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$R-X+\ddot{N}{{H}_{3}}\to HX+RN{{H}_{2}}\xrightarrow[373K]{R-X}HX+{{R}_{2}}NH\xrightarrow[373K]{R-X}{{R}_{3}}N+HX\xrightarrow[373K]{R-X}{{R}_{4}}{{N}^{+}}{{X}^{-}}$

(a)- ${{1}^{\circ }}>{{2}^{\circ }}>{{3}^{\circ }}>N{{H}_{3}}$

(b)- ${{2}^{\circ }}>{{3}^{\circ }}>{{1}^{\circ }}>N{{H}_{3}}$

(c)- ${{2}^{\circ }}>{{1}^{\circ }}>{{3}^{\circ }}>N{{H}_{3}}$

(d)- ${{2}^{\circ }}>{{1}^{\circ }}>N{{H}_{3}}>{{3}^{\circ }}$

Answer
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The basicity of an amine in an aqueous solution primarily depends upon the stability of the ammonium cation or the conjugate acid formed by accepting a proton from water. The stability of the ammonium cation depends upon a combination of the following three factors:

(i)- + I-Effect of the alkyl group

(ii)- Extend of H-bonding with water molecules

(iii)- Steric effects of the alkyl groups.

The stability of the ammonium cation due to H-bonding depends upon the number of H-atoms present on the N-atom. So, the ${{1}^{\circ }}$ will form three hydrogen bonding because of three H atoms, ${{2}^{\circ }}$ will for two hydrogen bonding because of two hydrogen atom, and ${{3}^{\circ }}$ will form one hydrogen bonding because of one hydrogen atom.

However, in the case of ammonium cation derived from tertiary amines, there is some steric repulsion to H-bonding, and hence the stability further decreases.

From the above discussion, we can conclude that it is a combination of +I-Effect, H-bonding, and steric effects are factors for determining the stability of ammonium cation. All these factors are favorable for ${{2}^{\circ }}$ amine and hence the ${{2}^{\circ }}$amine is the most stable and the strongest base. If the alkyl group is small like $C{{H}_{3}}$ then there is no steric hindrance to H-bonding, hence the stability due to hydrogen bonding predominates over the stability due to +I-Effect of the $C{{H}_{3}}$ group hence ${{1}^{\circ }}$ is a stronger base than the ${{3}^{\circ }}$.

Hence the stability order or basicity order is:

${{2}^{\circ }}>{{1}^{\circ }}>{{3}^{\circ }}>N{{H}_{3}}$

${{2}^{\circ }}>{{3}^{\circ }}>{{1}^{\circ }}>N{{H}_{3}}$

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