(a) What conditions must be met if a molecule with polar bonds is nonpolar? (b) What geometries will signify nonpolar molecules for \(\mathrm{AB}_{2}, \mathrm{AB}_{3},\) and \(\mathrm{AB}_{4}\) geometries?

Polarity refers to the separation of electric charges within a molecule, leading to a net electric dipole moment. A molecule can have polar bonds, meaning that the atoms in those bonds have a notable difference in electronegativity – the ability of an atom to attract electrons in a bond towards itself. However, a molecule itself can still be nonpolar if the net electric dipole moment is zero because the polar bonds are arranged symmetrically, canceling out each other's effects.

A molecule with polar bonds can be nonpolar if the following conditions are met: 1. The central atom has no lone pairs of electrons. 2. The surrounding atoms are identical or have the same electronegativity. 3. The molecule has a symmetrical geometry, leading to the cancellation of the net electric dipole moment arising from the polar bonds. It's important to note that if these conditions are not satisfied, then the molecule will be polar.

Now we will discuss different geometries of molecules with the general formulas AB₂, AB₃, and AB₄, and determine if they can form nonpolar molecules given the presence of polar bonds. 1. AB₂: The simplest geometry for AB₂ is linear, where the two B atoms are opposite each other (180° angle) with respect to the central A atom. With this configuration, the polar bonds' dipoles point in opposite directions and cancel each other, resulting in a nonpolar molecule. 2. AB₃: One possible geometry for AB₃ is trigonal planar, with bond angles of 120°. In this case, if all three B atoms are the same and no lone pairs are present on the central atom A, the dipoles of each polar bond would cancel out due to symmetry, and the molecule would be nonpolar. Another possible geometry is trigonal pyramidal, which will always result in a polar molecule. 3. AB₄: For AB₄, the geometry can either be tetrahedral or square planar. In the case of a tetrahedral geometry, if all four B atoms are the same and no lone pairs are present on the central atom A, the polar bonds' dipoles would cancel out each other due to symmetry, making the molecule nonpolar. In contrast, a molecule with square planar geometry will always result in a polar molecule. In conclusion, molecules with the general formulas AB₂, AB₃, and AB₄ can be nonpolar with the following geometries, provided that the polar bonds' dipoles cancel out due to symmetry: linear for AB₂, trigonal planar for AB₃, and tetrahedral for AB₄. However, other possible geometries, such as trigonal pyramidal and square planar, will result in polar molecules.