Problem 163 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation constant of monobasic acids A, B and \(\mathrm{C}\) are \(10^{-4}, 10^{-6}\) and \(10^{-10}\) respectively. The concentration of each monobasic acid is \(0.1 \mathrm{M}\). Which of the following has been arranged in increasing order of \(\mathrm{pH}\) ? (a) \(C

Step-by-Step Solution

Answer

Option (c) \(A

Step 1: Understanding pH and acid dissociation

The acidity of a solution is expressed by its pH value, which is related to the concentration of hydrogen ions, \([\mathrm{H}^+]\), in solution. For a weak monobasic acid with dissociation constant \(K_a\), the pH is primarily determined by the dissociation constant and the concentration of the acid.

Step 2: Relate dissociation constant to pH

The relation for a weak acid HA is written as: \[\mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^-\] and its dissociation constant is given by \(K_a = \frac{{[\mathrm{H}^+][\mathrm{A}^-]}}{{[\mathrm{HA}]}}\). In case of weak acids, \( [\mathrm{H}^+] = \sqrt{K_a \cdot C}\) where \(C\) is the concentration of the acid. Therefore, \(\mathrm{pH} = -\log([\mathrm{H}^+])\).

Step 3: Calculate [H+] for each acid

For acid A with \([K_a = 10^{-4}]\) and acid concentration (C) \( = 0.1\) M, \([\mathrm{H}^+] = \sqrt{10^{-4} \times 0.1} = 10^{-3}\). Similarly, for acid B, \([\mathrm{H}^+] = \sqrt{10^{-6} \times 0.1} = 10^{-4}\) and for acid C, \([\mathrm{H}^+] = \sqrt{10^{-10} \times 0.1} = 10^{-6}\).

Step 4: Determine the pH values

Calculate the pH for each acid: - For acid A, \(\mathrm{pH} = -\log(10^{-3}) = 3\).- For acid B, \(\mathrm{pH} = -\log(10^{-4}) = 4\).- For acid C, \(\mathrm{pH} = -\log(10^{-6}) = 6\).

Step 5: Arrange in order of increasing pH

Arrange the pH values in increasing order: \( \mathrm{pH_A} = 3 < \mathrm{pH_B} = 4 < \mathrm{pH_C} = 6 \). Therefore, the increasing order of pH for the acids is A < B < C.

Key Concepts

Understanding Weak Acid Dissociation

Weak acids, unlike strong acids, do not completely dissociate in solution. This partial dissociation means they exist in equilibrium with their ions in solution. You can imagine weak acid molecules like

• HA, which dissociates into
• \( H^+ \) ions and
• its conjugate base \( A^- \).

The equilibrium can be represented as: \[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \]
This concept is crucial when calculating the pH of a weak acid solution because the amount of dissociation directly affects the concentration of \( H^+ \) ions, contributing to the acidity, or pH, of the solution.
Weak acids are characterized by their lower dissociation compared to strong acids, so they reach this equilibrium without converting all of the original acid. This is why calculations involving weak acids focus more on the equilibrium expressions than on initial concentrations.

The Role of the Dissociation Constant

The dissociation constant, \( K_a \), is a numerical value that describes the strength of a weak acid in terms of its ability to donate protons (\( H^+ \) ions). The equation for a weak acid HA and its dissociation constant is: \[ K_a = \frac{{[\text{H}^+][\text{A}^-]}}{{[\text{HA}]}} \]
This equation reveals the relationship between the concentration of undissociated acid, HA, and its dissociated ions, \( H^+ \) and \( A^- \).

• A higher \( K_a \) value indicates a stronger acid with more dissociation occurring.
• A lower \( K_a \) value shows a weaker acid with less dissociation.

Understanding \( K_a \) gives insight into how much a weak acid will dissociate in solution, which is crucial for calculating pH. In practical terms, you can use \( K_a \) along with the acid's concentration \( C \) to determine the concentration of \( H^+ \) ions, which is central to finding the pH.

Calculating Hydrogen Ion Concentration

For weak acids, calculating the hydrogen ion concentration involves both the dissociation constant, \( K_a \), and the initial concentration of the acid, \( C \). Here is the formula that connects these variables: \[ [\text{H}^+] = \sqrt{K_a \cdot C} \]
This formula allows you to find the \( [H^+] \) concentration, which is pivotal because the pH of a solution can be directly calculated from this value using the formula: \[ \text{pH} = -\log([\text{H}^+]) \]

• With a higher \( [H^+] \), the pH value decreases, indicating a more acidic solution.
• Conversely, a lower \( [H^+] \) results in a higher pH, indicating less acidity.

This step is where you truly see the connection between equilibrium chemistry and acidity. By understanding this relationship, you can predict how changes in \( K_a \) or the initial concentration will affect the pH and the acidity of the solution.

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Key Concepts

Understanding Weak Acid Dissociation

Weak acids, unlike strong acids, do not completely dissociate in solution. This partial dissociation means they exist in equilibrium with their ions in solution. You can imagine weak acid molecules like

• HA, which dissociates into
• \( H^+ \) ions and
• its conjugate base \( A^- \).

The equilibrium can be represented as: \[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \]
This concept is crucial when calculating the pH of a weak acid solution because the amount of dissociation directly affects the concentration of \( H^+ \) ions, contributing to the acidity, or pH, of the solution.
Weak acids are characterized by their lower dissociation compared to strong acids, so they reach this equilibrium without converting all of the original acid. This is why calculations involving weak acids focus more on the equilibrium expressions than on initial concentrations.

The Role of the Dissociation Constant

The dissociation constant, \( K_a \), is a numerical value that describes the strength of a weak acid in terms of its ability to donate protons (\( H^+ \) ions). The equation for a weak acid HA and its dissociation constant is: \[ K_a = \frac{{[\text{H}^+][\text{A}^-]}}{{[\text{HA}]}} \]
This equation reveals the relationship between the concentration of undissociated acid, HA, and its dissociated ions, \( H^+ \) and \( A^- \).

• A higher \( K_a \) value indicates a stronger acid with more dissociation occurring.
• A lower \( K_a \) value shows a weaker acid with less dissociation.

Understanding \( K_a \) gives insight into how much a weak acid will dissociate in solution, which is crucial for calculating pH. In practical terms, you can use \( K_a \) along with the acid's concentration \( C \) to determine the concentration of \( H^+ \) ions, which is central to finding the pH.

Calculating Hydrogen Ion Concentration

For weak acids, calculating the hydrogen ion concentration involves both the dissociation constant, \( K_a \), and the initial concentration of the acid, \( C \). Here is the formula that connects these variables: \[ [\text{H}^+] = \sqrt{K_a \cdot C} \]
This formula allows you to find the \( [H^+] \) concentration, which is pivotal because the pH of a solution can be directly calculated from this value using the formula: \[ \text{pH} = -\log([\text{H}^+]) \]

• With a higher \( [H^+] \), the pH value decreases, indicating a more acidic solution.
• Conversely, a lower \( [H^+] \) results in a higher pH, indicating less acidity.

This step is where you truly see the connection between equilibrium chemistry and acidity. By understanding this relationship, you can predict how changes in \( K_a \) or the initial concentration will affect the pH and the acidity of the solution.