_a& =45\mathrm{kJ}/\mathrm{mol};\mathrm{\Delta }E=-25\mathrm{kJ}/\mathrm{mol}\\ \text{ (b) }{E}_a& =35\mathrm{kJ}/\mathrm{mol};\mathrm{\Delta }E=-10\mathrm{kJ}/\mathrm{mol}\\ \text{ (c) }{E}_a& =55\mathrm{kJ}/\mathrm{mol};\mathrm{\Delta }E=10\mathrm{kJ}/\mathrm{mol}\end{array}$

Short answer

Based on the given activation energies, and assuming that all collision factors are the same, the reactions can be ranked from slowest to fastest as follows: $(c)Ea=55\mathrm{kJ}/\mathrm{mol}>(a)Ea=45\mathrm{kJ}/\mathrm{mol}>(b)Ea=35\mathrm{kJ}/\mathrm{mol}$. Therefore, the order is (c), (a), (b).

Step-by-step solution

Recall the Arrhenius Equation

The Arrhenius equation relates the rate constant (k) of a reaction to its activation energy (Ea) and the temperature (T): $$k=A{e}^{\frac{-Ea}{RT}}$$ where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature. The higher the activation energy, the slower the reaction rate.

Analyze the Given Activation Energies

We are given the activation energies for the three reactions: (a) Ea = 45 kJ/mol (b) Ea = 35 kJ/mol (c) Ea = 55 kJ/mol Higher activation energy indicates a slower reaction rate. Comparing the activation energies of these reactions, reaction (c) has the highest activation energy, so it should be the slowest. Reaction (b) has the lowest activation energy and hence the fastest rate.

Rank the Reactions

Comparing the given activation energies and ignoring the energy changes since collision factors are assumed to be the same, we can rank the reactions in terms of their speed as follows: 1. (b) Ea = 35 kJ/mol 2. (a) Ea = 45 kJ/mol 3. (c) Ea = 55 kJ/mol Thus, the reactions can be ranked from slowest to fastest as: (c), (a), (b).

Key concepts

Arrhenius Equation

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 $$k=A{e}^{\frac{-Ea}{RT}}$$ 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

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.

Rate Constant

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.