Problem 6 - Analytical Chemistry

Question

A chemical deviation to Beer's law may occur if the concentration of an absorbing species is affected by the position of an equilibrium reaction. Consider a weak acid, HA, for which \(K_{\mathrm{a}}\) is \(2 \times 10^{-5}\). Construct Beer's law calibration curves of absorbance versus the total concentration of weak acid \(\left(C_{\text {total }}=[\mathrm{HA}]+\left[\mathrm{A}^{-}\right]\right),\) using values for \(C_{\text {total }}\) of \(1.0 \times 10^{-5}, 3.0 \times 10^{-5}, 5.0 \times 10^{-5}, 7.0 \times 10^{-5}, 9.0 \times 10^{-5}, 11 \times 10^{-5}\), and \(13 \times 10^{-5} \mathrm{M}\) for the following sets of conditions and comment on your results: (a) \(\varepsilon_{\mathrm{HA}}=\varepsilon_{\mathrm{A}^{-}}=2000 \mathrm{M}^{-1} \mathrm{~cm}^{-1} ;\) unbuffered solution. (b) \(\varepsilon_{\mathrm{HA}}=2000 \mathrm{M}^{-1} \mathrm{~cm}^{-1} ; \varepsilon_{\mathrm{A}^{-}}=500 \mathrm{M}^{-1} \mathrm{~cm}^{-1} ;\) unbuffered solution. (c) \(\varepsilon_{\mathrm{HA}}=2000 \mathrm{M}^{-1} \mathrm{~cm}^{-1} ; \varepsilon_{\mathrm{A}^{-}}=500 \mathrm{M}^{-1} \mathrm{~cm}^{-1} ;\) solution buffered to a \(\mathrm{pH}\) of 4.5 Assume a constant pathlength of \(1.00 \mathrm{~cm}\) for all samples.

Step-by-Step Solution

Answer

The calibration curves show varying slopes; buffering stabilizes deviations.

Step 1: Understanding the Equilibrium

For a weak acid HA, it dissociates into H+ and A- according to: \[\mathrm{HA} \rightleftharpoons \mathrm{H}^{+} + \mathrm{A}^{-}\] The equilibrium constant \(K_a\) is given as \(2 \times 10^{-5}\). This determines the concentrations of \(\mathrm{HA}, \mathrm{H}^{+},\) and \(\mathrm{A}^{-}\).

Step 2: Calculate Concentrations for HA and A⁻

Use the expression for the equilibrium constant:\[K_a = \frac{[\mathrm{H}^{+}][\mathrm{A}^{-}]}{[\mathrm{HA}]}\]Assuming that \([\mathrm{H}^{+}] = x\), we have:\[K_a = \frac{x^2}{C_\mathrm{total} - x}\] Now, solve for \(x\) to get concentrations \([\mathrm{HA}] = C_\mathrm{total} - x\) and \([\mathrm{A}^{-}] = x\).

Step 3: Calculate Absorbance for Unbuffered Solution with Same ε

For part (a), where \(\varepsilon_{\mathrm{HA}} = \varepsilon_{\mathrm{A}^{-}} = 2000\):The total absorbance \(A\) is:\[A = \varepsilon_{\mathrm{HA}} [\mathrm{HA}] + \varepsilon_{\mathrm{A}^{-}} [\mathrm{A}^{-}]\]Since \(\varepsilon_{\mathrm{HA}} = \varepsilon_{\mathrm{A}^{-}}\):\[A = 2000(C_{total})\]The absorbance is directly proportional to \(C_{total}\).

Step 4: Calculate Absorbance with Different ε Values

For part (b), where \(\varepsilon_{\mathrm{HA}} = 2000\) and \(\varepsilon_{\mathrm{A}^{-}} = 500\):\[A = 2000([\mathrm{HA}]) + 500([\mathrm{A}^{-}])\]Substitute \([\mathrm{HA}] = C_{total} - x\) and \([\mathrm{A}^{-}] = x\) from Step 2, to calculate the different absorbance values for each \(C_{total}\).

