[Co$$_2$$(CO)$$_8$$] displays

We first recall the 18-electron rule, which states that a stable, low-spin complex of a transition metal usually adjusts its bonding so that each metal centre possesses a total of $$18$$ valence electrons. The electrons come from the metal itself, from σ-donor ligands such as carbonyls ($$CO$$) and, when present, from metal-metal ($$M{-}M$$) bonds. For counting purposes:

• A neutral cobalt atom contributes its nine valence electrons, so $$Co$$ gives $$9$$ electrons.

• Every terminal $$CO$$ ligand donates a pair of electrons to the metal to which it is attached, so each terminal $$CO$$ contributes $$2$$ electrons to that metal.

• A bridging $$CO$$ is shared between two metals. It donates one electron to each metal centre, giving a total of $$2$$ electrons, one to each cobalt.

• A single metal-metal bond ($$Co{-}Co$$) contributes one electron to each of the two bonded metals (two electrons in all).

With these points in mind, let us evaluate the electron count for every structural description offered in the options.

Option A No $$Co{-}Co$$ bond, six terminal $$CO$$ and two bridging $$CO$$. For each cobalt we have:

$$ \begin{aligned} \text{Electrons} &= 9 \;(\text{from } Co) + 6\times2 \;(\text{from 6 terminal } CO) + 2\times1 \;(\text{from 2 bridging } CO) \\ &= 9 + 12 + 2 = 23 \end{aligned} $$

Twenty-three electrons far exceed the 18-electron limit, so such a formulation is impossible. Hence Option A is rejected.

Option B No $$Co{-}Co$$ bond, four terminal $$CO$$ and four bridging $$CO$$. Now each cobalt receives

$$ \begin{aligned} \text{Electrons} &= 9 \;(\text{from } Co) + 4\times2 \;(\text{from 4 terminal } CO) + 4\times1 \;(\text{from 4 bridging } CO) \\ &= 9 + 8 + 4 = 21 \end{aligned} $$

Again the total is greater than 18, so Option B is also impossible.

Option C One $$Co{-}Co$$ bond, six terminal $$CO$$ and two bridging $$CO$$. Here each cobalt counts

$$ \begin{aligned} \text{Electrons} &= 9 \;(\text{from } Co) + 6\times2 \;(\text{from 6 terminal } CO) + 2\times1 \;(\text{from 2 bridging } CO) + 1 \;(\text{from the } Co{-}Co \text{ bond})\\ &= 9 + 12 + 2 + 1 = 24 \end{aligned} $$

The total again exceeds 18, so at first glance this appears unfavourable. However, we must notice that six terminal $$CO$$ and two bridging $$CO$$ together already constitute all eight carbonyl ligands in the formula $$[Co_2(CO)_8]$$. A closer look shows that the count of terminal $$CO$$ ligands per cobalt is actually three, not six: each terminal $$CO$$ is attached to only one metal centre. Let us correct the calculation.

Per cobalt in Option C we really have:

$$ \begin{aligned} \text{Terminal } CO &: 3 \quad (\text{because 6 terminal CO shared equally})\\ \text{Bridging } CO &: 2\\[4pt] \text{Electrons} &= 9 + 3\times2 + 2\times1 + 1 \\ &= 9 + 6 + 2 + 1 = 18 \end{aligned} $$

Now the 18-electron requirement is exactly satisfied, so this structure is perfectly reasonable.

Option D One $$Co{-}Co$$ bond, four terminal $$CO$$ and four bridging $$CO$$. Each cobalt then has:

$$ \begin{aligned} \text{Electrons} &= 9 + 2\times2 \;(\text{because 4 terminal shared equally}) + 4\times1 + 1 \\ &= 9 + 4 + 4 + 1 = 18 \end{aligned} $$

This option also meets the 18-electron rule, so we need an additional criterion to differentiate Options C and D. Experimental crystallographic data resolve the matter: solid $$[Co_2(CO)_8]$$ indeed contains a single $$Co{-}Co$$ bond, with six terminal and two μ₂-bridging carbonyl ligands. Only two carbonyls bridge the two cobalt atoms; the remaining six are terminal.

Therefore the observed structure matches Option C.

Hence, the correct answer is Option C.