Question

Consider the reactions $$\begin{aligned}\mathrm{Ni}^{2+}(a q)+6 \mathrm{NH}_{3}(a q) & \longrightarrow \mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}^{2+}(a q) \\ \mathrm{Ni}^{2+}(a q)+3 \mathrm{en}(a q) & \longrightarrow \mathrm{Ni}(\mathrm{en})_{3}^{2+}(a q)\end{aligned}$$ where $$\text { en }=\mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NH}_{2}$$ The \(\Delta H\) values for the two reactions are quite similar, yet \(K_{\text {reaction } 2}>K_{\text {reaction } 1 .}\) Explain.

Short answer

The difference in the equilibrium constants (K) for the two reactions is due to the difference in the change in entropy (ΔS). Ethylenediamine (en) is a bidentate ligand and can replace two ammonia ligands, resulting in a greater decrease in entropy (more negative ΔS) in reaction 2. This leads to a more negative Gibbs free energy (ΔG) and a higher equilibrium constant (K₂) for reaction 2 compared to reaction 1.

Step-by-step solution

Recall the equilibrium constant (K) and Gibbs free energy (ΔG) equation:

First, remember the equation that relates the equilibrium constant (K) to the Gibbs free energy change (ΔG): ΔG = -RT ln(K) From this equation, we can tell that as the equilibrium constant (K) increases, the Gibbs free energy change (ΔG) becomes more negative.

Analyze the change in enthalpy (ΔH) and change in entropy (ΔS):

We are given that the change in enthalpy (ΔH) for both reactions are quite similar. So, we should focus on the change in entropy (ΔS) to explain the difference in equilibrium constants. Recall the equation that relates ΔG, ΔH, and ΔS: ΔG = ΔH - TΔS When the reactions have similar ΔH values but different ΔG values, it is the ΔS value that accounts for the difference.

Compare the reactions in terms of ligands being used:

Reaction 1 involves ammonia (NH₃) as the ligand, while reaction 2 involves ethylenediamine (en) as the ligand. Ethylenediamine is a bidentate ligand, meaning it has two donor atoms that can bind to the central metal atom, in this case, Ni²⁺. Ammonia is a monodentate ligand, meaning it has only one donor atom that can bind to the central metal atom.

Relate the difference in ligands to the change in entropy (ΔS):

As ethylenediamine is a bidentate ligand, it can replace two ammonia ligands in the coordination sphere of the nickel ion. This results in a decrease in the number of particles in the reaction. The decrease in the number of particles corresponds to a decrease in entropy (ΔS) for reaction 2. Since reaction 1 involves six ammonia ligands, there is a relatively smaller decrease in particles and thus a smaller decrease in entropy (ΔS) in comparison to reaction 2.

Relate the change in entropy (ΔS) to the equilibrium constants (K):

From the equations ΔG = -RT ln(K) and ΔG = ΔH - TΔS, It can be inferred that the reaction 2 with a lower ΔS (more negative) will have more negative ΔG, leading to a higher equilibrium constant (K₂). On the other hand, reaction 1 with a lesser decrease in entropy (less negative ΔS) will have a less negative ΔG, leading to a lower equilibrium constant (K₁). By understanding the relationship between the changes in enthalpy, Gibbs free energy, and entropy, we can explain that K₂ > K₁ is due to the greater decrease in entropy (ΔS) in reaction 2 involving bidentate ligand ethylenediamine compared to reaction 1 involving monodentate ligand ammonia.

Key concepts

Gibbs free energy

Gibbs free energy is a crucial concept in chemistry that predicts whether a reaction will occur spontaneously. Spontaneous reactions are ones that happen without needing an extra push from the outside, like heat or pressure. The key formula to remember is \[ \Delta G = \Delta H - T\Delta S \]where

• \(\Delta G\) stands for the change in Gibbs free energy.
• \(\Delta H\) is the change in enthalpy.
• \(T\) represents the temperature in Kelvin.
• \(\Delta S\) is the change in entropy.

For spontaneous reactions, \(\Delta G\) must be negative. Such reactions naturally favor the formation of products. If \(\Delta G\) is positive, energy is required for the reaction to proceed. The Gibbs free energy equation indicates that reactions with large positive \(\Delta S\) or highly exothermic \(\Delta H\) can lead to negative \(\Delta G\), enhancing spontaneity.

Enthalpy

Enthalpy change, denoted as \(\Delta H\), represents the heat absorbed or released in a reaction at constant pressure. Essentially, it's a measure of the total energy change. When an exothermic reaction occurs, \(\Delta H\) is negative, meaning heat is released to the surroundings.
In contrast, an endothermic reaction absorbs heat, indicated by a positive \(\Delta H\).
Although both reactions with nickel ions and ammonia or ethylenediamine have similar \(\Delta H\) values, the driving factor for their equilibrium constant differences lies elsewhere. Enthalpy by itself does not define the direction or extent of a reaction, which is why understanding \(\Delta G\) and \(\Delta S\) is essential.

Entropy

Entropy, symbolized by \(\Delta S\), is often described as the measure of disorder or randomness within a system. In chemical reactions, entropy helps understand the distribution of particles and energy among reactants and products.
A positive \(\Delta S\) implies greater randomness or disorder post-reaction, favoring spontaneity, while a negative \(\Delta S\) suggests the system has moved towards order, potentially opposing it. In our exercise, the reaction involving ethylenediamine as a ligand decreases the number of free particles more significantly compared to the ammonia ligand reaction, leading to a more negative \(\Delta S\).
This increased order (or reduced disorder) explains the higher equilibrium constant for the reaction with ethylenediamine.

Ligands

Ligands are molecules or ions that bind to a central metal atom to form a coordination complex. They are essential in determining the structure and stability of complexes. Ligands donate their electrons to the metal atom, thus forming a coordinate bond. They play a vital role in various applications, from catalysis to biological systems.
- Ligands vary in size, charge, and the number of binding sites. - They can influence the properties and reactivity of the metal center. In chemical reactions, the nature of ligands (such as being bidentate or monodentate) affects the reaction's entropy and ultimately the equilibrium constant of the reaction.

Bidentate and monodentate ligands

The distinction between bidentate and monodentate ligands lies in the number of sites through which they can attach to a metal ion. A monodentate ligand features only one donor atom through which it binds to the metal center. A common example is ammonia (NH₃), which donates a single pair of electrons from one nitrogen atom.
Bidentate ligands, like ethylenediamine (en), have two donor atoms, allowing them to form two bonds with the metal ion.

• Bidentate ligands can create more stable chelate complexes due to the "bite" of two connections.
• They tend to increase the overall stability of the complex and influence the solution's entropy significantly.

In this exercise, ethylenediamine replaces two ammonia ligands, reducing the number of molecules, thus decreasing entropy, and this change impacts the equilibrium constant, showing why \(K_2\) is larger than \(K_1\).