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

In an octahedral complex with the general formula \(\mathrm{MA_4B_2}\), there are two possibilities for geometrical isomers: cis and trans arrangements. The cis isomer has the two B ligands adjacent to each other, whereas the trans isomer has the two B ligands opposite each other.

Now, let's draw the cis and trans isomer of the complex \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{NO}_{2}\right)_{2}\). Cis isomer: In this arrangement, the two \(\mathrm{NO}_2\) ligands are adjacent to each other. \( \require{AMScd} \begin{CD} \ @. \mathrm{NO}_2 @. \mathrm{NO}_2 \\ @. \ | \\ \mathrm{NH}_3 - \mathrm{Co} @{-}@ _{- \mathrm{NH}_3} \\ @. \ | \\ @. \mathrm{NH}_3 @. \mathrm{NH}_3 \end{CD} \) Trans isomer: In this arrangement, the two \(\mathrm{NO}_2\) ligands are opposite to each other. \( \require{AMScd} \begin{CD} \ @. \mathrm{NH}_3 @. \mathrm{NH}_3 \\ @. \ | \\ \mathrm{NO}_2 - \mathrm{Co} @{-}@ _{- \mathrm{NH}_3} \\ @. \ | \\ @. \mathrm{NH}_3 @. \mathrm{NO}_2 \end{CD} \)

In the given complex, only the \(\mathrm{NO}_2\) ligand can exhibit linkage isomerism, as it can bind to the cobalt ion through either the nitrogen (N) or the oxygen (O) atom.

Now, let's draw the linkage isomers for each geometrical isomer. Cis isomers: 1. Both \(\mathrm{NO}_2\) ligands bind through nitrogen (cis-NN isomer) 2. One \(\mathrm{NO}_2\) ligand binds through nitrogen, and the other binds through oxygen (cis-NO isomer) \( \require{AMScd} \begin{CD} \ @. \mathrm{N}<- \mathrm{O}_2 @. \mathrm{N}<- \mathrm{O}_2 \\ @. \ | \\ \mathrm{NH}_3 - \mathrm{Co} @{-}@ _{- \mathrm{NH}_3} \\ @. \ | \\ @. \mathrm{NH}_3 @. \mathrm{NH}_3 \end{CD} \qquad \require{AMScd} \begin{CD} \ @. \mathrm{N}<- \mathrm{O}_2 @. \mathrm{O}-> \mathrm{N}<- \mathrm{O} \\ @. \ | \\ \mathrm{NH}_3 - \mathrm{Co} @{-}@ _{- \mathrm{NH}_3} \\ @. \ | \\ @. \mathrm{NH}_3 @. \mathrm{NH}_3 \end{CD} \) Trans isomers: 1. Both \(\mathrm{NO}_2\) ligands bind through nitrogen (trans-NN isomer) 2. One \(\mathrm{NO}_2\) ligand binds through nitrogen, and the other binds through oxygen (trans-NO isomer) \( \require{AMScd} \begin{CD} \ @. \mathrm{NH}_3 @. \mathrm{NH}_3 \\ @. \ | \\ \mathrm{N}<- \mathrm{O}_2 - \mathrm{Co} @{-}@ _{- \mathrm{NH}_3} \\ @. \ | \\ @. \mathrm{NH}_3 @. \mathrm{N}<- \mathrm{O}_2 \end{CD} \qquad \require{AMScd} \begin{CD} \ @. \mathrm{NH}_3 @. \mathrm{NH}_3 \\ @. \ | \\ \mathrm{N}<- \mathrm{O}_2 - \mathrm{Co} @{-}@ _{- \mathrm{NH}_3} \\ @. \ | \\ @. \mathrm{NH}_3 @. \mathrm{O}-> \mathrm{N}<- \mathrm{O} \end{CD} \) In total, there are four possible isomers for the complex \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{NO}_{2}\right)_{2}\): cis-NN, cis-NO, trans-NN, and trans-NO.