The molecules $\mathrm{BF}_{3}, \mathrm{CF}_{4}, \mathrm{CO}_{2}, \mathrm{PF}_{5},\( and \)\mathrm{SF}_{6}$ are all nonpolar, even though they contain polar bonds. Why?

A polar molecule is a molecule in which there is an uneven distribution of electron density, leading to regions of partial positive and negative charges within the molecule. A nonpolar molecule, on the other hand, has an even distribution of electron density, and there are no regions with partial charges.

The shape of the molecule plays a crucial role in determining its polarity. In some cases, polar bonds within a molecule can "cancel out" if the molecule has a symmetric geometry. This results in a nonpolar molecule, even though it contains polar bonds.

Now, let us analyze the geometries of the given molecules to determine why they are nonpolar despite having polar bonds: 1. BF3: Boron trifluoride has a trigonal planar geometry, meaning the three F atoms are around the central B atom in a flat, triangular arrangement. This symmetry causes the polarity of the individual B-F bonds to cancel out, leaving the molecule as nonpolar. 2. CF4: Carbon tetrafluoride has a tetrahedral geometry, meaning the four F atoms are symmetrically distributed around the central C atom in a three-dimensional arrangement. This symmetric distribution causes the polarity of the individual C-F bonds to cancel out, leaving the molecule nonpolar. 3. CO2: Carbon dioxide has a linear geometry, meaning the two O atoms are located on opposite sides of the central C atom in a straight line. This symmetry causes the polarity of the individual C-O bonds to cancel out, leaving the molecule nonpolar. 4. PF5: Phosphorus pentafluoride has a trigonal bipyramidal geometry, meaning the five F atoms are symmetrically distributed around the central P atom in a three-dimensional arrangement. This symmetric distribution causes the polarity of the individual P-F bonds to cancel out, leaving the molecule nonpolar. 5. SF6: Sulfur hexafluoride has an octahedral geometry, meaning the six F atoms are symmetrically distributed around the central S atom in a three-dimensional arrangement. This symmetric distribution causes the polarity of the individual S-F bonds to cancel out, leaving the molecule nonpolar.

In conclusion, the molecules BF3, CF4, CO2, PF5, and SF6 are nonpolar despite containing polar bonds because their molecular geometries are symmetric, and this symmetry causes the polarity of the individual bonds to "cancel out."