Question

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

Short answer

The higher equilibrium constant (K) for Reaction 2 compared to Reaction 1 can be attributed to the entropic effects associated with the bidentate nature of the ethylenediamine ligands. The formation of the complex ion in Reaction 2 involves fewer molecules than in Reaction 1, leading to a larger entropy increase (∆S) for Reaction 2. This results in a more negative Gibbs free energy change and hence a higher equilibrium constant for Reaction 2, even though both reactions have similar enthalpy changes.

Step-by-step solution

Understand the reactions

First, let's understand the two given reactions: Reaction 1: Nickel ions (Ni²⁺) react with ammonia molecules (NH₃) to form a complex ion with a hexaammine complex ion (Ni(NH₃)₆²⁺). Reaction 2: Nickel ions (Ni²⁺) react with ethylenediamine molecules (en) to form a complex ion with a triethylenediamine complex ion (Ni(en)₃²⁺). Notice that both reactions involve the formation of complex ions. Reaction 1 has six ammonia molecules bonded to nickel, while Reaction 2 has three ethylenediamine molecules bonded to nickel.

Enthalpy and Entropy

Recall that the equilibrium constant (K) of a reaction is related to its Gibbs free energy change (∆G) by the equation: \(\Delta G = -RT \ln K\) where R is the gas constant, and T is the temperature in Kelvin. Gibbs free energy change is also related to enthalpy change (∆H) and entropy change (∆S) by the equation: \(\Delta G = \Delta H - T \Delta S\) From these relations, we can understand that the equilibrium constant depends on both the enthalpy and the entropy changes of the reactions. A more negative ∆G or a higher ∆S will result in a higher equilibrium constant, all else being equal.

Role of Ligands

Both ammonia and ethylenediamine are ligands that can form coordinate covalent bonds with metal ions like nickel in these reactions. However, it's crucial to notice that ethylenediamine is a bidentate ligand because it can form two coordinate covalent bonds through its two nitrogen atoms with the central metal ion. On the other hand, ammonia is a monodentate ligand, as it can only form one bond with the central metal ion. In reaction 2, each of the ethylenediamine molecules wraps around the nickel ion, occupying two coordination sites simultaneously, while ammonia surrounds the nickel ion in reaction 1.

Explaining the higher equilibrium constant for Reaction 2

The higher equilibrium constant for the reaction with ethylenediamine (Reaction 2) compared to the one with ammonia (Reaction 1) can be attributed to the entropic effects associated with the bidentate nature of the ethylenediamine ligands. In Reaction 2, the formation of the complex ion involves fewer molecules compared to Reaction 1, which leads to a larger entropy increase (∆S) for Reaction 2 than for Reaction 1. As we know, an increase in entropy change promotes a more negative Gibbs free energy change, which in turn leads to a higher equilibrium constant (K) for Reaction 2. Even though both reactions have similar enthalpy changes, the higher equilibrium constant for Reaction 2 mainly arises because of its higher entropy change resulting from the bidentate nature of the ethylenediamine ligand, leading to the formation of fewer molecules upon complex formation.

Key concepts

Chelation and Ligands

In the world of chemistry, chelation is a fascinating process where a single ligand forms multiple bonds with a metal ion, creating a more stable structure known as a chelate. Ligands are molecules or ions that can donate at least one pair of electrons to a metal ion to form a coordinate covalent bond.

Let's take a closer look at why ligands like ethylenediamine ('en') are so special. Ethylenediamine is a bidentate ligand, which means it has two 'arms' that can simultaneously grab onto a metal ion, in this case, nickel (Ni²⁺). This multi-point attachment leads to a more complex and stable structure compared to ligands that attach at only one point, like ammonia (NH₃).

In the reaction detailed in the exercise, we see that forming a chelate with bidentate ligands like ethylenediamine results in fewer particles in the final mix. This has important implications on the reaction's Gibbs free energy and equilibrium constant, which we'll explore in the upcoming sections.

Gibbs Free Energy

Understanding Gibbs free energy (abla G) is critical in predicting whether a chemical reaction will occur spontaneously. It provides insights into the energetics of a reaction, tying together enthalpy (abla H), entropy (abla S), and the equilibrium constant (K).

The equation that relates Gibbs free energy to the equilibrium constant is given by abla G = -RT K, where R is the gas constant and T is the temperature in Kelvin. A spontaneous reaction, one that occurs without external energy input, is characterized by a negative abla G value. As such, a higher equilibrium constant indicates a more negative abla G and a greater extent of reaction. Bidentate ligands, due to their chelation ability, often lead to reactions with a more negative abla G, enhancing the formation of complex ions, as is the case in the exercise provided.

Equilibrium Constant

The equilibrium constant (K) expresses the balance between the reactants and products in a reversible chemical reaction at equilibrium. It’s a dimensionless number that gives us an idea of the position of equilibrium; a large K value implies that the equilibrium lies far to the right, favoring the products, while a small K value says that the reactants are favored.

In the textbook exercise, the fact that reaction 2 with ethylenediamine has a greater equilibrium constant than reaction 1 with ammonia implies that the product, Ni(en)₃²⁺, is much more favored than Ni(NH₃)₆²⁺. Despite similar enthalpies, the difference in the number of particles involved due to chelation impacts the entropy change, which then affects the equilibrium constant. As we know, a more positive change in entropy contributes to a more negative Gibbs free energy, hence a higher K.

Enthalpy and Entropy in Reactions

Enthalpy (abla H) and entropy (abla S) are essential concepts in understanding chemical reactions. Enthalpy represents the heat content of a system, and reactions can either release heat (exothermic) or absorb it (endothermic). On the other hand, entropy measures the randomness or disorder within a system.

The intriguing interplay between enthalpy and entropy decides the spontaneity of a reaction. The equation abla G = abla H - Tabla S links Gibbs free energy, with both enthalpy and entropy. In the case of complex ion formation with chelating ligands, chelation often causes a larger increase in disorder because fewer particles are generated as products. Even if the enthalpy changes (abla H) of two reactions are similar, as with our exercise problem, the reaction with the greater entropy change (abla S) will likely have a more favorable (abla G), and therefore, a larger equilibrium constant. This is a prime example of entropy's significant role in the stability of chelated complexes and the overall reaction's equilibrium.