Step 1: Understanding the Concept:
The basic strength of amines in aqueous solution is determined by a combination of three factors:
- Inductive effect: $+I$ effect of alkyl groups increases electron density on Nitrogen.
- Solvation effect: Extent of hydrogen bonding with water (decreases from $1^\circ$ to $3^\circ$).
- Steric hindrance: Bulkiness of alkyl groups blocking the approach of protons.
Step 2: Detailed Explanation:
1. For methyl substituted amines in water, the cumulative effect of these factors results in a specific order.
2. Secondary amine $(CH_3)_2NH$ is always the most basic because it has a good balance of $+I$ effect and solvation.
3. Primary amine $CH_3NH_2$ is more basic than tertiary $(CH_3)_3N$ because, although the tertiary has more $+I$ effect, it is severely hindered and poorly solvated in water.
4. $NH_3$ has no alkyl groups to provide $+I$ effect, making it the weakest base.
The order is: $2^\circ > 1^\circ > 3^\circ > NH_3$.
Step 3: Final Answer:
The order is $(CH_3)_2NH > CH_3NH_2 > (CH_3)_3N > NH_3$.