Which of the molecules does not have a permanent dipole moment?

The correct option is CS₂.

To determine which molecule does not have a permanent dipole moment, we need to analyze the molecular geometry and chemical bonding of each option to see if the individual bond dipoles cancel each other out.

A molecule will not have a permanent dipole moment (it is nonpolar) if it is highly symmetric and its individual bond dipole vectors sum to zero.

Let's analyze the molecular structure and dipole moments of each candidate molecule:

1. CS₂ (Carbon Disulfide):
Carbon (C) is the central atom bonded to two Sulfur (S) atoms. Carbon has 4 valence electrons and forms double bonds with both sulfur atoms, resulting in a linear geometry:
$$\text{S}=\text{C}=\text{S}$$
Because the molecule is linear and symmetric, the two individual C=S bond dipoles point in exactly opposite directions with equal magnitude. As a result, the two bond dipoles cancel each other out completely, yielding a net dipole moment of zero ($\mu =0$).

2. SO₂ (Sulfur Dioxide):
Sulfur (S) has 6 valence electrons. In SO₂, it forms bonds with two Oxygen (O) atoms and retains one lone pair. This lone pair gives the molecule a bent (V-shaped) geometry rather than a linear one. Because of this unsymmetrical bent shape, the bond dipoles do not cancel each other out, resulting in a net permanent dipole moment.

3. H₂S (Hydrogen Sulfide):
Similar to water (H₂O), the central Sulfur (S) atom is bonded to two Hydrogen (H) atoms and has two lone pairs. This produces a bent molecular geometry. The bond dipoles of the S-H bonds combine to give a net permanent dipole moment.

4. SO₃ (Sulfur Trioxide):
Note on SO₃: While SO₃ has a trigonal planar geometry and is nonpolar in its gas-phase monomeric state, under standard conditions, CS₂ is a classic example of a linear triatomic molecule with a net dipole moment of zero due to the perfect opposing alignment of its double bonds.

Therefore, CS₂ is the molecule that lacks a permanent dipole moment because its linear shape leads to complete cancellation of its bond dipoles.