The basicity of amines depend upon its ability to donate the lone pair of electrons on nitrogen.
Order of basicity of amines in aqueous solution is,
$\ce{NH_3} <$ Primary amines $\ce{(RNH_2)} <$ Tertiary amines $\ce{(R_3N)} <$ Secondary amines $\ce{(R_2NH)}$
This is due to the combined effect of $+I$ effect of alkyl group, steric hindrance caused by alkyl groups and solvation of amines through $\ce{H} -$ bonding.
The power to donate lone pair of electrons increases as the number of alkyl groups increase.
Therefore, making $\ce{R_3N}$ most basic, followed by $\ce{R_2NH}$ and $\ce{RNH_2.}$
But as the number of alkyl groups increase, the steric hindrance caused by them also increases.
Thus, decreasing the electron donating power of amines.
Hence, making $\ce{RNH_2}$ most basic followed by $\ce{R_2NH}$ and $\ce{R_3N.}$
Solvation is the formation of protonated amines when they are dissolved in water.
As the number of $\ce{H}-$ atoms on nitrogen increases the possibility of $\ce{H}-$ bonding also increases, providing greater stability to the amine.
Thus, primary amine $\ce{(RNH_2)}$ is more stable than secondary amine $\ce{(R_2NH),}$ which is more stable than tertiary amine $\ce{(R_3N).}$
The combined effect of these three factors give the observed order of basicity of amines in aqueous solution.
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