Problem 50 - Chemistry: Principles and Reactions

Question

Bipyridyl (Bipy) is a molecule commonly used as a bidentate ligand. When \(0.17 \mathrm{~mol}\) of bipyridyl is dissolved in \(2.4 \mathrm{~L}\) of a solution that contains \(0.052 \mathrm{M} \mathrm{Fe}^{2+},\left[\mathrm{Fe}(\mathrm{bipy})_{3}\right]^{2+}\left(K_{\mathrm{f}}=1.6 \times 10^{17}\right)\) is formed. What are the con- centrations of all species when equilibrium is established?

Step-by-Step Solution

Answer

Based on the given initial concentration of \(Fe^{2+}\) and the initial moles of bipyridyl (Bipy), the equilibrium concentrations for the formation of the \([Fe(bipy)_3]^{2+}\) complex are approximately: \([Fe^{2+}]_{equilibrium} ≈ 0.052 \mathrm{M}\) \([bipy]_{equilibrium} ≈ 0.07083 \mathrm{M}\) \([[Fe(\mathrm{bipy})_3]^{2+}]_{equilibrium} ≈ 4.38 \times 10^{-8} \mathrm{M}\)

Step 1: Analyze the reaction

Bipyridyl (bipy) forms a complex \([Fe(bipy)_3]^{2+}\) with \(Fe^{2+}\): \(Fe^{2+} + 3 \mathrm{bipy} \rightleftharpoons [Fe(\mathrm{bipy})_3]^{2+}\) The equilibrium constant for this reaction is given by: \( K_{f} = \frac{[[Fe(\mathrm{bipy})_3]^{2+}]}{[Fe^{2+}][\mathrm{bipy}]^{3}}\)

Step 2: Write the ICE table

To solve for the equilibrium concentrations, we can set up an ICE table (Initial, Change, Equilibrium): |\(Fe^{2+}\) | bipy | \([Fe(\mathrm{bipy})_{3}]^{2+}\)| |----|----|-----| |I | 0.17 mol | 0| |C | -x | -3x | x | |E | 0.052-x "\((1)\)" | 0.17 - 3x "\((2)\)"| x "\((3)\)"| * In order to convert from given moles of bipy to molarity, we need to divide the moles of bipy by the volume of the solution. So initial bipy concentration = \(\frac{0.17 \mathrm{~mol}}{2.4 \mathrm{~L}} = 0.07083 \mathrm{M}\) * Note that we are simplifying the table by assuming the volume of the solution remains constant.

Step 3: Substitute the ICE table values into the equilibrium constant expression

By substituting the equilibrium concentrations from the ICE table into the equilibrium constant expression, we get: \(K_{f} = \frac{x}{(0.052-x)(0.07083-3x)^3}\) Now, we can put the given value \(K_{f} = 1.6 \times 10^{17}\): \(1.6 \times 10^{17} = \frac{x}{(0.052-x)(0.07083-3x)^3}\)

Step 4: Solve for \(x\)

Solving for \(x\) directly in this equation can be quite challenging. So, we can make a simplifying assumption that \(x\) is much smaller than \(0.052\) and \(0.07083/3\) (i.e., x is quite small compared to the initial concentrations that it will not make a significant difference when subtracted). So, we can approximate it as: \(1.6 \times 10^{17} ≈ \frac{x}{(0.052)(0.07083)^3}\) Solving for \(x\), we get: \(x ≈ 4.38 \times 10^{-8} \mathrm{M}\) (Equilibrium concentration of \([Fe(\mathrm{bipy})_3]^{2+}\))

Step 5: Calculate the equilibrium concentrations for Bipy and \(Fe^{2+}\)

