Problem 26 - Analytical Chemistry

Question

The stoichiometry of a metal-ligand complex, \(\mathrm{ML}_{n}\), is determined by the method of continuous variations. A series of solutions is prepared in which the combined concentrations of \(\mathrm{M}\) and \(\mathrm{L}\) are held constant at \(5.15 \times 10^{-4} \mathrm{M}\). The absorbances of these solutions are measured at a wavelength where only the metal-ligand complex absorbs. Using the following data, determine the formula of the metal-ligand complex. $$ \begin{array}{ccc} \text { mole fraction } \mathrm{M} & \text { mole fraction } \mathrm{L} & \text { absorbance } \\ \hline 1.0 & 0.0 & 0.001 \\ 0.9 & 0.1 & 0.126 \\ 0.8 & 0.2 & 0.260 \\ 0.7 & 0.3 & 0.389 \\ 0.6 & 0.4 & 0.515 \\ 0.5 & 0.5 & 0.642 \\ 0.4 & 0.6 & 0.775 \\ 0.3 & 0.7 & 0.771 \\ 0.2 & 0.8 & 0.513 \\ 0.1 & 0.9 & 0.253 \\ 0.0 & 1.0 & 0.000 \end{array} $$

Step-by-Step Solution

Answer

The formula of the metal-ligand complex is \( \text{ML}_3 \).

Step 1: Understand the Method of Continuous Variations

The method of continuous variations involves preparing a series of solutions where the total concentration of reactants (metal, \( \text{M} \), and ligand, \( \text{L} \)) is constant. The absorbance at a wavelength where only the metal-ligand complex \( \text{ML}_n \) absorbs is measured, and the maximum absorbance corresponds to the optimal stoichiometry.

Step 2: Identify Maximum Absorbance

Look at the absorbance values in the data provided. The maximum absorbance is found at the combination where the mole fraction of \( \text{M} \) is 0.4 and \( \text{L} \) is 0.6, with an absorbance of 0.775.

Step 3: Determine the Stoichiometric Ratio

The fraction of 0.4 for \( \text{M} \) and 0.6 for \( \text{L} \) suggests a ratio of \( \text{M} \) to \( \text{L} \) of 2:3. This ratio indicates the stoichiometry of the metal-ligand complex is \( \text{ML}_3 \), meaning each metal ion combines with three ligand molecules.

Key Concepts

Metal-Ligand Complex

A metal-ligand complex is a chemical compound consisting of a central metal atom or ion bonded to surrounding molecules or ions known as ligands. These ligands can be ions, or neutral molecules, capable of donating a pair of electrons to the metal, forming a coordination bond. The primary role of the ligand is to stabilize the metal ion and can modify the properties of the metal such as its reactivity, solubility, and color.

• In our context, the metal-ligand complex is in the form of \( \text{ML}_n \), which indicates that one metal ion \( \text{M} \) is surrounded by \( n \) ligand \( \text{L} \) molecules.
• The stoichiometry (ratio) of these complexes is crucial as it determines the number of ligand atoms directly interacting with the metal ion.
• The properties and function of a metal-ligand complex heavily depend on its stoichiometry, which in our exercise indicates \( \text{ML}_3 \), meaning three ligands per metal ion.

Method of Continuous Variations

This experimental technique, commonly known as the Job's method, is used to determine the stoichiometry of a chemical complex formed between two species. It involves varying the ratio of two reactants while keeping their total concentration constant. The resulting property, often absorbance, is then measured for each mixture.

• The goal is to identify the proportion at which the change in the property is maximized, indicating the stoichiometric ratio of the components in the complex.
• In this method, we prepare multiple mixtures, varying the mole fraction of each reactant until we identify the mixture that gives the highest absorbance value, reflecting the most efficient binding of elements into a complex.
• The method of continuous variations is especially useful for systems where the direct chemical analysis is challenging due to overlapping or similar chemical species.

For the exercise at hand, it was found that the maximum absorbance occurred when the mole fraction of metal was 0.4 and ligand was 0.6, leading to the determination of a metal-ligand ratio of \( 2:3 \).

Absorbance Measurement

Absorbance measurement is a technique used in spectroscopy to determine the concentration of a solute within a solution by measuring the amount of light absorbed. It's based on Beer-Lambert Law, which relates absorbance \( A \) to the concentration \( c \), path length \( l \), and molar absorptivity \( \varepsilon \):
\[ A = \varepsilon cl \]

• The absorbance is typically measured at a specific wavelength where the species of interest (here the metal-ligand complex) maximally absorbs light, minimizing interference from other species.
• In the given exercise, the wavelength chosen is one where only the metal-ligand complex absorbs significantly, thereby allowing accurate determination of the complex's formation.
• By plotting absorbance against the mole fraction of the components, we can observe the behavior of the system and determine the optimal stoichiometry of the complex, as was done to identify the \( \text{ML}_3 \) ratio.

In practice, these measurements are essential for understanding solution behaviors and for determining the quantitative properties of chemical substances.