To identify the compounds with a peroxide linkage (an -O-O- single bond), we will determine the structure of each given oxyacid. A useful first step is to calculate the oxidation state of the central atom (sulfur) in each compound, assuming the usual oxidation state of -2 for oxygen. If the calculated oxidation state exceeds the maximum possible oxidation state for sulfur (+6), it strongly indicates the presence of a peroxide linkage, where oxygen atoms have an oxidation state of -1.

Step-by-step analysis of each option:

A: $\mathrm{H}_2 \mathrm{~S}_2 \mathrm{O}_7$ (Disulfuric acid or Oleum)

1. Calculate the oxidation state of S: Let the oxidation state of S be $x$. $2(+1) + 2(x) + 7(-2) = 0$ $2 + 2x - 14 = 0$ $2x = 12 Rightarrow x = +6$
2. Analyze the oxidation state: The oxidation state of sulfur is +6, which is the maximum possible oxidation state for sulfur. This does not suggest a peroxide linkage.
3. Determine the structure: Disulfuric acid has a structure with an S-O-S linkage, formed by linking two sulfuric acid molecules with the removal of one water molecule. $$\begin{array}{ccc} \mathrm{O} & & \mathrm{O} \\ \|\| & & \|\| \\ \mathrm{HO}-\mathrm{S} & -\mathrm{O}- & \mathrm{S}-\mathrm{OH} \\ \|\| & & \|\| \\ \mathrm{O} & & \mathrm{O} \end{array}$$ There is no -O-O- linkage. Thus, option A is incorrect.

B: $\mathrm{H}_2 \mathrm{~S}_2 \mathrm{O}_8$ (Peroxodisulfuric acid or Marshall's acid)

1. Calculate the oxidation state of S: Assuming O is -2. Let the oxidation state of S be $x$. $2(+1) + 2(x) + 8(-2) = 0$ $2 + 2x - 16 = 0$ $2x = 14 Rightarrow x = +7$
2. Analyze the oxidation state: The calculated oxidation state of +7 is impossible for sulfur, as its maximum oxidation state is +6. This indicates the presence of a peroxide linkage. If we assume one peroxide bond (-O-O-), there are two oxygen atoms with an oxidation state of -1 and six with -2. The calculation becomes: $2(+1) + 2(x) + 6(-2) + 2(-1) = 0$ $2 + 2x - 12 - 2 = 0 Rightarrow 2x = 12 Rightarrow x = +6$ This is a valid oxidation state for sulfur.
3. Determine the structure: The structure contains a peroxide bridge between the two sulfur atoms. $$\begin{array}{ccc} \mathrm{O} & & \mathrm{O} \\ \|\| & & \|\| \\ \mathrm{HO}-\mathrm{S} & -\mathrm{O}-\mathrm{O}- & \mathrm{S}-\mathrm{OH} \\ \|\| & & \|\| \\ \mathrm{O} & & \mathrm{O} \end{array}$$ The structure contains a peroxide (-O-O-) linkage. Thus, option B is correct.

C: $\mathrm{H}_2 \mathrm{~S}_2 \mathrm{O}_5$ (Disulfurous acid)

1. Calculate the oxidation state of S: Let the oxidation state of S be $x$. $2(+1) + 2(x) + 5(-2) = 0$ $2 + 2x - 10 = 0$ $2x = 8 Rightarrow x = +4$ (average oxidation state)
2. Analyze the oxidation state: The oxidation state +4 is a valid state for sulfur. This does not suggest a peroxide linkage.
3. Determine the structure: The structure of disulfurous acid contains a direct S-S bond. $$\begin{array}{ccc} \mathrm{O} & & \mathrm{O} \\ \|\| & & \|\| \\ \mathrm{HO}-\mathrm{S} & - & \mathrm{S}-\mathrm{OH} \\ & & | \\ & & \mathrm{O} \end{array}$$ There is no -O-O- linkage. Thus, option C is incorrect.

D: $\mathrm{H}_2 \mathrm{SO}_5$ (Peroxomonosulfuric acid or Caro's acid)

1. Calculate the oxidation state of S: Assuming O is -2. Let the oxidation state of S be $x$. $2(+1) + x + 5(-2) = 0$ $2 + x - 10 = 0 Rightarrow x = +8$
2. Analyze the oxidation state: The calculated oxidation state of +8 is impossible for sulfur. This indicates the presence of a peroxide linkage. With one peroxide group, there are two oxygen atoms with an oxidation state of -1 and three with -2. $2(+1) + x + 3(-2) + 2(-1) = 0$ $2 + x - 6 - 2 = 0 Rightarrow x = +6$ This is a valid oxidation state.
3. Determine the structure: The structure is similar to sulfuric acid, but one -OH group is replaced by an -OOH group. $$\begin{array}{c} \mathrm{O} \\ \|\| \\ \mathrm{HO}-\mathrm{S}-\mathrm{O}-\mathrm{O}-\mathrm{H} \\ \|\| \\ \mathrm{O} \end{array}$$ The structure contains a peroxide (-O-O-) linkage. Thus, option D is correct.

Conclusion

The compounds containing a peroxide linkage are $\mathrm{H}_2 \mathrm{~S}_2 \mathrm{O}_8$ and $\mathrm{H}_2 \mathrm{SO}_5$.