Problem 78 - Chemistry: The Central Science

Question

The acid-base indicator bromcresol green is a weak acid. The yellow acid and blue base forms of the indicator are present in equal concentrations in a solution when the \(\mathrm{pH}\) is \(4.68\). What is the \(\mathrm{p} K_{a}\) for bromcresol green?

Step-by-Step Solution

Answer

Using the Henderson-Hasselbalch equation, we find that the pKa is equal to the pH when the concentrations of the acid and base forms are equal. Given that the pH is 4.68, the pKa for bromcresol green is also 4.68.

Step 1: Understand the given information

We are given that the yellow acid (HA) and blue base (A-) forms of the bromcresol green are equally present in a solution when the pH is 4.68. We can express this as: $$[\mathrm{HA}] = [\mathrm{A-}]$$

Step 2: Applying the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation relates the pH, pKa, and the concentrations (in molar) of the acid and base forms of the indicator: $$\mathrm{pH} = \mathrm{p}K_{a} + \log \frac{[\mathrm{base}]}{[\mathrm{acid}]}$$ Since the concentration of the acid and base forms are equal, $$\mathrm{pH} = \mathrm{p}K_{a} + \log \frac{[\mathrm{A-}]}{[\mathrm{HA}]} = \mathrm{p} K_{a} + \log 1$$ And, as the log of 1 is 0, we get that: $$\mathrm{pH} = \mathrm{p} K_{a}$$

Step 3: Substitute the given pH value and solve for pKa

Now let's substitute the given pH value: $$4.68 = \mathrm{p} K_{a}$$ Since the pH is equal to the pKa in this case, we find that the pKa is 4.68. Therefore, the pKa for bromcresol green is: $$\mathrm{p} K_{a} = 4.68$$

Key Concepts

Understanding Bromcresol Green

Bromcresol green is a common acid-base indicator used in chemistry to visually signal the acidity or basicity of a solution. Its role as an indicator comes from its ability to change color depending on the pH level of the surrounding environment. Specifically, bromcresol green appears yellow in acidic solutions and turns blue in basic solutions. This color change occurs because of the different ionic forms it takes: the yellow form is its acid form (HA), and the blue form is its base form (A-).
Bromcresol green can effectively indicate the pH range typically between 3.8 and 5.4. When discussing its chemical behavior, one should note that at a pH of 4.68, both its yellow and blue forms are equally present. This pH value is critical because it indicates the pKa of bromcresol green, which is a measure of its acidity. Understanding this helps chemists know exactly when the indicator undergoes its color change, providing a precise point of reference in experiments.
In summary, bromcresol green is highly valuable in the laboratory because it provides a simple, visible cue that helps scientists determine the pH level of a solution quickly and effectively. Its color change is a direct consequence of it being a weak acid, where each form dominates at different pH levels.

Exploring the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a pivotal tool in chemistry for calculating the pH of a buffer solution. It expresses the pH as the sum of the pKa (a constant for the acid) and the logarithm of the ratio of the concentration of the base form of the acid to its acid form. The equation is written as:
\[ \text{pH} = \text{p}K_a + \log \frac{[\text{A}^-]}{[\text{HA}]} \]
This equation becomes particularly useful when dealing with acid-base indicators, like bromcresol green, especially at the point where the concentration of the acid equals that of the base. In this scenario, the logarithmic term becomes zero, simplifying the expression to:
\[ \text{pH} = \text{p}K_a \]
By setting the concentrations equal, as in the cases of color indicators, we determine that at pH equal to pKa, the system is at an equilibrium point for the indicator's form transition. This property enhances our ability to determine the precise pKa value for the substance, directly reflecting when the color shift occurs in indicators like bromcresol green.
The Henderson-Hasselbalch equation is beneficial because it allows the determination of pKa from empirical pH observations, linking observable data to intrinsic chemical properties.

The Role of Acid-Base Indicators

Acid-base indicators serve as crucial visual aids in chemistry experiments for determining the pH of solutions. These organic compounds display different colors in their acidic and basic forms, helping to identify the pH range through direct observation. This color change is due to the establishment of equilibrium between the ionizable forms of the indicator while in solution, exemplified by bromcresol green's shift from yellow to blue.
Indicators are carefully selected based on their color-change range, which should align with the specific pH range required for a particular experimental goal. For example, bromcresol green changes color in a pH range from 3.8 to 5.4, making it suitable for titrations and experiments within this range. They work on the principle that at a given pH equal to the indicator's pKa, the amounts of its acid and base forms are balanced, effectively marking the midpoint of its color change interval.
In practice, acid-base indicators are invaluable for titrations and other analytical methods where a clear, immediate indication of pH is imperative. Their simplicity and effectiveness allow for rapid assessment and adjustment of reaction conditions, benefiting both educational settings and research.