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In this question, we explore the differences between metal coordination by monodentate and bidentate ligands. Formation constants, \(K_{t}\), for \(\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}(\mathrm{aq})\) and \(\left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+}(\mathrm{aq})\) are as follows: $$\begin{aligned} &\mathrm{Ni}^{2+}(\mathrm{aq})+6 \mathrm{NH}_{3}(\mathrm{aq}) \longrightarrow\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}(\mathrm{aq})\\\ &K_{f}=10^{8} \end{aligned}$$ $$\mathrm{Ni}^{2+}(\mathrm{aq})+3 \mathrm{en}(\mathrm{aq}) \longrightarrow\left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+}(\mathrm{aq}) \quad \quad K_{\mathrm{f}}=10^{18}$$ The difference in \(K_{f}\) between these complexes indicates a higher thermodynamic stability for the chelated complex, caused by the chelate effect. Recall that \(K\) is related to the standard free energy of the reaction by \(\Delta_{r} G^{\circ}=-R T \ln K\) and \(\Delta_{r} G^{\circ}=\Delta_{r} H^{\circ}-T \Delta_{r} S^{\circ} .\) We know from experiment that \(\Delta_{r} H^{\circ}\) for the NH seaction is \(-109 \mathrm{kJ} / \mathrm{mol}-\mathrm{rxn},\) and \(\Delta_{r} H^{\circ}\) for the ethylenediamine reaction is \(-117 \mathrm{kJ} / \mathrm{mol}\) -rxn. Is the difference in \(\Delta_{r} H^{\circ}\) sufficient to account for the \(10^{10}\) difference in \(K_{f} ?\) Comment on the role of entropy in the second reaction.

Step by step solution

01

Calculate the Standard Free Energy for Both Complexes

Use the formula \( \Delta_{r} G^{\circ} = -RT \ln K_{f} \). Given that \( R = 8.314 \text{ J/mol路K} \) and \( T = 298 \text{ K} \), calculate \( \Delta_{r} G^{\circ} \) for both complexes:For \( \left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+} \),\[ \Delta_{r} G^{\circ}_{NH_{3}} = -8.314 \times 298 \times \ln(10^{8}) \]For \( \left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+} \),\[ \Delta_{r} G^{\circ}_{en} = -8.314 \times 298 \times \ln(10^{18}) \]Calculate these values.

02

Insert Logarithmic Values for Free Energy Calculation

For the \( \left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+} \) complex:\( \ln(10^{8}) = 18.42 \), so\[ \Delta_{r} G^{\circ}_{NH_{3}} = -8.314 \times 298 \times 18.42 \approx -45.8 \text{ kJ/mol} \] For the \( \left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+} \) complex:\( \ln(10^{18}) = 41.46 \), so\[ \Delta_{r} G^{\circ}_{en} = -8.314 \times 298 \times 41.46 \approx -102.9 \text{ kJ/mol} \]

03

Analyze the Enthalpy Contributions

Given \( \Delta_{r} H^{\circ}_{NH_{3}} = -109 \text{ kJ/mol} \) and \( \Delta_{r} H^{\circ}_{en} = -117 \text{ kJ/mol} \), the difference in \( \Delta_{r} H^{\circ} \) is \(-117 + 109 = -8 \text{ kJ/mol} \), which is not enough to account for the \( 57.1 \text{ kJ/mol} \) difference in \( \Delta_{r} G^{\circ} \) (\(-102.9 + 45.8 = -57.1\)).

04

Determine the Role of Entropy

Use \( \Delta_{r} G^{\circ} = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ} \) to find the entropy contribution. For the NH鈧� complex:\[ \Delta_{r} S^{\circ}_{NH_{3}} = \frac{-45.8 + 109}{298} \approx 0.212 \text{ kJ/mol路K} \]For the \( \mathrm{en} \) complex:\[ \Delta_{r} S^{\circ}_{en} = \frac{-102.9 + 117}{298} \approx 0.048 \text{ kJ/mol路K} \]This significant entropy gain in the chelated \( \mathrm{en} \) complex suggests entropy plays a crucial role in enhancing stability through the chelate effect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Metal Coordination

