Total Number of Compounds Containing Bridging Oxo Group

To determine the total number of compounds having at least one bridging oxo group ($\text{X}-\text{O}-\text{X}$ linkage), let us analyze the structure of each given molecule:

1. $\text{N}_2\text{O}_3$ (Dinitrogen trioxide):

   • The predominant stable form is asymmetrical: $\text{O}=\text{N}-\text{NO}_2$, which contains a direct $\text{N}-\text{N}$ bond and no bridging oxygen atom ($\text{N}-\text{O}-\text{N}$).
   • (Note: A minor symmetrical isomer $\text{O}=\text{N}-\text{O}-\text{N}=\text{O}$ exists at low temperatures, which contains an $\text{N}-\text{O}-\text{N}$ bridge).

2. $\text{N}_2\text{O}_5$ (Dinitrogen pentoxide):

   • Structure: $\text{O}_2\text{N}-\text{O}-\text{NO}_2$
   • It contains one bridging oxo group ($\text{N}-\text{O}-\text{N}$). (Yes)

3. $\text{P}_4\text{O}_6$ (Tetraphosphorus hexoxide):

   • Cage structure where each phosphorus atom is linked to three other phosphorus atoms via oxygen atoms.
   • It contains $6$ bridging oxo groups ($\text{P}-\text{O}-\text{P}$). (Yes)

4. $\text{P}_4\text{O}_7$ (Tetraphosphorus heptoxide):

   • Structure derived from $\text{P}_4\text{O}_6$ by adding one terminal oxygen ($\text{P}=\text{O}$).
   • It retains $6$ bridging oxo groups ($\text{P}-\text{O}-\text{P}$). (Yes)

5. $\text{H}_4\text{P}_2\text{O}_5$ (Pyrophosphorous acid):

   • Structure: $\text{HO(H)(O)P}-\text{O}-\text{P(O)(H)OH}$
   • It contains one bridging oxo group ($\text{P}-\text{O}-\text{P}$). (Yes)

6. $\text{H}_5\text{P}_3\text{O}_{10}$ (Triphosphoric acid):

   • Structure: \text{HO(O)_2P}-\text{O}-\text{P(O)(OH)}-\text{O}-\text{PO(OH)}_2
   • It contains $2$ bridging oxo groups ($\text{P}-\text{O}-\text{P}$). (Yes)

7. $\text{H}_2\text{S}_2\text{O}_3$ (Thiosulfuric acid):

   • Structure: $\text{HO}-\text{S}(=\text{O})(=\text{S})-\text{OH}$
   • It contains a sulfur-sulfur double bond ($\text{S}=\text{S}$) and no bridging oxo group ($\text{S}-\text{O}-\text{S}$). (No)

8. $\text{H}_2\text{S}_2\text{O}_5$ (Pyrosulfurous acid / Disulfurous acid):

   • Structure: $\text{HO}-\text{S}(=\text{O})-\text{S}(=\text{O})_2-\text{OH}$
   • It contains a direct sulfur-sulfur bond ($\text{S}-\text{S}$) and no bridging oxo group ($\text{S}-\text{O}-\text{S}$). (No)

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Conclusion:

The molecules containing at least one bridging oxo group are: $\text{N}_2\text{O}_5, \text{P}_4\text{O}_6, \text{P}_4\text{O}_7, \text{H}_4\text{P}_2\text{O}_5, \text{H}_5\text{P}_3\text{O}_{10}$

Thus, the total count is 5 (or 6, if the symmetrical isomer of $\text{N}_2\text{O}_3$ is included).