Step 5: Calculate pH for Buffered Solution and Adjust Concentrations

For part (c), using Henderson-Hasselbalch equation:\[ \text{pH} = \text{pK}_a + \log \left(\frac{[\mathrm{A}^{-}]}{[\mathrm{HA}]}\right) = 4.5 \]Calculate \([\mathrm{HA}]\) and \([\mathrm{A}^{-}]\) using the pH value. Then check if equilibrium concentrations are significantly affected. If not, use \([\mathrm{HA}] = C_{total]\) and find absorbance as before: \[A = 2000([\mathrm{HA}]) + 500([\mathrm{A}^{-}])\]

Step 6: Analyze and Compare Results for Each Scenario

Compare the slope of the calibration curves from each condition. (a) When \(\varepsilon_{\mathrm{HA}} = \varepsilon_{\mathrm{A}^{-}}\), the slope is constant. (b) When \(\varepsilon_{\mathrm{HA}} eq \varepsilon_{\mathrm{A}^{-}}\), the slope varies slightly due to the influence of weak acid dissociation. (c) Buffering minimizes deviations, making the calibration curve more consistent with Beer's law.

Key Concepts

Chemical Equilibrium

In chemistry, equilibrium refers to the point where the concentrations of reactants and products remain constant over time in a reversible reaction. For a weak acid, such as HA, it dissociates into H⁺ ions and A⁻ ions. This dissociation reaches a chemical equilibrium, represented by the reaction \(\mathrm{HA} \rightleftharpoons \mathrm{H}^{+} + \mathrm{A}^{-}\). The equilibrium constant (\(K_a\)) for this process is a key factor. It quantifies the balance between the reactants and products at equilibrium.
The equilibrium constant \(K_a\) is expressed as:
\[K_a = \frac{[\mathrm{H}^{+}][\mathrm{A}^{-}]}{[\mathrm{HA}]}\]
Thus, knowing \(K_a\) helps determine the concentrations of HA, H⁺, and A⁻ in solution. In this exercise, \(K_a = 2 \times 10^{-5}\), indicating that the weak acid partially dissociates. Understanding chemical equilibrium is vital in predicting the behavior of weak acids in different conditions and in constructing accurate calibration curves.

Weak Acids

Weak acids are classified by their ability to partially dissociate into ions in solution, unlike strong acids which fully dissociate. The partial dissociation is represented by an equilibrium constant parameter \(K_a\).
Because of their incomplete dissociation, weak acids have both the undissociated acid (HA) and its ions (H⁺ and A⁻) present in the solution. This partial ionization is crucial for the calculation of species concentrations when dealing with absorbance and calibration curves.
Common examples of weak acids include acetic acid (CH₃COOH) and formic acid (HCOOH). In scenarios like Beer's law exercises, understanding the weak acid’s dissociation allows for accurate calculations of pH and equilibrium concentrations. It also impacts the absorbance in spectrophotometric measurements, especially when the molar absorptivity of the acid and its conjugate base differ.

Absorbance Calculation

The absorbance of a solution indicates how much light is absorbed as it passes through the solution. According to Beer's Law, absorbance depends on the molar absorptivity (ε), the path length of light through the solution (usually in cm), and the concentration of the absorbing species.
The formula is expressed as:
\[ A = \varepsilon l c \]
where:

• \(A\) is absorbance
• \(\varepsilon\) is molar absorptivity
• \(l\) is the path length
• \(c\) is the concentration of the absorbing species

In unbuffered solutions where \(\varepsilon_{\mathrm{HA}} = \varepsilon_{\mathrm{A}^{-}}\), the absorbance is directly proportional to concentration. However, when \(\varepsilon_{\mathrm{HA}} eq \varepsilon_{\mathrm{A}^{-}}\), the different absorptivities play a crucial role in the absorbance calculation, as seen in parts (b) and (c) of the exercise.

Calibration Curves

Calibration curves are graphical representations that show the relationship between the concentration of a substance and the absorbance of light at a specific wavelength. By plotting absorbance versus concentration, one can determine the concentration of an unknown sample based on its absorbance.
When plotting a calibration curve for a weak acid in water, different conditions such as buffering and variances in molar absorptivity need to be considered. In the exercise, three different conditions were explored:

• When \(\varepsilon_{\mathrm{HA}} = \varepsilon_{\mathrm{A}^{-}}\), a direct relationship is maintained, providing a straight-line calibration curve.
• If \(\varepsilon_{\mathrm{HA}} eq \varepsilon_{\mathrm{A}^{-}}\), the slope changes because of the varying contributions of HA and A⁻ to the absorbance.
• A buffered solution at a given pH stabilizes equilibrium and results in a more predictable calibration curve, closer to ideal linearity.

Calibration curves are fundamental tools necessary for accurately quantitative analysis in laboratories, specifically when dealing with weak acids subjected to varying conditions.