Problem 27 - Analytical Chemistry

Question

The stoichiometry of a metal-ligand complex, \(\mathrm{ML}_{n}\), is determined by the mole-ratio method. A series of solutions are prepared in which the metal's concentration is held constant at \(3.65 \times 10^{-4} \mathrm{M}\) and the ligand's concentration is varied from \(1 \times 10^{-4} \mathrm{M}\) to \(1 \times 10^{-3} \mathrm{M}\). Using the following data, determine the stoichiometry of the metal-ligand complex. $$ \begin{array}{cccc} \text { [ligand] (M) } & \text { absorbance } & \text { [ligand] (M) } & \text { absorbance } \\ \hline 1.0 \times 10^{-4} & 0.122 & 6.0 \times 10^{-4} & 0.752 \\ 2.0 \times 10^{-4} & 0.251 & 7.0 \times 10^{-4} & 0.873 \\ 3.0 \times 10^{-4} & 0.376 & 8.0 \times 10^{-4} & 0.937 \\ 4.0 \times 10^{-4} & 0.496 & 9.0 \times 10^{-4} & 0.962 \\ 5.0 \times 10^{-4} & 0.625 & 1.0 \times 10^{-3} & 1.002 \end{array} $$

Step-by-Step Solution

Answer

The stoichiometry of the metal-ligand complex is 1:1 (ML).

Step 1: Overview of the Mole-Ratio Method

The mole-ratio method is used to determine the stoichiometry of a complex by plotting absorbance against the ratio of ligand to metal concentration. The point at which a break or change in the slope occurs indicates the stoichiometric ratio of metal to ligand in the complex.

Step 2: Calculate the Mole Ratio

For each ligand concentration in the table, calculate the mole ratio (ligand:metal) using the formula \( \text{mole ratio} = \frac{[\text{ligand}]}{[\text{metal}]} \), where \([\text{metal}] = 3.65 \times 10^{-4} \text{ M}\).

Step 3: Fill in The Table with Mole Ratios

Using the formula, calculate and fill in the mole ratios:- \(1.0 \times 10^{-4} / 3.65 \times 10^{-4} = 0.274\)- \(2.0 \times 10^{-4} / 3.65 \times 10^{-4} = 0.548\)- \(3.0 \times 10^{-4} / 3.65 \times 10^{-4} = 0.822\)- \(4.0 \times 10^{-4} / 3.65 \times 10^{-4} = 1.096\)- \(5.0 \times 10^{-4} / 3.65 \times 10^{-4} = 1.370\)- \(6.0 \times 10^{-4} / 3.65 \times 10^{-4} = 1.644\)- \(7.0 \times 10^{-4} / 3.65 \times 10^{-4} = 1.918\)- \(8.0 \times 10^{-4} / 3.65 \times 10^{-4} = 2.192\)- \(9.0 \times 10^{-4} / 3.65 \times 10^{-4} = 2.466\)- \(1.0 \times 10^{-3} / 3.65 \times 10^{-4} = 2.740\)

Step 4: Plot Absorbance Against Mole Ratio

Create a plot of absorbance (y-axis) versus the mole ratio of ligand to metal (x-axis) using the data from the table. This plot is used to identify the stoichiometry.

Step 5: Analyze the Absorbance Plot

Examine the plot for a noticeable change or plateau in absorbance, which indicates the formation of the completed complex. The x-value at this point is the ligand-to-metal ratio for the complex.

Step 6: Determine the Stoichiometry

From the plot, the absorbance appears to plateau at a mole ratio of approximately 1:1, suggesting that the stoichiometry of the metal-ligand complex is 1 metal ion per ligand (ML).

Key Concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that refers to the calculation of reactants and products in chemical reactions. In our context, it helps us understand the exact ratio of components in a metal-ligand complex. When forming such a complex, knowing the stoichiometry allows chemists to predict how much of each component is necessary to form the desired compound. The mole-ratio method is particularly useful here. By carefully measuring how absorbance changes with varying ligand concentrations, one can determine the stoichiometric relationship between metal ions and ligands. This method hinges on the variation in absorbance as this provides a direct indication of the completion of the complex formation. Through stoichiometric analysis, we can ascertain the precise ligand-to-metal ratio necessary to form a stable and effective compound.

Metal-Ligand Complex

In chemical science, a metal-ligand complex is an assembly consisting of a central metal atom or ion attached to surrounding molecules or ions, called ligands. These ligands are bound to the metal via coordinate covalent bonds. The essence of a metal-ligand complex lies in its ability to stabilize the metal ion through these interactions. The stoichiometry, such as in the formula \(\mathrm{ML}_n\), suggests the number \(n\) of ligand molecules that coordinate with each metal ion. Understanding these complexes is crucial given that they form the basis of many chemical processes, including catalysis and molecular recognition. In our exercise, the metal-ligand complex's formation is monitored through the mole-ratio method, revealing the point at which the metal ion is fully coordinated, indicating the stoichiometric formula of the complex.

Absorbance

Absorbance is a measure of the amount of light absorbed by a solution, and it is commonly used to track concentrations in chemical analysis. When a ligand binds to a metal ion to form a complex, the resultant compound often has different absorbance properties compared to the unbound metal ion or ligand. The Beer-Lambert law relates absorbance \(A\) to the concentration of the absorbing species as follows: \(A = \varepsilon \cdot c \cdot l\), where \(\varepsilon\) is the molar absorptivity, \(c\) is the concentration, and \(l\) is the path length. In our scenario, absorbance data is plotted against the mole ratio to observe where a significant change or plateau occurs, indicating the saturation point where the complex formation has reached completion. This absorbance peak provides insight into the stoichiometric ratio of the metal-ligand, allowing for a better understanding of complex stability and composition.

Ligand Concentration

Ligand concentration plays a critical role in determining the stoichiometry of a metal-ligand complex. By adjusting the concentration of the ligand while keeping the metal concentration constant, one can observe how the binding interaction varies. Ligands are typically small molecules that donate electron pairs to the metal, creating a stable complex. In the mole-ratio method, a series of solutions with varying ligand concentrations are prepared. The relationship between ligand concentration and absorbance reveals information about the coordination number, influencing the ultimate stoichiometry of the complex. Precise knowledge of ligand concentration is essential not only for understanding complex formation but also for successful experimental design in coordination chemistry. This careful control over concentration helps discover the correct stoichiometric formula, essential for applications in fields like drug design and catalysis.