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Chapter 10: Problem 26

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. $$ \begin{array}{ccc} \text { mole fraction } \mathrm{M} & \text { mole fraction } \mathrm{L} & \text { absorbance } \\ \hline 1.0 & 0.0 & 0.001 \\ 0.9 & 0.1 & 0.126 \\ 0.8 & 0.2 & 0.260 \\ 0.7 & 0.3 & 0.389 \\ 0.6 & 0.4 & 0.515 \\ 0.5 & 0.5 & 0.642 \\ 0.4 & 0.6 & 0.775 \\ 0.3 & 0.7 & 0.771 \\ 0.2 & 0.8 & 0.513 \\ 0.1 & 0.9 & 0.253 \\ 0.0 & 1.0 & 0.000 \end{array} $$

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

Bonert and Pohl reported results for the atomic absorption analysis of several metals in the caustic suspensions produced during the manufacture of soda by the ammonia-soda process. \(^{31}\) (a) The concentration of Cu is determined by acidifying a \(200.0-\mathrm{mL}\) sample of the caustic solution with \(20 \mathrm{~mL}\) of concentrated \(\mathrm{HNO}_{3}\), adding \(1 \mathrm{~mL}\) of \(27 \% \mathrm{w} / \mathrm{v} \mathrm{H}_{2} \mathrm{O}_{2},\) and boiling for \(30 \mathrm{~min} .\) The resulting solution is diluted to \(500 \mathrm{~mL}\) in a volumetric flask, filtered, and analyzed by flame atomic absorption using matrix matched standards. The results for a typical analysis are shown in the following table. $$ \begin{array}{ccc} \text { solution } & \mathrm{mg} \mathrm{Cu} / \mathrm{L} & \text { absorbance } \\ \hline \text { blank } & 0.000 & 0.007 \\ \text { standard } 1 & 0.200 & 0.014 \\ \text { standard } 2 & 0.500 & 0.036 \\ \text { standard } 3 & 1.000 & 0.072 \\ \text { standard } 4 & 2.000 & 0.146 \\ \text { sample } & & 0.027 \end{array} $$ Determine the concentration of \(\mathrm{Cu}\) in the caustic suspension. (b) The determination of \(\mathrm{Cr}\) is accomplished by acidifying a \(200.0-\mathrm{mL}\) sample of the caustic solution with \(20 \mathrm{~mL}\) of concentrated \(\mathrm{HNO}_{3}\), adding \(0.2 \mathrm{~g}\) of \(\mathrm{Na}_{2} \mathrm{SO}_{3}\) and boiling for \(30 \mathrm{~min}\). The Cr is isolated from the sample by adding \(20 \mathrm{~mL}\) of \(\mathrm{NH}_{3}\), producing a precipitate that includes the chromium as well as other oxides. The precipitate is isolated by filtration, washed, and transferred to a beaker. After acidifying with \(10 \mathrm{~mL}\) of \(\mathrm{HNO}_{3}\), the solution is evaporated to dryness. The residue is redissolved in a combination of \(\mathrm{HNO}_{3}\) and \(\mathrm{HCl}\) and evaporated to dryness. Finally, the residue is dissolved in \(5 \mathrm{~mL}\) of \(\mathrm{HCl}\), filtered, diluted to volume in a 50 -mL volumetric flask, and analyzed by atomic absorption using the method of standard additions. The atomic absorption results are summarized in the following table. $$ \begin{array}{lcc} {\text { sample }} & \mathrm{mg} \mathrm{Cr}_{\text {added }} / \mathrm{L} & \text { absorbance } \\ \hline \text { blank } & & 0.001 \\ \text { sample } & & 0.045 \\ \text { standard addition } 1 & 0.200 & 0.083 \\ \text { standard addition } 2 & 0.500 & 0.118 \\ \text { standard addition } 3 & 1.000 & 0.192 \end{array} $$ Report the concentration of \(\mathrm{Cr}\) in the caustic suspension.

In the DPD colorimetric method for the free chlorine residual, which is reported as \(\mathrm{mg} \mathrm{Cl}_{2} / \mathrm{L},\) the oxidizing power of free chlorine converts the colorless amine \(\mathrm{N}, \mathrm{N}\) -diethyl- \(p\) -phenylenediamine to a colored dye that absorbs strongly over the wavelength range of \(440-580 \mathrm{nm}\). Analysis of a set of calibration standards gave the following results. $$ \begin{array}{cc} \mathrm{mg} \mathrm{Cl}_{2} / \mathrm{L} & \text { absorbance } \\ \hline 0.00 & 0.000 \\ 0.50 & 0.270 \\ 1.00 & 0.543 \\ 1.50 & 0.813 \\ 2.00 & 1.084 \end{array} $$ A sample from a public water supply is analyzed to determine the free chlorine residual, giving an absorbance of \(0.113 .\) What is the free chlorine residual for the sample in \(\mathrm{mg} \mathrm{Cl}_{2} / \mathrm{L}\) ?

