Draw all geometrical and linkage isomers of \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{NO}_{2}\right)_{2}\).

The coordination number is the total number of ligands attached to the central atom. In this complex, there are 4 ammonia (NH3) molecules and 2 nitrite (NO2) ions attached to the cobalt atom (Co), giving a coordination number of 6. For complexes with a coordination number of 6, the typical geometric configuration is octahedral, where the six ligands are situated around the central atom in a symmetrical manner.

In an octahedral complex, the ligands at positions 90° from each other are adjacent (cis arrangement) and those at 180° from each other are opposite (trans arrangement). Let's find all possible cis and trans configurations of NH3 and NO2 ligands: 1. Cis arrangement: Place both NO2 ligands in adjacent positions (90°) with respect to each other. The remaining four positions will be occupied by NH3 ligands. There is only one possible cis arrangement: Co-ONO-NH3 | \ NH3-Co-NH3 | / NH3-NO2 2. Trans arrangement: Place both NO2 ligands in opposite positions (180°) with respect to each other. The remaining four positions will be occupied by NH3 ligands. There is only one possible trans arrangement: Co-ONO-NH3 | \ NH3-Co-NH3 | / NH3-ONO

The ligand nitrite (NO2) can bond to the central atom through either the nitrogen (N) or oxygen (O) atom, resulting in different linkage isomers. First, let's rewrite our geometrical isomers with the general NO2 formula. Cis arrangement: Co-NO2-NH3 | \ NH3-Co-NH3 | / NH3-NO2 Trans arrangement: Co-NO2-NH3 | \ NH3-Co-NH3 | / NH3-NO2 Now, let's find all possible linkage isomers for both the cis and trans arrangements: Cis arrangement: a) Both NO2 ligands connected through nitrogen (N): Co-N(=O)O-NH3 | \ NH3-Co-NH3 | / NH3-N(=O)O b) One NO2 connected through nitrogen (N), and one connected through oxygen (O): Co-N(=O)O-NH3 | \ NH3-Co-NH3 | / NH3-O=N-O c) Both NO2 ligands connected through oxygen (O): Co-O=N-O-NH3 | \ NH3-Co-NH3 | / NH3-O=N-O Trans arrangement: a) Both NO2 ligands connected through nitrogen (N): Co-N(=O)O-NH3 | \ NH3-Co-NH3 | / NH3-N(=O)O b) One NO2 connected through nitrogen (N), and one connected through oxygen (O): Co-N(=O)O-NH3 | \ NH3-Co-NH3 | / NH3-O=N-O c) Both NO2 ligands connected through oxygen (O): Co-O=N-O-NH3 | \ NH3-Co-NH3 | / NH3-O=N-O In conclusion, there are three cis linkage isomers and three trans linkage isomers for the complex \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{NO}_{2}\right)_{2}\), making a total of six possible isomers.