Step 1: Understanding the Concept:
A dipole moment \( (\mu) \) occurs when there is a separation of charge within a molecule due to differences in electronegativity between bonded atoms.
The dipole moment of a whole molecule is the vector sum of all individual bond dipole moments.
Even if individual bonds are polar, a molecule can have a zero net dipole moment if its geometry is perfectly symmetrical, causing the vectors to cancel each other out.
Molecules without lone pairs on the central atom and identical surrounding atoms are typical candidates for zero dipole moment.
Step 2: Key Formula or Approach:
We evaluate the geometry of each molecule:
- Linear, Trigonal Planar, and Tetrahedral shapes with identical peripheral atoms result in \( \mu = 0 \).
- Bent or Pyramidal shapes (usually containing lone pairs) lead to \( \mu \neq 0 \).
Step 3: Detailed Explanation:
Let's analyze the molecules in option (d):
1. \( CH_4 \) (Methane): Carbon is sp\(^3\) hybridized. It has 4 bond pairs and 0 lone pairs. The geometry is tetrahedral. Because all four C-H bonds are identical and arranged symmetrically, the bond dipoles cancel completely. Hence, \( \mu = 0 \).
2. \( BF_3 \) (Boron Trifluoride): Boron is sp\(^2\) hybridized. It has 3 bond pairs and 0 lone pairs. The geometry is trigonal planar. The three polar B-F bonds are at \( 120^\circ \) angles to each other. The vector sum of these three equal dipoles is zero. Hence, \( \mu = 0 \).
3. \( CO_2 \) (Carbon Dioxide): Carbon is sp hybridized. It forms two double bonds with Oxygen. The molecule is linear (\( O=C=O \)). The two C=O bond dipoles are equal and opposite, cancelling each other out. Hence, \( \mu = 0 \).
Now, let's see why others fail:
- \( NH_3, NF_3, PF_3 \): These have a pyramidal geometry due to a lone pair on the central atom. The bond dipoles do not cancel.
- \( H_2O \): It has a bent (V-shaped) geometry due to two lone pairs. The O-H dipoles do not cancel.
Since only set (d) contains exclusively non-polar molecules, it is the correct answer.
Step 4: Final Answer:
The molecules with zero dipole moment are \( CH_4, BF_3, \) and \( CO_2 \).
This is Option (d).