Give reasons \[{\left( {C{H_3}} \right)_2}NH\] is more basic than $\left( {C{H_3}} \right)N$ in an aqueous solution.

Hint: The organic molecules can be differentiated according to the basicity based on the character of basicity. The basic character that needs to be observed here is the ability to accept protons and stability level of the organic molecule.

Complete answer:
The structure of \[{\left( {C{H_3}} \right)_2}NH\] is similar known as the ${2^o}$ amine salt structure, as in the internal $N$ group there are two molecules of $C{H_3}$ associated. Therefore, this molecule forms the ionic structure of the salt where there is excess electron on the nitrogen residue. The $\left( {C{H_3}} \right)N$ is known as ${3^o}$ carbon as there are three molecules of $C{H_3}$ associated with the internal $N$ residue. There is an inductive effect which $ + I$ caused as a result of $C{H_3}$ residues present in the molecule.
The less inductive character means more stability as the $N$ residue is itself electronegative pulling the electron towards itself. The basic nature depends on the ability to accept hydrogen residues with the given molecules. The higher degree of hydration is possible in ${2^o}$ amine salt and the resultant molecule is more stable as compared to that of the ${3^o}$ amine salts when the hydration takes place.
In the presence of an aqueous solution, the hydration of the organic molecules can take place as there is presence of free protons. The stability is because even after the hydration of $\left( {C{H_3}} \right)N$ there is no stability because of a positive charge on the $N$ residue. But the hydrogenation of the other organic molecule offers more stability in that level.
Higher stability of the salt means that the reactivity would be less in the stable salt as compared to that of the other amine salt. This proves that the molecule \[{\left( {C{H_3}} \right)_2}NH\] has more basicity as compared to that of $\left( {C{H_3}} \right)N$ based on the character for accepting protons.

Note: The level of stability is important in the comparison and the inductive effect plays an important role in this process. The resulting electron density is important for the stability of the organic molecule and the formation of charge also takes part in the same process.