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Arrange the following in the increasing order of their basic strength:
${C}_{2}{H}_{5}N{H}_{2}, {C}_{6}{H}_{5}N{H}_{2}, N{H}_{3}, {C}_{6}{H}_{5}C{H}_{2}N{H}_{2} and {({C}_{2}{H}_{5})}_{2}NH$.

Answer
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Hint: Basicity can be referred to as "electron pair instability", and this electron pair instability increases as the charge density on the compound increases. This means that the basicity should or does increase with the increased negative charge on the group. And this increased charge on the group is due to the presence of the electron releasing group attached to the main group.

Complete step by step answer:
Basicity of a compound can be defined as its ability to attract a pair of electrons.
To determine the basicity of a compound, we need to see what type of substituent group is attached to the compound (amine is this case).

If the substituent group attached is an electron withdrawing group (EWG), then they decrease the basicity of the compound due to their -I and -R effect. The more the number of EWG attached, the less will be the basicity of the compound. For example, $-N{O}_{2}$ group is an EWG due to its -R effect.

If the substituent group attached is an electron releasing group (ERG), then they increase the basicity of the compound due to their +I and +R effect. The more the number of ERG attached, the more will be the basicity of the compound. For example, $-C{H}_{3}$ group is an ERG due to its +I effect. This is because the +I effect increases the electron density on the compound and due to this increase, the basicity increases.

Now let us look at the compounds given to us.
i) In ${C}_{2}{H}_{5}N{H}_{2}$, the $-{C}_{2}{H}_{5}$ group shows +I effect and thus is more basic than ammonia itself due to the presence of a +I group. And due to the same reason ${({C}_{2}{H}_{5})}_{2}NH$ is more basic than ${C}_{2}{H}_{5}N{H}_{2}$ and $N{H}_{3}$.

ii) In ${C}_{6}{H}_{5}N{H}_{2}$, we have a benzene ring attached to the nitrogen atom of amine. The lone pair present on the nitrogen results in resonance around the benzene ring. Due to this resonance, the electron density around the nitrogen atom decreases and hence basicity also decreases. Aniline is the least basic compound as it shows resonance.

iii) In ${C}_{6}{H}_{5}C{H}_{2}N{H}_{2}$, the $-N{H}_{2}$ is directly attached to the $-C{H}_{2}$ group which is shows +I effect. But the +I effect of $-C{H}_{2}$ gets distributed among both the amine group and the benzene ring. So due to this distribution of the +I effect, there is a less increase in the basicity of the compound than ${C}_{2}{H}_{5}N{H}_{2}$. Therefore, ${C}_{6}{H}_{5}C{H}_{2}N{H}_{2}$ is more basic than ${C}_{6}{H}_{5}N{H}_{2}$ and $N{H}_{3}$ but is less basic than ${C}_{2}{H}_{5}N{H}_{2}$ and ${({C}_{2}{H}_{5})}_{2}NH$.

Hence the increasing order of the compounds according to their basic strength is as follows.
${C}_{6}{H}_{5}N{H}_{2}$ < $N{H}_{3}$ < ${C}_{6}{H}_{5}C{H}_{2}N{H}_{2}$ < ${C}_{2}{H}_{5}N{H}_{2}$ < ${({C}_{2}{H}_{5})}_{2}NH$

Note: The aryl amines are known to be less basic than the alkyl-substituted amines. This is the reason that the electron density which is provided by the nitrogen atom gets distributed throughout the aromatic ring due to resonance.