Problem 121 - Chemistry: An Atoms-Focused Approach

Question

Estimate the \(K_{\mathrm{a}}\) values of the following indicators: a. Bromphenol blue, whose transition color occurs at a pH of about 3.8 b. Bromcresol green, whose transition color occurs at a \(\mathrm{pH}\) of about 4.3 c. Alizarin yellow \(\mathrm{R}\), whose transition color occurs at pH of about 10.9

Step-by-Step Solution

Answer

Question: Estimate the \(K_{\mathrm{a}}\) values for the following indicators: a) Bromphenol blue, b) Bromcresol green, and c) Alizarin yellow \(\mathrm{R}\). Answer: The estimated \(K_{\mathrm{a}}\) values for the three indicators are: a. Bromphenol blue: \(K_{\mathrm{a}} \approx 1.58 \times 10^{-4}\), b. Bromcresol green: \(K_{\mathrm{a}} \approx 5.01 \times 10^{-5}\), and c. Alizarin yellow \(\mathrm{R}\): \(K_{\mathrm{a}} \approx 1.26 \times 10^{-11}\).

Step 1: Bromphenol blue estimation:

For bromphenol blue, the transition color occurs at a pH value of 3.8. To calculate its \(K_a\) value, use the formula \(K_{\mathrm{a}} = 10^{-\mathrm{pH}}\). Plug in the given pH value: \(K_{\mathrm{a}} = 10^{-3.8}\).

Step 2: Bromphenol blue \(K_{\mathrm{a}}\) value:

Calculate the \(K_{\mathrm{a}}\) value for bromphenol blue: \(K_{\mathrm{a}} = 10^{-3.8} \approx 1.58 \times 10^{-4}\).

Step 3: Bromcresol green estimation:

For bromcresol green, the transition color occurs at a pH value of 4.3. Use the same formula: \(K_{\mathrm{a}} = 10^{-\mathrm{pH}}\). Plug in the given pH value: \(K_{\mathrm{a}} = 10^{-4.3}\).

Step 4: Bromcresol green \(K_{\mathrm{a}}\) value:

Calculate the \(K_{\mathrm{a}}\) value for bromcresol green: \(K_{\mathrm{a}} = 10^{-4.3} \approx 5.01 \times 10^{-5}\).

Step 5: Alizarin yellow \(\mathrm{R}\) estimation:

For alizarin yellow \(\mathrm{R}\), the transition color occurs at a pH value of 10.9. Use the same formula: \(K_{\mathrm{a}} = 10^{-\mathrm{pH}}\). Plug in the given pH value: \(K_{\mathrm{a}} = 10^{-10.9}\).

Step 6: Alizarin yellow \(\mathrm{R}\) \(K_{\mathrm{a}}\) value:

Calculate the \(K_{\mathrm{a}}\) value for alizarin yellow \(\mathrm{R}\): \(K_{\mathrm{a}} = 10^{-10.9} \approx 1.26 \times 10^{-11}\). Therefore, the estimated \(K_{\mathrm{a}}\) values for the three indicators are: a. Bromphenol blue: \(K_{\mathrm{a}} \approx 1.58 \times 10^{-4}\) b. Bromcresol green: \(K_{\mathrm{a}} \approx 5.01 \times 10^{-5}\) c. Alizarin yellow \(\mathrm{R}\): \(K_{\mathrm{a}} \approx 1.26 \times 10^{-11}\)

Key Concepts

pH Calculations

The concept of pH is fundamental in chemistry and represents the acidity or basicity of an aqueous solution. It is defined as the negative logarithm of the hydrogen ion concentration in a solution: \[ \text{pH} = -\log_{10}[H^+] \] The pH scale typically ranges from 0 to 14. A pH less than 7 indicates acidic conditions, while a pH greater than 7 indicates a basic or alkaline environment. A pH of exactly 7 is considered neutral.
pH calculations are essential when working with acid-base reactions. For weak acids, the pH can be determined from their acid dissociation constant, or \( K_a \). This constant measures the strength of an acid in solution, with larger \( K_a \) values indicating stronger acids.
To find \( K_a \) from pH, you use the formula \( K_a = 10^{-\text{pH}} \). By inserting the pH value into this formula, you can calculate the \( K_a \) for the acid. This is particularly useful for estimating the acid strength of indicators, which change color at specific pH values.

Acid-Base Indicators

Acid-base indicators are substances that change color based on the pH of the solution they are in. These indicators are weak acids or bases that have distinctly different colors in their protonated and deprotonated forms. This property allows them to act as visual cues for the acidity or basicity of a solution.
Some common indicators include:

• Bromphenol Blue: Changes color around pH 3.8.
• Bromcresol Green: Changes color around pH 4.3.
• Alizarin Yellow R: Changes color around pH 10.9.

These color changes correspond to the pH at which the indicator's protonated and deprotonated forms are at equilibrium. This is why indicators are invaluable in titration processes, where they help determine the endpoint by showing a clear color change. The pH at which this color change occurs can be used to estimate the \( K_a \) value for the indicator, as illustrated with the provided examples.

Equilibrium Constants

Equilibrium constants are crucial in understanding chemical reactions that reach a balance between reactants and products. There are different equilibrium constants for various chemical processes, but the acid dissociation constant, \( K_a \), is key in acid-base chemistry.
The value of \( K_a \) is derived from the equilibrium expression for the dissociation of a weak acid: \[ HA \rightleftharpoons H^+ + A^- \] The equilibrium constant is expressed as: \[ K_a = \frac{[H^+][A^-]}{[HA]} \]
\( K_a \) provides insight into the acid's ability to donate protons. High \( K_a \) values suggest strong acids, while low \( K_a \) values indicate weak acids. In practice, knowing the \( K_a \) helps predict the extent of dissociation of the acid in solution, which in turn influences the solution's pH.
For the exercise, the calculation of \( K_a \) values for indicators like Bromphenol Blue involves determining the pH at which half of the indicator is dissociated. This aligns with the equilibrium point of its color change, thereby giving an estimate of its \( K_a \).