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EDTA forms colored complexes with a variety of metal ions that may serve as the basis for a quantitative spectrophotometric method of analysis. The molar absorptivities of the EDTA complexes of \(\mathrm{Cu}^{2+}, \mathrm{Co}^{2+}\), and \(\mathrm{Ni}^{2+}\) at three wavelengths are summarized in the following table (all values of \(\varepsilon\) are in \(\left.\mathrm{M}^{-1} \mathrm{~cm}^{-1}\right).\) $$ \begin{array}{cccc} \text { metal } & \varepsilon_{462.9} & \varepsilon_{732.0} & \varepsilon_{378.7} \\ \hline \mathrm{Co}^{2+} & 15.8 & 2.11 & 3.11 \\ \mathrm{Cu}^{2+} & 2.32 & 95.2 & 7.73 \\ \mathrm{Ni}^{2+} & 1.79 & 3.03 & 13.5 \end{array} $$ Using this information determine the following: (a) The concentration of \(\mathrm{Cu}^{2+}\) in a solution that has an absorbance of 0.338 at a wavelength of \(732.0 \mathrm{nm}\). (b) The concentrations of \(\mathrm{Cu}^{2+}\) and \(\mathrm{Co}^{2+}\) in a solution that has an absorbance of 0.453 at a wavelength of \(732.0 \mathrm{nm}\) and 0.107 at a wavelength of \(462.9 \mathrm{nm}\) (c) The concentrations of \(\mathrm{Cu}^{2+}, \mathrm{Co}^{2+},\) and \(\mathrm{Ni}^{2+}\) in a sample that has an absorbance of 0.423 at a wavelength of \(732.0 \mathrm{nm}, 0.184\) at a wavelength of \(462.9 \mathrm{nm}\), and 0.291 at a wavelength of \(378.7 \mathrm{nm}\). The pathlength, \(b\), is \(1.00 \mathrm{~cm}\) for all measurements.

Step by step solution

01

Understanding the Law

We need to use Beer's Law, which states that the absorbance (A) is directly proportional to the concentration (c), pathlength (b), and molar absorptivity (蔚). The formula is given by: \( A = \varepsilon \cdot c \cdot b \). In this problem, the pathlength \( b \) is 1 cm, simplifying the equation to \( A = \varepsilon \cdot c \).

02

Solve (a) for Cu虏鈦� Only

Given that the absorbance \( A = 0.338 \) at \( 732.0 \ \mathrm{nm} \) for \( \mathrm{Cu}^{2+} \), with \( \varepsilon = 95.2 \ \mathrm{M}^{-1}\mathrm{cm}^{-1} \) from the table. Using \( A = \varepsilon \cdot c \), we have: \[ c = \frac{A}{\varepsilon} = \frac{0.338}{95.2} = 0.00355 \ \mathrm{M} \].

03

System of Equations for (b)

We now have two absorbances at different wavelengths, forming a system: \( A_{732.0} = 0.453 \) and \( A_{462.9} = 0.107 \). Using \( A = \varepsilon \cdot c \), we set up:1. \( 95.2 \cdot c_{\mathrm{Cu}^{2+}} + 2.11 \cdot c_{\mathrm{Co}^{2+}} = 0.453 \)2. \( 2.32 \cdot c_{\mathrm{Cu}^{2+}} + 15.8 \cdot c_{\mathrm{Co}^{2+}} = 0.107 \).

04

Solving System of Equations for (b)

By solving these two equations simultaneously (using substitution or elimination methods), we first express one concentration in terms of the other, plug into the second equation, solve for one concentration, then use it to find the other:- Solving gives \( c_{\mathrm{Cu}^{2+}} = 0.00437 \ \mathrm{M} \)- Plug the value into one of the equations to get \( c_{\mathrm{Co}^{2+}} = 0.00606 \ \mathrm{M} \).

05

Solve (c) with Three Unknowns

With three wavelengths and associated absorbances, we have:1. \( 95.2 \cdot c_{\mathrm{Cu}^{2+}} + 2.11 \cdot c_{\mathrm{Co}^{2+}} + 3.03 \cdot c_{\mathrm{Ni}^{2+}} = 0.423 \)2. \( 2.32 \cdot c_{\mathrm{Cu}^{2+}} + 15.8 \cdot c_{\mathrm{Co}^{2+}} + 1.79 \cdot c_{\mathrm{Ni}^{2+}} = 0.184 \)3. \( 7.73 \cdot c_{\mathrm{Cu}^{2+}} + 3.11 \cdot c_{\mathrm{Co}^{2+}} + 13.5 \cdot c_{\mathrm{Ni}^{2+}} = 0.291 \).

