Question

Which of the following reactions would you expect to proceed at a faster rate at room temperature? Why? (Hint: Think about which reaction would have the lower activation energy.) $$ \begin{aligned} 2 \mathrm{Ce}^{4+}(a q)+\mathrm{Hg}_{2}^{2+}(a q) & \longrightarrow 2 \mathrm{Ce}^{3+}(a q)+2 \mathrm{Hg}^{2+}(a q) \\ \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) & \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) \end{aligned} $$

Short answer

Reaction (2), \(H_{3}O^{+}(a q)+OH^{-}(a q) \longrightarrow 2 H_{2}O(l)\), is expected to proceed at a faster rate at room temperature because it has a lower activation energy due to its reactants being comparatively more stable than those in Reaction (1). The acidic and basic counterparts readily form water molecules with little need for an intermediate activation energy, making the reaction faster.

Step-by-step solution

Reaction (1)

Reactants: \(2 Ce^{4+}(a q)\) and \(Hg_{2}^{2+}(a q)\) Products: \(2 Ce^{3+}(a q)\) and \(2 Hg^{2+}(a q)\)

Reaction (2)

Reactants: \(H_{3}O^{+}(a q)\) and \(OH^{-}(a q)\) Products: \(2 H_{2}O(l)\) #Step 2: Analyze Reactant Stability and Activation Energies# For each reaction, we will evaluate the stability of the reactants, which helps us estimate the activation energy.

Reaction (1) Reactant Stability

\(Ce^{4+}\) is a highly oxidized state of the cerium metal, which wants to reduce to a lower energy state, such as \(Ce^{3+}\). \(Hg_{2}^{2+}\) is a fairly stable mercury ion, but it will become more stable when it reduces to \(Hg^{2+}\).

Reaction (2) Reactant Stability

\(H_{3}O^{+}\) and \(OH^{-}\) are the acidic and basic counterparts that exist with water as solvent. When they meet, they readily form water molecules (\(H_{2}O\)), with little need for an intermediate activation energy. #Step 3: Compare Activation Energies Enabled by Reactant Stability#

Which reaction has a lower activation energy?

Looking at the differences in reactant stabilities, it seems that Reaction (2) requires less activation energy for the reactants to turn into the products. The \(H_{3}O^{+}\) and \(OH^{-}\) ions, being already at a comparable energy size, will need less activation energy to form \(H_{2}O\), whereas the highly oxidized state of cerium in Reaction (1) potentially requires a higher activation energy. #Step 4: Determine Which Reaction Proceeds Faster#

Faster Reaction at Room Temperature

Since Reaction (2) presumably has a lower activation energy due to reactant stability, it will proceed at a faster rate at room temperature compared to Reaction (1).

Key concepts

Activation Energy

Activation energy is the initial energy necessary to initiate a chemical reaction. Think of it like the hump a car has to get over before it can coast downhill. The lower the hump, the easier it is for the reaction to start. In a classroom situation, if we compare it to starting a class discussion, some topics (like a popular movie) are easy to start talking about, requiring very little encouragement (low activation energy), while others (such as quantum mechanics) might need a lot of prompting (high activation energy).

In the context of the exercise, we consider whether Reaction (1) or Reaction (2) requires less encouragement to get going. Since Reaction (2) involves common ions in water that are just waiting to snap together to form water molecules, it's like our popular movie discussion - easy to start with a low activation energy required.

Reactant Stability

Reactant stability is the measure of how content molecules or ions are in their current state. Happy, stable molecules are like satisfied students; they're not looking to change. However, unstable molecules are eager for a reaction, much like antsy students that can't wait to move to a new class. In our exercise,

Cerium’s High Charged State

makes it somewhat like an agitated student, it has a lot of potential to change by dropping to a lower energy state, which represents its desire to transform into a more stable ion, thus requiring a higher activation energy.

On the other hand, in

Acid-base Neutralization

, the reactants are like students in a temporary elective; they are quite ready to move on to their 'home' class of water. As a result, the activation energy needed for acid-base neutralization is lower because the reactants are less stable and more eager to react.

Cerium Oxidation States

Cerium, like other elements, can wear different 'jackets' - referred to as oxidation states. The state of its 'jacket' tells us how many electrons it has gained or lost. In the higher oxidation state of

\( Ce^{4+} \)

, cerium has lost four electrons. This is akin to wearing a tight jacket; not very comfortable, and it's eager to take it off to reach a more comfortable state, such as the

\( Ce^{3+} \) state.

This transition represents a decrease in energy, similar to a student finding a more comfortable position in a chair. However, reaching that comfortable state requires energy upfront (the activation energy), which is higher for the cerium to change its 'jacket' from

\( Ce^{4+} \) to

\( Ce^{3+} \) compared to the ions in the acid-base neutralization, which have a smaller energy barrier to combine into water.

Acid-Base Neutralization

Acid-base neutralization is like a dance between protons and hydroxide ions. The acids are like dancers holding out protons (represented by

\( H_3O^+ \) ions),

and the bases are like dancers eager to accept them (represented by

\( OH^- \) ions).

They are drawn to each other naturally, and when they meet, they form water, exiting the dance floor. This process doesn't require much energy - the dancers are already on the floor and ready to pair up. So the activation energy for the dance (or the reaction) to proceed is low, hence acid-base neutralization is often fast and spontaneous, much like an impromptu dance party in a courtyard that doesn't take much to get going.