Which of the following is associated with decrease in pK_b value of amines? 

Increase in basic strength

The correct answer is option 2. Increase in basic strength.

To understand why an increase in basic strength is associated with a decrease in the \(pK_b\) value of amines, let's delve into the concept of pKb and how it relates to basicity.

 \(pK_b\) and Basicity:

\(pK_b\): The pKb of an amine is defined as the negative logarithm of its base dissociation constant \((K_b)\). It is a measure of the basic strength of the amine. Lower \(pK_b\) values indicate stronger bases (higher basicity).

Base Dissociation Constant \((K_b)\): This constant describes the equilibrium constant for the dissociation of the amine in water to form the conjugate acid and hydroxide ion:

\(\text{Amine} + \text{H}_2\text{O} \rightleftharpoons \text{Amine-H}^+ + \text{OH}^- \)

The \(K_b\) is given by:

\(\text{Kb} = \frac{[\text{Amine-H}^+][\text{OH}^-]}{[\text{Amine}]} \)

Relationship Between Basic Strength and \(pK_b\):

Basic Strength: Refers to the ability of an amine to accept a proton \((H^+)\) from water to form the conjugate acid (Amine-\(H^+\)).

Lower \(pK_b\): Indicates a stronger base because it means that the equilibrium lies more towards the formation of the conjugate acid (Amine-\(H^+\)) and hydroxide ion \((OH^-)\). This implies that the amine more readily accepts protons from water.

Factors Affecting Basic Strength and \(pK_b\):

Number of Alkyl Groups: Amines with more alkyl groups attached to the nitrogen are generally more basic. This is because alkyl groups donate electron density through induction and hyperconjugation, stabilizing the negative charge on the nitrogen and making it more available to accept a proton.

Resonance Effects:

Aromatic amines (like aniline) have lower basicity compared to alkyl amines due to resonance delocalization of the lone pair of electrons on the nitrogen into the aromatic ring. This reduces the availability of the lone pair for protonation, increasing the \(pK_b\) value.

Electron-Donating Groups:

Substituents that donate electron density to the nitrogen (such as alkyl groups) increase the basic strength and decrease the \(pK_b\)value.

Electron-Withdrawing Groups: 

Substituents that withdraw electron density from the nitrogen (like nitro groups) decrease the basic strength and increase the \(pK_b\) value.

Why Increase in Basic Strength Decreases :

Proton Acceptance: A stronger base (higher basic strength) indicates that the amine readily accepts protons from water to form the conjugate acid. This equilibrium shift towards the conjugate acid formation corresponds to a lower \(pK_b\) value because more of the amine molecules are protonated in solution.

Acid-Base Equilibrium: The lower the \(pK_b\), the stronger the base, because it reflects a higher concentration of the amine in its basic form relative to its conjugate acid form in aqueous solution.

Conclusion:

An increase in basic strength of amines leads to a decrease in their pKb values. This is because stronger bases more effectively accept protons \((H^+)\) in aqueous solution, resulting in a greater proportion of the amine existing in its basic form. Therefore, when comparing the options provided, an increase in basic strength is directly associated with a decrease in the \(pK_b\)value of amines.