Final answer:
The stronger base between CH3NH2 and (CH3)2NH is (CH3)2NH due to its greater inductive effect from two methyl groups, making its conjugate acid (CH3)2NH₂+ the weaker acid compared to CH3NH3+.
Step-by-step explanation:
The comparison between CH3NH2 (methylamine) and (CH3)2NH (dimethylamine) as bases and their respective conjugate acids (CH3)2NH₂+ and CH3NH3+ involves understanding their strength in accepting protons. Methylamine is somewhat a weaker base compared to dimethylamine because it has only one methyl group that can donate electron density to the nitrogen, compared to dimethylamine's two methyl groups which provide a greater inductive effect and hence, more stabilization of the conjugate acid.
The conjugate acid of the stronger base, in this case, (CH3)2NH₂+, is weaker because a strong base has a weaker tendency to accept a proton back and hence, a weaker conjugate acid. Conversely, CH3NH3+, being the conjugate acid of the weaker base CH3NH2, is comparatively the stronger acid as it has a stronger tendency to donate a proton.