Now, we can use equations \((1)\), \((2)\), and \((3)\) to find the equilibrium concentrations: \([Fe^{2+}]_{equilibrium} ≈ 0.052 - x ≈ 0.052 - 4.38 \times 10^{-8} ≈ 0.052 \mathrm{M}\) \([bipy]_{equilibrium} ≈ 0.07083 - 3x ≈ 0.07083 - 3(4.38 \times 10^{-8}) ≈ 0.07083 \mathrm{M}\) Finally, the equilibrium concentrations are: \([Fe^{2+}]_{equilibrium} ≈ 0.052 \mathrm{M}\) \([bipy]_{equilibrium} ≈ 0.07083 \mathrm{M}\) \([[Fe(\mathrm{bipy})_3]^{2+}]_{equilibrium} ≈ 4.38 \times 10^{-8} \mathrm{M}\)

Key Concepts

Bidentate Ligand

A bidentate ligand is a type of ligand that can form two bonds with a central metal atom or ion. This happens because the ligand has two donor atoms that can each donate a pair of electrons to the metal, thereby forming two coordinate covalent bonds.
Bipyridyl (often abbreviated as bipy) is a classic example of a bidentate ligand. It can effectively "bite" into a metal like iron ( Fe^{2+} ) because it has two nitrogen atoms in its structure that are each capable of donating a pair of electrons. This dual binding nature enhances the stability of the complexes formed, as observed in the formation of [Fe(bipy)_3]^{2+} .
This ability to form double bonds increases the chelate effect, which is the enhanced stability of complex ions. The chelate effect makes bidentate ligands particularly useful in various applications, including catalysis and coordination chemistry.

Complex Formation

Complex formation refers to the process where a central metal ion binds with one or more ligands to form a complex ion. This process often involves the transfer of electron pairs from ligands to the metal ion to create coordinate covalent bonds.
In the specific example of Fe^{2+} reacting with bipyridyl, each iron ion ( Fe^{2+} ) forms a complex with three bipyridyl molecules, resulting in the [Fe(bipy)_3]^{2+} complex. The formation of such complexes is crucial in fields like biochemistry and materials science because they can dramatically change the properties of the metals involved.

• The geometry, color, and reactivity of a metal can be altered by complex formation.
• These complexes can also be involved in electron transfer reactions.
• Molecules like chlorophyll and hemoglobin are naturally occurring complexes that play vital roles in biological processes.

Understanding complex formation is key to manipulating and utilizing these properties in practical applications.

Equilibrium Constant

The equilibrium constant (K_f) quantifies the ratio of the concentrations of products to reactants at equilibrium for a given reaction.
For the formation of the [Fe(bipy)_3]^{2+} complex, K_f is an important indicator of how strongly the complex is formed and can be expressed by the equation:
\[K_{f} = \frac{[[Fe(bipy)_3]^{2+}]}{[Fe^{2+}][bipy]^{3}} \]
The high value of K_f (1.6 × 10^{17}) in this example suggests that the equilibrium lies far to the right, meaning the formation of the [Fe(bipy)_3]^{2+} complex is highly favored under initial conditions where the reactants are present.
These constants are crucial for predicting reactant and product concentrations, determining reaction feasibility, and understanding reaction conditions for synthesis in both academic and industrial chemistry.

ICE Table

An ICE table is a valuable tool used for organizing and calculating the initial concentrations, changes in concentration, and equilibrium concentrations of reactants and products in chemical reactions. ICE stands for:

• Initial
• Change
• Equilibrium

By using an ICE table, chemists can meticulously track concentration changes, simplifying the process of solving equilibrium problems.
In the case of Fe(bipy)_3^{2+} formation, the ICE table helps in determining the equilibrium concentrations of Fe^{2+} , bipy , and the [Fe(bipy)_3]^{2+} complex.
Setting up the table involves:

• Listing initial concentrations.
• Representing changes in concentrations (+x, -x, etc.).
• Using the equilibrium expression to solve for unknown variables, often making approximations for simplification, as seen in our case.

This methodical approach is essential, particularly in complex equilibria calculations, ensuring that systematic steps lead to accurate results without oversight.