06

Solving System of Equations for (c)

Using substitution or matrix methods (like Gaussian elimination) to solve the three equations gives:- \( c_{\mathrm{Cu}^{2+}} = 0.00437 \ \mathrm{M} \)- \( c_{\mathrm{Co}^{2+}} = 0.00960 \ \mathrm{M} \)- \( c_{\mathrm{Ni}^{2+}} = 0.01382 \ \mathrm{M} \).

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Key Concepts

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

EDTA Complexes

EDTA, or Ethylenediaminetetraacetic acid, is a molecule widely used in chemistry for forming complexes with metal ions. A complex is a compound consisting of a central metal atom or ion bonded to surrounding molecules or anions, known as ligands. EDTA is an outstanding ligand due to its ability to form stable complexes with a range of metal ions, including those of transition metals such as copper (\(\mathrm{Cu}^{2+}\)). EDTA is a hexadentate ligand meaning it can form six bonds with a metal ion, thus strongly holding the metal ion and often changing its solubility or reactivity.
In spectrophotometric analysis, EDTA-metal complexes can absorb light at specific wavelengths. This coloring forms the foundation for quantitative analysis because each metal-EDTA complex will have a characteristic molar absorptivity, reflecting how strongly it absorbs light.
Forming complexes with EDTA provides a reliable way of detecting and measuring metal ions in various solutions by exploiting these unique properties.

Beer's Law

Beer's Law, also known as the Beer-Lambert Law, is a core principle in the field of spectrophotometry. It describes the linear relationship between the absorbance of light and the properties of the material through which the light is traveling. The law is mathematically expressed as:
\[ A = \varepsilon \cdot c \cdot b \] where:

• \(A\) is the measured absorbance (no units).
• \(\varepsilon\) is the molar absorptivity (in \(\mathrm{M^{-1} \ cm^{-1}}\)), also known as molar extinction coefficient.
• \(c\) is the concentration of the solution (in \(\mathrm{M}\) or mol/L).
• \(b\) is the pathlength that the light travels through the sample (in cm).

In the given exercise, Beer's Law is used to calculate the concentration of metal ions in a solution by measuring the absorbance at specified wavelengths. Since pathlength \(b\) is typically held constant and known, the equation simplifies the direct calculation of the concentration if the molar absorptivity and absorbance are known.

Molar Absorptivity

Molar absorptivity, denoted by \(\varepsilon\), is a crucial parameter in spectrophotometry. It signifies how strongly a particular substance absorbs light at a given wavelength. Molar absorptivity is expressed in units of \(\mathrm{M^{-1} \ cm^{-1}}\), indicating that it is an intrinsic property of the substance and independent of concentration.
A high molar absorptivity means the substance strongly absorbs light, while a low value suggests weaker absorption. For example, in our exercise, copper's EDTA complex (\(\mathrm{Cu}^{2+}\)) at 732.0 nm shows a high molar absorptivity of 95.2 \(\mathrm{M^{-1} \ cm^{-1}}\), signifying strong light absorption at this wavelength.
Knowing the molar absorptivity allows for using Beer's Law to deduce the concentration of a solute by measuring the absorbance. Each metal-EDTA complex exhibits different molar absorptivities at various wavelengths, facilitating the analysis of solutions containing multiple metal ions.

System of Equations

A system of equations in the context of spectrophotometric analysis involves using Beer's Law. It allows us to study multiple species (here, different metal ions) that each contribute to the absorbance at various wavelengths. When given absorbance values at multiple wavelengths, a system of equations can be created for each wavelength and its corresponding absorbance.
For example, when analyzing a mixture of \(\mathrm{Cu}^{2+}\), \(\mathrm{Co}^{2+}\), and \(\mathrm{Ni}^{2+}\), simultaneous equations are established:

• Each equation corresponds to a different wavelength and the sum of absorbances from each metal complex.
• Each metal ion's contribution is expressed as its concentration multiples by its molar absorptivity at that wavelength.

Solving these equations often involves methods like substitution or matrix calculations. The solution yields the concentrations of each metal ion in the sample. This approach extends Beer's Law to complex mixtures, enabling the determination of individual component concentrations within a mixture.