Question

The molecule methylamine \(\left(\mathrm{CH}_{3} \mathrm{NH}_{2}\right)\) can act as a monodentate ligand. The following are equilibrium reactions and the thermochemical data at \(298 \mathrm{~K}\) for reactions of methylamine and en with \(\mathrm{Cd}^{2+}(a q)\) : \(\mathrm{Cd}^{2+}(a q)+4 \mathrm{CH}_{3} \mathrm{NH}_{2}(a q) \rightleftharpoons\left[\mathrm{Cd}\left(\mathrm{CH}_{3} \mathrm{NH}_{2}\right)_{4}\right]^{2+}(a q)\) \(\Delta H^{\circ}=-57.3 \mathrm{~kJ} ; \quad \Delta S^{\circ}=-67.3 \mathrm{~J} / \mathrm{K} ; \quad \Delta G^{\circ}=-37.2 \mathrm{~kJ}\) $$ \mathrm{Cd}^{2+}(a q)+2 \operatorname{en}(a q) \rightleftharpoons\left[\mathrm{Cd}(\mathrm{en})_{2}\right]^{2+}(a q) $$ \(\Delta H^{\circ}=-56.5 \mathrm{~kJ} ; \quad \Delta S^{\circ}=+14.1 \mathrm{~J} / \mathrm{K} ; \quad \Delta G^{\circ}=-60.7 \mathrm{~kJ}\) (a) Calculate \(\Delta G^{\circ}\) and the equilibrium constant \(K\) for the following ligand exchange reaction: \(\left[\mathrm{Cd}\left(\mathrm{CH}_{3} \mathrm{NH}_{2}\right)_{4}\right]^{2+}(a q)+2 \operatorname{en}(a q) \rightleftharpoons\) $$ \left[\mathrm{Cd}(\mathrm{en})_{2}\right]^{2+}(a q)+4 \mathrm{CH}_{3} \mathrm{NH}_{2}(a q) $$ (b) Based on the value of \(K\) in part \((a)\), what would you conclude about this reaction? What concept is demonstrated? (c) Determine the magnitudes of the enthalpic \(\left(\Delta H^{\circ}\right)\) and the entropic \(\left(-T \Delta S^{\circ}\right)\) contributions to \(\Delta G^{\circ}\) for the ligand exchange reaction. Explain the relative magnitudes. (d) Based on information in this exercise and in the "A Closer Look" box on the chelate effect, predict the sign of \(\Delta H^{\circ}\) for the following hypothetical reaction: \(\left[\mathrm{Cd}\left(\mathrm{CH}_{3} \mathrm{NH}_{2}\right)_{4}\right]^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons\) $$ \left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}(a q)+4 \mathrm{CH}_{3} \mathrm{NH}_{2}(a q) $$

Short answer

The equilibrium constant \(K\) for the ligand exchange reaction: \(\left[\mathrm{Cd}\left(\mathrm{CH}_{3}\mathrm{NH}_{2}\right)_{4}\right]^{2+}(a q)+2 \operatorname{en}(a q) \rightleftharpoons \left[\mathrm{Cd}(\mathrm{en})_{2}\right]^{2+}(a q)+4 \mathrm{CH}_{3}\mathrm{NH}_{2}(a q)\) is 9.69 × \(10^{3}\) and \(\Delta G^{\circ}\) is -23.5 kJ/mol. Since the equilibrium constant \(K\) is greater than 1, the reaction favors the formation of products, demonstrating the concept of the chelate effect. The enthalpy change \(\Delta H^{\circ}\) for the ligand exchange reaction is 0.8 kJ/mol, and the entropic contribution \(-T\Delta S^{\circ}\) is -24.27 kJ/mol. The positive enthalpy change for the hypothetical reaction implies that the reaction would be less favorable.

Step-by-step solution

Calculate ΔG° and K for the ligand exchange reaction

The ligand exchange reaction is: \(\left[\mathrm{Cd}\left(\mathrm{CH}_{3}\mathrm{NH}_{2}\right)_{4}\right]^{2+}(a q)+2 \operatorname{en}(a q) \rightleftharpoons \left[\mathrm{Cd}(\mathrm{en})_{2}\right]^{2+}(a q)+4 \mathrm{CH}_{3}\mathrm{NH}_{2}(a q)\) We can obtain the ΔG° for the ligand exchange reaction using the given reactions: \(\Delta G^{\circ}_{ligand} = \Delta G^{\circ}_{Cd(en)_{2}} - \Delta G^{\circ}_{Cd(CH_{3}NH_{2})_{4}} = -60.7 \, \text{kJ/mol} - (-37.2 \, \text{kJ/mol}) = -23.5 \, \text{kJ/mol}\) To calculate the equilibrium constant K, we'll use the relation: \(K = e^{-\frac{\Delta G^{°}}{RT}}\) Where R is the gas constant (8.314 J/mol·K), T is the temperature in Kelvin (298 K), and ΔG° we calculated above. Plug in the values: \(K = e^{-\frac{-23.5 \times 10^{3} \, \text{J/mol}}{8.314 \, \text{J/mol·K} \times 298 \, \text{K}}} = 9.69 \times 10^{3}\)

Conclusions based on the value of K

Since the value of K is greater than 1, we can conclude that the ligand exchange reaction favors the formation of the products, i.e., \([\mathrm{Cd}(\mathrm{en})_{2}]^{2+}(a q) + 4 \mathrm{CH}_{3}\mathrm{NH}_{2}(a q)\). This demonstrates the concept of chelate effect, where the formation of multidentate ligand (en) complexes is favored.

