Analysis of statements regarding H\(_2\)O, NH\(_3\), and CH\(_4\):
- Statement (A): Central atoms in H\(_2\)O, NH\(_3\), and CH\(_4\) are sp\(^3\) hybridized.
- H\(_2\)O: Oxygen is sp\(^3\) hybridized, resulting in a bent geometry due to two lone pairs.
- NH\(_3\): Nitrogen is sp\(^3\) hybridized, resulting in a trigonal pyramidal geometry due to one lone pair.
- CH\(_4\): Carbon is sp\(^3\) hybridized, resulting in a tetrahedral geometry with no lone pairs.
- Statement (B): Bond angles H–O–H, H–N–H, and H–C–H are approximately 104.5°, 107.5°, and 109.5°, respectively.
- H\(_2\)O bond angle is ~104.5° due to lone pair-lone pair repulsion.
- NH\(_3\) bond angle is ~107.5° due to the influence of one lone pair.
- CH\(_4\) bond angle is 109.5° consistent with a perfect tetrahedral structure.
- Statement (C): Dipole moments increase in the order CH\(_4\)<NH\(_3\)<H\(_2\)O.
- CH\(_4\): Exhibits no net dipole moment due to its symmetrical tetrahedral structure.
- NH\(_3\): Possesses a dipole moment owing to its trigonal pyramidal shape.
- H\(_2\)O: Has the largest dipole moment due to its bent shape and strong hydrogen bonding interactions.
- Statement (D): H\(_2\)O and NH\(_3\) are Lewis acids, and CH\(_4\) is a Lewis base.
- Lewis Acid: An electron pair acceptor. H\(_2\)O and NH\(_3\) are not typically Lewis acids; they function as electron pair donors.
- Lewis Base: An electron pair donor. CH\(_4\) is not a Lewis base as it lacks available lone pairs for donation.
- Statement (E): An aqueous solution of NH\(_3\) is basic, with NH\(_3\) acting as a Lowry-Bronsted acid and H\(_2\)O as a Lowry-Bronsted base.
- In water, NH\(_3\) acts as a Bronsted-Lowry base by accepting a proton from H\(_2\)O.
- H\(_2\)O acts as a Bronsted-Lowry acid by donating a proton to NH\(_3\).
Following the evaluation of each statement, statements (A), (B), and (C) are determined to be true. Thus, the correct conclusion is: A, B, and C only.