Considering the basic strength of amines in an aqueous solution, which one has the smallest pK$$_b$$ value?

We begin by recalling the mathematical link between the equilibrium constant for basicity and the tabulated quantity pKb.

The definition is $$pK_b=-\log K_b\;,$$ so a smaller numerical value of $$pK_b$$ corresponds to a larger $$K_b$$ and therefore to a stronger base in water.

Now we examine the four amines.

First, let us write each structure clearly and note the factors that influence its basic strength:

1. Option A : $$(CH_3)_2NH$$ (dimethylamine, a secondary amine)
2. Option B : $$CH_3NH_2$$ (methylamine, a primary amine)
3. Option C : $$(CH_3)_3N$$ (trimethylamine, a tertiary amine)
4. Option D : $$C_6H_5NH_2$$ (aniline, an aryl primary amine)

Three main effects control basic strength in water:

(i) The +I (electron-donating inductive) effect of alkyl groups increases electron density on the nitrogen atom and makes proton acceptance easier.
(ii) Steric hindrance around the nitrogen may make solvation of the conjugate acid difficult; poorer solvation lowers the effective basicity in aqueous solution.
(iii) Resonance, if it withdraws the lone pair from nitrogen (as in aniline), decreases basicity.

Applying these points one by one:

• In aniline $$C_6H_5NH_2$$, the lone pair on nitrogen is partially delocalised into the benzene ring through resonance. This sharply decreases the availability of the pair for protonation, so aniline is the weakest base among the four.

• Between the three aliphatic amines, the inductive effect grows when we attach more methyl groups: two methyl groups in dimethylamine push more electron density than one in methylamine, and three in trimethylamine push the most.

• However, after protonation, the cation $$\left((CH_3)_3NH\right)^+$$ of trimethylamine is bulky. Water molecules find it hard to surround and stabilise this ion. Because solvation is poor, the net basicity of trimethylamine is lower than what its inductive effect alone would suggest.

• Dimethylamine $$ (CH_3)_2NH $$ strikes an optimal balance: it enjoys a strong +I effect from two methyl groups and, being less crowded than the tertiary cation, it is still very well solvated. Consequently its equilibrium constant $$K_b$$ is the largest, and hence its $$pK_b$$ is the smallest.

• Methylamine $$CH_3NH_2$$ has only one +I methyl group, so its electron donation and basicity are less than that of dimethylamine.

Putting the arguments together, the basic strength order in aqueous solution is

$$ (CH_3)_2NH \;>\; CH_3NH_2 \;>\; (CH_3)_3N \;>\; C_6H_5NH_2. $$

Translating this into pKb values (remember, smaller means stronger), we get

$$ pK_b\big((CH_3)_2NH\big) \;<\; pK_b\big(CH_3NH_2\big) \;<\; pK_b\big((CH_3)_3N\big) \;<\; pK_b\big(C_6H_5NH_2\big). $$

Therefore, the amine with the smallest pKb value is dimethylamine, shown in the list as $$(CH_3)NH$$ (Option A).

Hence, the correct answer is Option A.