Lin and Brown described a quantitative method for methanol based on its effect on the visible spectrum of methylene blue. \({ }^{23}\) In the absence of methanol, methylene blue has two prominent absorption bands at 610 \(\mathrm{nm}\) and \(663 \mathrm{nm}\), which correspond to the monomer and the dimer, respectively. In the presence of methanol, the intensity of the dimer's absorption band decreases, while that for the monomer increases. For concentrations of methanol between 0 and \(30 \% \mathrm{v} / \mathrm{v},\) the ratio of the two absorbance, \(A_{663} / A_{610}\), is a linear function of the amount of methanol. Use the following standardization data to determine the \(\% \mathrm{v} / \mathrm{v}\) methanol in a sample if \(A_{610}\) is 0.75 and \(A_{663}\) is 1.07 . $$ \begin{array}{cccc} \% \mathrm{v} / \mathrm{v} \text { methanol } & A_{663} / A_{610} & \% \mathrm{v} / \mathrm{v} \text { methanol } & A_{663} / A_{610} \\ \hline 0.0 & 1.21 & 20.0 & 1.62 \\ 5.0 & 1.29 & 25.0 & 1.74 \\ 10.0 & 1.42 & 30.0 & 1.84 \\ 15.0 & 1.52 & & \end{array} $$

Saito describes a quantitative spectrophotometric procedure for iron based on a solid-phase extraction using bathophenanthroline in a poly(vinyl chloride) membrane. \({ }^{22}\) In the absence of \(\mathrm{Fe}^{2+}\) the membrane is colorless, but when immersed in a solution of \(\mathrm{Fe}^{2+}\) and \(\mathrm{I}^{-},\) the membrane develops a red color as a result of the formation of an \(\mathrm{Fe}^{2+}\) -bathophenanthroline complex. A calibration curve determined using a set of external standards with known concentrations of \(\mathrm{Fe}^{2+}\) gave a standardization relationship of $$ A=\left(8.60 \times 10^{3} \mathrm{M}^{-1}\right) \times\left[\mathrm{Fe}^{2+}\right] $$ What is the concentration of iron, in \(\mathrm{mg} \mathrm{Fe} / \mathrm{L},\) for a sample with an absorbance of 0.100 ?

The concentration of acetylsalicylic acid, \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4},\) in aspirin tablets is determined by hydrolyzing it to the salicylate ion, \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{O}_{2}^{-},\) and determining its concentration spectrofluorometrically. A stock standard solution is prepared by weighing \(0.0774 \mathrm{~g}\) of salicylic acid, \(\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2}\), into a 1-L volumetric flask and diluting to volume. A set of calibration standards is prepared by pipeting \(0,2.00,4.00,6.00,8.00,\) and 10.00 \(\mathrm{mL}\) of the stock solution into separate \(100-\mathrm{mL}\) volumetric flasks that contain \(2.00 \mathrm{~mL}\) of \(4 \mathrm{M} \mathrm{NaOH}\) and diluting to volume. Fluorescence is measured at an emission wavelength of \(400 \mathrm{nm}\) using an excitation wavelength of \(310 \mathrm{nm}\) with results shown in the following table. $$ \begin{array}{cc} \text { mL of stock solution } & \text { emission intensity } \\ \hline 0.00 & 0.00 \\ 2.00 & 3.02 \\ 4.00 & 5.98 \\ 6.00 & 9.18 \\ 8.00 & 12.13 \\ 10.00 & 14.96 \end{array} $$ Several aspirin tablets are ground to a fine powder in a mortar and pestle. A 0.1013 -g portion of the powder is placed in a 1-L volumetric flask and diluted to volume with distilled water. A portion of this solution is filtered to remove insoluble binders and a 10.00 -mL aliquot transferred to a 100 -mL volumetric flask that contains \(2.00 \mathrm{~mL}\) of \(4 \mathrm{M}\) \(\mathrm{NaOH}\). After diluting to volume the fluorescence of the resulting solution is 8.69 . What is the \(\% \mathrm{w} / \mathrm{w}\) acetylsalicylic acid in the aspirin tablets?
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