Which of the Phosphorus oxoacid can create silver mirror from AgNO$$_3$$ solution?

We need to identify which phosphorus oxoacid can reduce $$AgNO_3$$ to metallic silver (creating a silver mirror, i.e., giving a positive Tollen's test). A phosphorus oxoacid can act as a strong reducing agent if it contains a direct $$P-H$$ bond because the hydrogen bonded directly to phosphorus can be released, providing electrons for the reduction process. Oxoacids with $$P-H$$ bonds are strong enough reducing agents to reduce $$Ag^+$$ ions to metallic silver. Metaphosphoric acid $$HPO_{3n}$$ (polymetaphosphoric acid) has the structure where all hydrogen atoms are bonded to oxygen as $$O-H$$ groups and no direct $$P-H$$ bond; therefore it cannot reduce $$Ag^+$$ to silver.

Pyrophosphorous acid $$H_4P_2O_5$$, the pyro-form of phosphorous acid $$H_3PO_3$$, retains the direct $$P-H$$ bonds present in phosphorous acid. Since phosphorous acid is diprotic with one hydrogen bonded to phosphorus, the condensation of two $$H_3PO_3$$ molecules (losing water) produces $$H_4P_2O_5$$ with the same $$P-H$$ bonds, and these bonds give it the reducing power to reduce $$Ag^+$$ ions to metallic silver:

$$Ag^+(aq) + e^- \rightarrow Ag(s)$$

Thus $$H_4P_2O_5$$ can create a silver mirror from $$AgNO_3$$ solution. Hypophosphoric acid $$H_4P_2O_6$$ contains a direct $$P-P$$ bond and hydrogen atoms only bonded to oxygen as $$O-H$$ groups, so it lacks a $$P-H$$ bond and cannot reduce $$Ag^+$$. Pyrophosphoric acid $$H_4P_2O_7$$ (the pyro-form of phosphoric acid $$H_3PO_4$$) also has no $$P-H$$ bonds since all hydrogens are in $$O-H$$ groups, and therefore it cannot act as a reducing agent. Only $$H_4P_2O_5$$ (pyrophosphorous acid) contains $$P-H$$ bonds, giving it the necessary reducing power to reduce $$Ag^+$$ to metallic silver and produce a silver mirror. The correct answer is Option 2: $$H_4P_2O_5$$.