Problem 32 - Analytical Chemistry

Question

The following information is available for the extraction of \(\mathrm{Cu}^{2+}\) by \(\mathrm{CCl}_{4}\) and dithizone: \(K_{\mathrm{D}, \mathrm{c}}=7 \times 10^{4} ; \beta_{2}=5 \times 10^{22} ; K_{\mathrm{a}, \mathrm{HL}}=3 \times 10^{-5}\) \(K_{\mathrm{D}, \mathrm{HL}}=1.1 \times 10^{4} ;\) and \(n=2 .\) What is the extraction efficiency if a 100.0 -mL sample of an aqueous solution that is \(1.0 \times 10^{-7} \mathrm{M} \mathrm{Cu}^{2+}\) and \(1 \mathrm{M}\) in \(\mathrm{HCl}\) is extracted using \(10.0 \mathrm{~mL}\) of \(\mathrm{CCl}_{4}\) containing \(4.0 \times 10^{-4}\) \(M\) dithizone (HL)?

Step-by-Step Solution

Answer

The extraction efficiency is nearly 100%.

Step 1: Identify the relevant extraction equilibrium

We noted the provided parameters: \( K_{D, c} = 7 \times 10^4 \), \( \beta_2 = 5 \times 10^{22} \), \( K_{a, HL} = 3 \times 10^{-5} \), \( K_{D, HL} = 1.1 \times 10^4 \), and \( n = 2 \). \( K_D \) values relate to the distribution of Cu between phases, \( \beta_2 \) is the stability constant of the Cu-dithizone complex, and \( K_{a, HL} \) and \( K_{D, HL} \) relate to dithizone properties. These constants describe how Cu is partitioned between aqueous and organic phases.

Step 2: Determine the distribution coefficient for Cu

The distribution coefficient \( D \) can be calculated using \( K_D \), \( \beta_2 \), and other parameters. It is given by:\[ D = \frac{K_{D,c} \cdot \beta_2 \cdot K_{D,HL}^{n-1}}{1 + \frac{K_{a,HL} \cdot [H^+]}{K_{D,HL}}} \].Plug in the values: \( D = \frac{7 \times 10^4 \cdot 5 \times 10^{22} \cdot (1.1 \times 10^4)}{1 + \frac{3 \times 10^{-5} \cdot 1}{1.1 \times 10^4}} \). Solve for \( D \).

Step 3: Simplify the expression and calculate \( D \)

The denominator simplifies since \( [H^+] = 1\), making the expression:\[ 1 + \frac{3 \times 10^{-5}}{1.1 \times 10^4} \approx 1 \].Thus, \( D = \frac{7 \times 10^4 \cdot 5 \times 10^{22} \cdot 1.1 \times 10^4}{1} \), which calculates to \( D = 3.85 \times 10^{31} \).

Step 4: Calculate the extraction efficiency

The extraction efficiency \( \%E \) is given by the formula:\[ \%E = \frac{D \cdot V_o}{D \cdot V_o + V_a} \times 100 \],where \( V_o = 10 \) mL is the organic phase volume, and \( V_a = 100 \) mL is the aqueous volume. Substitute:\[ \%E = \frac{3.85 \times 10^{31} \cdot 10}{3.85 \times 10^{31} \cdot 10 + 100} \times 100 \].

Step 5: Simplify and solve for percentage extraction

Given the very large value of \( D \), the term \( 3.85 \times 10^{31} \cdot 10 \) in the expression is overwhelmingly large compared to 100, making:\[ \%E \approx \frac{3.85 \times 10^{32}}{3.85 \times 10^{32} + 100} \times 100 \approx 100\% \].

Key Concepts

Distribution Coefficient

The distribution coefficient, often denoted as \( D \), plays a crucial role in determining how a substance is partitioned between two immiscible liquid phases. It is a measure of the ratio of the concentration of a solute in two phases at equilibrium. In the context of extraction, knowing the distribution coefficient helps us predict how much of a solute will remain in the original phase and how much will move to the new phase.

To calculate \( D \) in an extraction process, relevant equilibrium constants like \( K_{D,c} \), \( \beta_2 \), and \( K_{D,HL} \) are used. The formula used is\[ D = \frac{K_{D,c} \cdot \beta_2 \cdot K_{D,HL}^{n-1}}{1 + \frac{K_{a,HL} \cdot [H^+]}{K_{D,HL}}} \]Here,

• \( K_{D,c} \) is the distribution constant for the solute copper (Cu) between the organic and aqueous layers.
• \( \beta_2 \) is the stability constant for the complex formed between the solute and the extractant.
• \( K_{D,HL} \) is the distribution constant for the extractant, dithizone, on its own.
• \( [H^+] \) is the concentration of hydrogen ions, relevant in this case because we're dealing with an acidic solution.

Extraction Equilibrium

Extraction equilibrium refers to the state where the solute distribution between two phases stops changing with time. At equilibrium, the rate of solute moving from the aqueous phase to the organic phase equals the rate of solute moving in the opposite direction.

The equilibrium is established based on the balance dictated by the distribution coefficient. In our case, copper ions are initially in an aqueous solution and are transferred into an organic phase (\( CCl_4 \) with dithizone).

Achieving equilibrium is essential because

• It ensures maximum solute transfer, optimizing extraction efficiency.
• It allows for accurate predictions using mathematical relationships.
• It provides the basis for calculating the extraction efficiency.

Understanding how equilibrium is reached and maintained is critical for designing efficient extraction systems.

Stability Constant

The stability constant, denoted as \( \beta \), indicates the stability of a complex ion formed in solution. In this context, \( \beta_2 \) represents the stability of the copper-dithizone complex.

This constant plays a pivotal role in determining how strongly an ion is bound within the complex, affecting the overall extraction efficiency. A higher stability constant implies a more stable complex, enhancing the likelihood of extracting the ion into the organic phase.

The stability constant is used in calculating the distribution coefficient, as seen in the formula\[ D = \frac{K_{D,c} \cdot \beta_2 \cdot K_{D,HL}^{n-1}}{1 + \frac{K_{a,HL} \cdot [H^+]}{K_{D,HL}}} \] This relationship demonstrates how tightly bound complexes can influence the movement of solutes between phases. Understanding stability constants is pivotal for assessing how coordination chemistry affects separation strategies.

Analytical Chemistry

Analytical chemistry involves methods and techniques to identify and quantify matter. It plays a significant role in designing, understanding, and optimizing extraction processes.

In this exercise, analytical chemistry principles guide the calculation of extraction efficiency. It involves assessing equilibrium constants and using them in calculations to understand solute distribution. The choice of solvent and extractant, dictated by chemical properties, is another aspect rooted in analytical chemistry.

• The approach requires a detailed understanding of how components interact at the molecular level.
• It leverages mathematical relationships to calculate expected outcomes, like the efficiency of extraction.
• Methods in analytical chemistry can help verify predicted outcomes through experimental results.

Thus, analytical chemistry integrates theory and experimental practice, ensuring that extraction processes are not only efficient but also verifiable in a lab setting.