The number of S = O and S - OH bonds present in peroxodisulphuric acid and pyrosulphuric acid respectively are:

First, we recall the molecular formulas of the two oxy-acids involved.

Peroxodisulphuric acid has the formula $$H_2S_2O_8$$, while pyrosulphuric acid has the formula $$H_2S_2O_7$$.

We now write the accepted structural formulae.

For peroxodisulphuric acid $$H_2S_2O_8$$, the two sulphur atoms are linked through a peroxide bridge. The complete formula is

$$HO{-}S(=O)_2{-}O{-}O{-}S(=O)_2{-}OH.$$

For pyrosulphuric acid $$H_2S_2O_7$$, the two sulphur atoms are linked through a single oxygen (there is no O-O bond). The complete formula is

$$HO{-}S(=O)_2{-}O{-}S(=O)_2{-}OH.$$

With the structures written, we can now count the individual bond types requested.

Counting in peroxodisulphuric acid

Each sulphur atom in $$HO{-}S(=O)_2{-}O{-}O{-}S(=O)_2{-}OH$$ shows:

• Two double bonds to oxygen, that is, two $$S = O$$ bonds.
• One single bond to a hydroxyl oxygen, that is, one $$S - OH$$ bond.

Because there are two identical sulphur centres, we simply multiply the numbers obtained for one sulphur atom by two.

Number of $$S = O$$ bonds:

$$2 \text{ (bonds per S)} \times 2 \text{ (S atoms)} = 4.$$

Number of $$S - OH$$ bonds:

$$1 \text{ (bond per S)} \times 2 \text{ (S atoms)} = 2.$$

So, peroxodisulphuric acid contains four $$S = O$$ bonds and two $$S - OH$$ bonds.

Counting in pyrosulphuric acid

In $$HO{-}S(=O)_2{-}O{-}S(=O)_2{-}OH$$ each sulphur atom again displays:

• Two $$S = O$$ bonds.
• One $$S - OH$$ bond.

Proceeding exactly as before, we multiply by two for the two sulphur atoms.

Number of $$S = O$$ bonds:

$$2 \times 2 = 4.$$

Number of $$S - OH$$ bonds:

$$1 \times 2 = 2.$$

Thus, pyrosulphuric acid also possesses four $$S = O$$ bonds and two $$S - OH$$ bonds.

Summarising our counts:

Peroxodisulphuric acid: $$(4,\,2).$$

Pyrosulphuric acid: $$(4,\,2).$$

Among the given choices, the ordered pair $$(4,\,2)$$ for each acid appears only in Option B.

Hence, the correct answer is Option B.