Determine the enthalpic (ΔH°) and entropic (-TΔS°) contributions

First, we need to calculate ΔH° and ΔS° for the ligand exchange reaction: \(\Delta H^{\circ}_{ligand} = \Delta H^{\circ}_{Cd(en)_{2}} - \Delta H^{\circ}_{Cd(CH_{3}NH_{2})_{4}} = -56.5 \, \text{kJ/mol} - (-57.3 \, \text{kJ/mol}) = 0.8 \, \text{kJ/mol}\) \(\Delta S^{\circ}_{ligand} = \Delta S^{\circ}_{Cd(en)_{2}} - \Delta S^{\circ}_{Cd(CH_{3}NH_{2})_{4}} = 14.1 \, \text{J/mol·K} - (-67.3 \, \text{J/mol·K}) = 81.4 \, \text{J/mol·K}\) Now, we can calculate -TΔS°: \(-T \Delta S^{\circ}_{ligand} = -(298 \, \text{K})(81.4 \, \text{J/mol·K}) = -24269 \, \text{J/mol} = -24.27 \, \text{kJ/mol}\)

Predict the sign of ΔH° for the hypothetical reaction

The hypothetical reaction is: \(\left[\mathrm{Cd}\left(\mathrm{CH}_{3}\mathrm{NH}_{2}\right)_{4}\right]^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons \left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}(a q)+4\mathrm{CH}_{3} \mathrm{NH}_{2}(a q)\) Now, the hypothetical reaction is similar to the ligand exchange reaction we analyzed earlier with the exception that the multidentate ligand en is replaced with the monodentate ligand NH3. Since replacing multidentate ligands with monodentate ligands generally weaken the bonds, it is reasonable to expect that the sign of ΔH° for this reaction will be positive. This would indicate that the hypothetical reaction is less favorable since energy would have to be absorbed for the exchange.

Key concepts

Chelate Effect

The chelate effect is a phenomenon observed in coordination chemistry where complexes with multidentate ligands, like en (ethylenediamine), are more stable than those with equivalent monodentate ligands, such as methylamine. Multidentate ligands, also known as chelating ligands, form stronger complexes because they bind to the central metal atom at multiple points, creating a ring structure.
This enhanced binding occurs for several reasons:

• Entropy Favorability: Forming a ring structure releases other molecules in the solution, increasing disorder.
• Enthalpic Considerations: The creation of multiple bonds can be energetically favorable, even if we only consider each bond's strength individually.

The greater stability of chelated complexes, as demonstrated by the ligand exchange reaction where \([ ext{Cd}( ext{en})_2]^{2+}\) forms preferentially, underscores the chelate effect's significance in chemistry.

Thermodynamics

Thermodynamics plays an essential role in understanding ligand exchange reactions. It involves the study of energy changes in chemical reactions. When considering reactions like the ligand exchange between \([ ext{Cd}( ext{CH}_3 ext{NH}_2)_4]^{2+}\) and \([ ext{Cd}( ext{en})_2]^{2+}\), we analyze both enthalpy (\(\Delta H^{\circ}\)) and entropy (\(\Delta S^{\circ}\)) to determine the spontaneity of a reaction with Gibbs free energy (\(\Delta G^{\circ}\)).
Key points include:

• Enthalpy Change (\(\Delta H^{\circ}\)): Reflects the heat absorbed or released.
• Entropy Change (\(\Delta S^{\circ}\)): Indicates the disorder or randomness.
• Gibbs Free Energy (\(\Delta G^{\circ}\)): A combination of enthalpy and entropy that reveals if a reaction is spontaneous at a given temperature.

Energy considerations help determine this exchange's favorability and support the understanding of why the formation of \([ ext{Cd}( ext{en})_2]^{2+}\) is preferred.

Enthalpy and Entropy

Enthalpy and entropy are critical concepts in ligand exchange reactions. Enthalpy (\(\Delta H^{\circ}\)) measures the heat absorbed or released, while entropy (\(\Delta S^{\circ}\)) involves changes in disorder.

In the reaction provided, the differences in these values for \([ ext{Cd}( ext{CH}_3 ext{NH}_2)_4]^{2+}\) and \([ ext{Cd}( ext{en})_2]^{2+}\) are insightful:

• \(n\Delta H^{\circ}_{ ext{ligand}} = 0.8 ext{ kJ/mol}\) indicates a minor enthalpy change, suggesting energy changes are slight when replacing methylamine with en.
• \(\Delta S^{\circ}_{ ext{ligand}} = 81.4 ext{ J/mol·K}\) shows a significant increase in entropy, illustrating the chelate effect where releasing four methylamine molecules increases the system's disorder.

The effects of both enthalpy and entropy determine the reaction's spontaneity, as reflected in Gibbs free energy calculations.

Equilibrium Constant

The equilibrium constant (\(K\)) is central in evaluating chemical reactions like ligand exchange. It helps quantify the reaction's tendency to form products over reactants at equilibrium. The calculated \(K\) for the ligand exchange is \(9.69 \times 10^3\), reflecting a strong preference for the formation of \([ ext{Cd}( ext{en})_2]^{2+}\).

Using the formula:
\[K = e^{-rac{\Delta G^{\circ}}{RT}}\]

• Where \(R\) is the gas constant (8.314 J/mol·K) and \(T\) is temperature in Kelvin (298 K).

This high \(K\) value indicates that, at equilibrium, the concentration of the products is much higher than that of the reactants, demonstrating the reaction's strong proclivity towards completion. Additionally, a \(K\) significantly greater than 1 supports the chelate effect, as it shows multidentate ligands forming more stable complexes.