Step 1: Understanding the Question:
The question asks for the order of basicity of various classes of alkyl amines explicitly evaluated in the gaseous phase.
Step 2: Detailed Explanation:
The basicity of an amine is strictly determined by the availability and ease of donation of the lone pair of electrons on the central nitrogen atom.
When amines are evaluated in the gaseous phase (or in a completely non-polar solvent), there are no solvent interactions. Therefore, complex phenomena like hydrogen bonding, solvation enthalpy, and steric hindrance during solvation are entirely completely absent.
The only electronic effect dictating the basicity is the Inductive Effect (+I effect) of the attached alkyl groups (-R).
Alkyl groups are electron-donating. They push electron density through the sigma bonds toward the nitrogen atom. This massively increases the electron density on the nitrogen, making the lone pair much more repelled and much more eager to attack a proton.
Therefore, the more alkyl groups attached to the nitrogen, the stronger the +I effect, and the stronger the resulting base.
Ammonia ($\text{NH}_3$): Zero alkyl groups (weakest base).
Primary amine ($\text{R-NH}_2$): One alkyl group (+I effect from 1 source).
Secondary amine ($\text{R}_2\text{NH}$): Two alkyl groups (+I effect from 2 sources).
Tertiary amine ($\text{R}_3\text{N}$): Three alkyl groups (+I effect from 3 sources) (strongest base).
The strict, unbroken basic strength order in the gaseous phase is:
$\text{NH}_3 < 1^\circ \text{ amine} < 2^\circ \text{ amine} < 3^\circ \text{ amine}$.
Step 3: Final Answer:
The expected order is $NH_{3}<R-NH_{2}<R_{2}NH<R_{3}N$, matching option (b).