Problem 69 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An acid type indicator, HIn differs in colour from its conjugate base \(\left(\mathrm{In}^{-}\right)\). The human eye is sensitive to colour differences only when the ratio \([\operatorname{In}] /[\mathrm{HIn}]\) is greater than 10 or smaller than \(0.1\). What should be the minimum change in the \(\mathrm{pH}\) of the solution to observe a complete colour change \(\left(K_{\mathrm{a}}=1.0 \times 10^{-5}\right)\) ? (a) \(0.0\) (b) \(1.0\) (c) \(2.0\) (d) \(5.0\)

Step-by-Step Solution

Answer

The minimum change in pH required to observe a complete colour change is 2.0.

Step 1 - Understanding the concept of the acid dissociation constant (Ka)

The acid dissociation constant, Ka, gives us a measure of the strength of an acid in solution. It is defined as the equilibrium constant for the reaction where the acid (HA) dissociates into its conjugate base (A-) and a hydrogen ion (H+). The formula for the dissociation constant is Ka = [H+][A-] / [HA]. We can relate the Ka to pH by the Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]), where pKa is the negative logarithm of Ka (-log(Ka)).

Step 2 - Calculating the pKa

To use the Henderson-Hasselbalch equation, we first calculate the pKa. The pKa is the negative logarithm of the acid dissociation constant (Ka). Since we are given that Ka = 1.0 x 10^-5, we find pKa using pKa = -log(Ka).

Step 3 - Determining the color change condition

The indicator color change is significant when the ratio [In-]/[HIn] is either greater than 10 or smaller than 0.1. We'll calculate the pH change needed for both conditions using the Henderson-Hasselbalch equation: pH = pKa + log([In-]/[HIn]). We will need to calculate the change in pH for when the ratio is 10 and when it is 0.1.

Step 4 - Calculating the minimum pH change for the ratio 10

For the ratio [In-]/[HIn] to be 10, we set up the equation: pH = pKa + log(10). Since log(10) equals 1, the pH at this ratio is simply pH = pKa + 1.

Step 5 - Calculating the minimum pH change for the ratio 0.1

For the ratio [In-]/[HIn] to be 0.1, we set up the equation: pH = pKa + log(0.1). Since log(0.1) equals -1, the pH at this ratio is pH = pKa - 1.

Step 6 - Finding the total pH range for color change

The complete color change occurs between the pH when the ratio [In-]/[HIn] is 10 and when it is 0.1. Therefore, the change in pH is the difference between the pH values found in Step 4 and Step 5, which is 2 units of pH (from pKa - 1 to pKa + 1).

Key Concepts

Understanding the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation beautifully links the pH of a solution with the pKa of the acid and the ratio of the concentrations of the conjugate base and the acid. It represents a rearranged form of the acid dissociation constant expression to solve for pH, where pH is the measure of acidity in the solution.

The Henderson-Hasselbalch equation is given by:
\[ \text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right) \]
This equation allows us to predict the pH of a buffer system, and by extension, it helps in identifying the color change of indicators in acid-base titrations. The pKa is a constant determined by the dissociation tendency of the particular acid, which in this case is given as \(1.0 \times 10^{-5}\) for its Ka value. Calculating the pKa as the negative logarithm of Ka, we can manipulate the Henderson-Hasselbalch equation to estimate the concentration ratios responsible for a visible color change in acid-base indicators.

Calculating pH Change in Acid-Base Solutions

To comprehend how pH change affects an acid-base indicator, first understand that the pH of a solution is a numerical scale used to specify the acidity or basicity. When acids or bases are added to a solution, the pH can shift, influencing the state of the acid-base indicators.

For practical calculations, we can determine the minimum pH change required for the significant color change by considering the ratio of the concentrations of the indicator's conjugate base to the acid form. Using the equation provided by the Henderson-Hasselbalch equation, we assess the pH at two points: when the ratio of concentration is 10 and when it is 0.1. The minimum change is the difference between these two pH values.

This process demystifies the mathematics behind titration curves and buffers while providing clarity in laboratory settings, ensuring comprehension beyond mere memorization of concepts. Through understanding how to calculate pH change, students and professionals can predict and control reactions in a myriad of chemical applications.

Indicator Color Change and pH Sensitivity

The magic of chemistry often lies in the visual—at least when it comes to acid-based indicators. These substances change color at certain pH levels, serving as a practical visual cue for the acidity or basicity of a solution.

They work because they are weak acids or bases that dissociate in solution, and their different forms have different colors. The human eye can discern these color changes when the ratio of the concentrations of both forms exceeds certain thresholds, typically when it is greater than 10 or smaller than 0.1.

The color change is closely related to the concept of pH sensitivity. In the exercise, the Ka value dictates the sensitivity of the indicator we are examining. By applying the Henderson-Hasselbalch equation, we calculated that a visible color change takes place when the pH shifts by at least 2 units. This knowledge is crucial for titration experiments where the end point is indicated by a color change.