(a) The \(\mathrm{P}_{4}, \mathrm{P}_{4} \mathrm{O}_{6}\) and $\mathrm{P}_{4} \mathrm{O}_{10}$ molecules have a common structural feature of four \(\mathrm{P}\) atoms arranged in a tetrahedron (Figures 22.27 and 22.28 ). Does this mean that the bonding between the \(\mathrm{P}\) atoms is the same in all these cases? Explain. (b) Sodium trimetaphosphate $\left(\mathrm{Na}_{3} \mathrm{P}_{3} \mathrm{O}_{9}\right)$ and sodium tetrametaphosphate \(\left(\mathrm{Na}_{4} \mathrm{P}_{4} \mathrm{O}_{12}\right)\) are used as water-softening agents. They contain cyclic $\mathrm{P}_{3} \mathrm{O}_{9}^{3-}\( and \)\mathrm{P}_{4} \mathrm{O}_{12}{ }^{4-}$ ions, respectively. Propose reasonable structures for these ions.

In order to determine the bonding between the P atoms in each molecule, let us consider the following: 1. P4: It is a tetrahedron with four P atoms. Each P atom forms single bonds with the other three P atoms. 2. P4O6: This molecule consists of a P4 tetrahedron, with two oxygen atoms bonded to each P atom, forming P=O bonds. Each P atom still forms single bonds with the other three P atoms. 3. P4O10: In this molecule, each P atom in the P4 tetrahedron is bonded to five other atoms: three P atoms and two oxygen atoms. The bonding between P atoms becomes weaker due to the presence of double bonds with oxygen atoms. The bonding between the P atoms in P4, P4O6, and P4O10 is not the same in all cases due to the differences in the bonding with the oxygen atoms.

To propose reasonable structures for the cyclic P3O9^3- and P4O12^4- ions, we need to consider their bonding. 1. P3O9^3-: In this ion, we have three P atoms and nine O atoms. Each P atom can form double bonds with two O atoms (P=O), resulting in six O atoms. This leaves three O atoms, which can connect the three P atoms, forming a cyclic structure with P-O-P single bonds. Structure for P3O9^3- ion: A cyclic structure with three P=O double bonds and three P-O-P single bonds. 2. P4O12^4-: In this ion, we have four P atoms and twelve O atoms. Similar to the P3O9^3- ion, each P atom can form double bonds with two O atoms (P=O), resulting in eight O atoms. This leaves four more O atoms, which can connect the four P atoms, forming a cyclic structure with P-O-P single bonds. Structure for P4O12^4- ion: A cyclic structure with four P=O double bonds and four P-O-P single bonds.