Step 1: Understanding basic strength.
The basic strength of a molecule is determined by the availability of the lone pair of electrons on the nitrogen atom, which can accept a proton (H+). The more available the lone pair is, the stronger the base. Alkyl groups attached to the nitrogen atom donate electron density, increasing the electron density on nitrogen and enhancing its basicity. However, as more alkyl groups are added, the electron density is more effectively delocalized, reducing the availability of the lone pair for protonation, which decreases the basic strength.
Step 2: Analysis of the options.
- (A) \( \text{NH}_3>\text{CH}_3\text{NH}_2>(\text{CH}_3)_2\text{NH>(\text{CH}_3)_3\text{N} \):} Correct. Ammonia (\( \text{NH}_3 \)) is the strongest base because it has the most available lone pair of electrons. As we add methyl groups (which are electron-donating), the basicity increases until we reach methylamine (\( \text{CH}_3\text{NH}_2 \)). However, as we add more methyl groups (\( (\text{CH}_3)_2\text{NH} \) and \( (\text{CH}_3)_3\text{N} \)), the basicity decreases due to reduced lone pair availability from nitrogen.
- (B) \( (\text{CH}_3)\text{NH_2>\text{CH}_3\text{NH}_2>(\text{CH}_3)_2\text{NH}>\text{NH}_3 \):} Incorrect. The basicity order is incorrect because ammonia (\( \text{NH}_3 \)) is stronger than methylamine (\( \text{CH}_3\text{NH}_2 \)).
- (C) \( \text{CH}_3\text{NH}_2>(\text{CH}_3)_2\text{NH>(\text{CH}_3)\text{NH}_2>\text{NH}_3 \):} Incorrect. This is incorrect because the order for methylamine and dimethylamine is reversed, and ammonia (\( \text{NH}_3 \)) is the strongest base.
Step 3: Conclusion.
The correct order of basicity is given by option (A): \( \text{NH}_3>\text{CH}_3\text{NH}_2>(\text{CH}_3)_2\text{NH}>(\text{CH}_3)_3\text{N} \).
\[
\boxed{\text{NH}_3>\text{CH}_3\text{NH}_2>(\text{CH}_3)_2\text{NH}>(\text{CH}_3)_3\text{N}}
\]
Final Answer: \( \text{NH}_3>\text{CH}_3\text{NH}_2>(\text{CH}_3)_2\text{NH}>(\text{CH}_3)_3\text{N} \)