The basic strength of amines in aqueous solution depends on two main factors: the electron-donating inductive effect of alkyl groups and the stability of the conjugate acid formed after protonation. A smaller pK_b value indicates a stronger base.

Step 1: Understand pK_b and basic strength
pK_b is the negative logarithm of the base dissociation constant (K_b). A smaller pK_b means a larger K_b, which means the base is stronger.

Step 2: Compare alkyl amines
Alkyl groups are electron-donating, which increases the electron density on the nitrogen atom, making it a stronger base. Among simple alkyl amines:

• Primary amine (CH₃NH₂) has one methyl group.
• Secondary amine ((CH₃)₂NH) has two methyl groups, so more electron-donating effect.
• Tertiary amine ((CH₃)₃N) has three methyl groups, but in aqueous solution, steric hindrance and solvation effects reduce its basicity compared to secondary amines.

The order of basic strength in water is: secondary > primary > tertiary.

Step 3: Consider aromatic amine
Aniline (C₆H₅NH₂) has a phenyl group attached to the nitrogen. The phenyl group is electron-withdrawing due to resonance, which decreases the electron density on nitrogen, making it a very weak base. It has a much larger pK_b compared to alkyl amines.

Step 4: Final comparison
Among the given options:

• (CH₃)₃N: tertiary amine, weaker due to steric and solvation effects.
• C₆H₅NH₂: aromatic amine, very weak base.
• CH₃NH₂: primary amine, moderately strong.
• (CH₃)₂NH: secondary amine, strongest among alkyl amines in water.

Therefore, (CH₃)₂NH has the smallest pK_b value.

Related Topics and Formulae

Basic Strength and pK_b:
The base dissociation constant K_b for an amine B is given by: $B+H_{2}O\rightleftharpoons {\mathrm{BH}}^{+}+{\mathrm{OH}}^{-},K_b=\frac{\left[{\mathrm{BH}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]}{\left[B\right]}$ pK_b = -log₁₀(K_b). A smaller pK_b means a stronger base.

Factors Affecting Basicity:
- Inductive Effect: Electron-donating groups (like alkyl groups) increase basicity; electron-withdrawing groups (like phenyl) decrease it.
- Solvation Effect: In water, the conjugate acid of primary amines is more solvated than tertiary amines due to more hydrogen bonding.
- Steric Hindrance: Bulky groups around nitrogen make it harder to protonate, reducing basicity.

Order in Aqueous Solution:
For alkyl amines: (CH₃)₂NH > CH₃NH₂ > (CH₃)₃N > NH₃
Aromatic amines are much weaker: C₆H₅NH₂ has pK_b ~ 9.38, while (CH₃)₂NH has pK_b ~ 3.27.