Question

The \(\mathrm{p} K_{\mathrm{s}}\) of a weak acid \(\mathrm{HA}\) is greater than the \(\mathrm{p} K_{\mathrm{b}}\) value of a weak base \(\mathrm{BOH}\). An aqueous solution of the salt \(\mathrm{AB}\) formed by the neutralization of this acid by the base will be (1) neutral (2) basic (3) alkaline (4) acidic if the solution is dilute

Short answer

Option 2: basic

Step-by-step solution

Understanding the problem

The problem involves comparing the dissociation constants of a weak acid and a weak base to determine the nature of the solution formed from their salt.

Define \(\mathrm{p} K_{\mathrm{s}}\) and \(\mathrm{p} K_{\mathrm{b}}\)

\(\mathrm{p} K_{\mathrm{s}}\) is the negative logarithm of the acid dissociation constant \(K_{\mathrm{a}}\) of the weak acid \(\mathrm{HA}\), and \(\mathrm{p} K_{\mathrm{b}}\) is the negative logarithm of the base dissociation constant \(K_{\mathrm{b}}\) of the weak base \(\mathrm{BOH}\).

Relationship between \( \mathrm{p} K_{\mathrm{s}}\) and \( \mathrm{p} K_{\mathrm{b}}\)

Given that \(\mathrm{p} K_{\mathrm{s}}\) of \(\mathrm{HA}\) is greater than \(\mathrm{p} K_{\mathrm{b}}\) of \(\mathrm{BOH}\). This means \(\mathrm{HA}\) is a weaker acid compared to the strength of the base \(\mathrm{BOH}\).

Nature of salt \(\mathrm{AB}\)

Salt \(\mathrm{AB}\) is formed by neutralization of \(\mathrm{HA}\) and \(\mathrm{BOH}\). Since the base \(\mathrm{BOH}\) is stronger than the acid \(\mathrm{HA}\), the resulting solution of the salt will be basic in nature.

Conclusion

From the given options, the solution of the salt \(\mathrm{AB}\) will be basic, as the base is stronger than the acid.

Key concepts

acid-base equilibria

When dealing with acids and bases, one of the key concepts to understand is acid-base equilibria. It involves the balance between acids and bases in a solution. An acid like \(HA\) donates hydrogen ions (protons), while a base like \(BOH\) accepts them. The strength of an acid or base is measured by how completely it dissociates in water.

For a weak acid and a weak base, they only partially dissociate. This partial dissociation is described by equilibrium constants: \(K_a\) for acids and \(K_b\) for bases. These constants tell us how well the acid or base can donate or accept protons, respectively.

The negative logarithms of these constants, \(pK_a\) and \(pK_b\), give a more straightforward numerical value to understand and compare. The problem mentioned above uses these \(pK_a\) and \(pK_b\) values to determine the nature of the solution.

dissociation constants

Dissociation constants are fundamental to understanding how acids and bases behave in water. For a weak acid \(HA\), it partially dissociates in water, establishing an equilibrium: \[ HA (aq) \leftrightarrow H^+ (aq) + A^- (aq). \] The equilibrium constant \(K_a\) expresses this behavior mathematically: \[ K_a = \frac{[H^+][A^-]}{[HA]} \]

Similarly, for a weak base \(BOH\), the equilibrium is: \[ BOH (aq) \leftrightarrow B^+ (aq) + OH^- (aq). \] And its constant \(K_b\) is: \[ K_b = \frac{[B^+][OH^-]}{[BOH]} \] A key relationship to remember is that the relative strengths of the acid and base can influence the final pH of the solution formed by their salt. The lower the \(pK_a\), the stronger the acid. The lower the \(pK_b\), the stronger the base.

In the exercise, \(pK_s\) of \(HA\) being greater than \(pK_b\) means \(HA\) is a weaker acid compared to \(BOH\) being a stronger base.

neutralization reactions

Neutralization reactions involve the reaction between an acid and a base to produce a salt and often water. For instance, when \(HA\) (a weak acid) reacts with \(BOH\) (a weak base), it forms the salt \(AB\) and water: \[ HA + BOH \rightarrow AB + H_2O. \]

The nature of the resulting solution depends on the relative strengths of the acid and base. In this case, since \(HA\) is a weaker acid (higher \(pK_s\)) and \(BOH\) is a stronger base (lower \(pK_b\)), the resultant salt solution \(AB\) will be basic. This is because the conjugate base of the weak acid and the conjugate acid of the weak base will affect the pH of the solution.

Since \(BOH\)'s conjugate acid is stronger, it will be more effective in maintaining a basic environment in the solution.