Metal coordination is a key concept in understanding how metal ions interact with ligands to form complex structures. In coordination chemistry, a metal ion binds to donor atoms from molecules or ions known as ligands. This process gives rise to metal complexes, which can vary greatly in stability and composition. The type of interaction between the metal and ligands determines the properties of the complex.
In general, metal ions can coordinate with multiple ligands simultaneously. These interactions exhibit a wide range of geometries, such as octahedral, tetrahedral, or square planar configurations. The stability and particular structure of a metal complex depend not only on the nature of the metal ion but also on the characteristics of the ligands involved. By studying coordination and the resulting complexes, chemists can predict and manipulate the behavior of metal ions in biological, industrial, and environmental contexts.
Understanding metal coordination is fundamental for appreciating the subtleties of reactions and processes involving metal-ligand interactions.

Monodentate Ligands

Monodentate ligands are a type of ligand that can form only a single bond with a central metal atom or ion in a coordination complex. These ligands possess one donor atom or group, which establishes the bond with the metal. Examples of monodentate ligands include ammonia ( H鈧� ) and chloride ions ( Cl鈦� ).
Because they make only one bond with the metal, monodentate ligands provide flexibility in the structure of complexes but may result in less stable formations compared to multidentate ligands. This is evident in the difference in stability constants (formation constants) when comparing complexes formed by monodentate and bidentate ligands.
For the ammonia complex [ Ni(NH鈧�)鈧� ]虏鈦� , the coordination involves six monodentate ammonia molecules binding to a single nickel ion, demonstrating the role these ligands play in forming simple, symmetrical complexes.

Bidentate Ligands

Bidentate ligands, as opposed to monodentate ligands, can form two bonds with a central metal ion. This ability to "bite" twice into the metal ion creates a chelate ring, which significantly enhances the stability of the metal complex through the chelate effect.
Ethylenediamine ( en ) is a classic example of a bidentate ligand, as seen in the complex [ Ni(en)鈧� ]虏鈦� . Here, each en molecule forms two bonds with the nickel ion, creating three chelate rings. The increased stability arises because the ligand's two donor atoms bonding to the same metal ion reduce the likelihood of dissociation compared to monodentate ligands.
The formation constant for the [ Ni(en)鈧� ]虏鈦� complex is substantially higher than that of the ammonium complex, demonstrating the role of bidentate ligands in creating more thermodynamically stable complexes.

Formation Constants

Formation constants, symbolized as K_{ f or K_{ t, ) denote the stability of metal-ligand complexes in a solution. These values are determined experimentally and indicate how readily the reaction products form from the reactants under equilibrium conditions.
A higher formation constant implies a stronger and more stable metal-ligand complex. For instance, the [ Ni(en)鈧� ]虏鈦� complex has a formation constant of 10^{ 18 ), far exceeding that of the [ Ni(NH鈧�)鈧� ]虏鈦� complex at 10^{ 8 ), indicating how the chelating effect due to bidentate ligands enhances stability.
The differences in formation constants among complexes highlight the varying contributions of enthalpy and entropy in their formation, proving useful in predicting reaction behaviors and designing specific complexes in chemical applications.

Standard Free Energy

Standard free energy ( 螖rG^ 掳 ) is a crucial factor in determining the spontaneity and stability of chemical reactions. It is related to the formation constant via the equation 螖rG^ 掳 = -RT ln K_{ f, ), where R is the gas constant and T is the temperature in Kelvin. The calculated standard free energy of the [ Ni(NH鈧�)鈧� ]虏鈦� and [ Ni(en)鈧� ]虏鈦� complexes shows a significant difference, emphasizing how bidentate ligands and the resulting chelate effect create a thermodynamically more favorable structure compared to monodentate ligands.
While the enthalpy changes for these reactions are similar, the huge difference in entropy, particularly for the [ Ni(en)鈧� ]虏鈦� complex, highlights why bidentate ligands form more stable complexes. The increased disorder achieved when three en molecules form chelate rings around a nickel ion greatly contributes to a lower 螖rG^ 掳 and thus higher stability.