Step 1 : Understanding the Question:
This problem asks for the oxidation state of sulfur in peroxydisulfuric acid, commonly known as Marshall's acid. Calculating oxidation states for complex acids can be tricky because the standard algebraic method (assuming $O = -2$ and $H = +1$) often fails if the molecule contains peroxide linkages ($-O-O-$). In such bonds, oxygen has an oxidation state of $-1$. This question specifically tests the student's knowledge of chemical structure versus simple algebraic calculation.
Step 2 : Key Formulas and approach:
We must respect the rule that an element cannot have an oxidation state higher than its group number. Sulfur is in Group 16, so its maximum possible oxidation state is $+6$. If the standard calculation yields a value higher than $+6$, it indicates the presence of a peroxide bond.
Our approach is:
1. Perform the standard algebraic calculation.
2. Recognize that the result ($+7$) is impossible.
3. Use the structural formula of $H_2S_2O_8$ to identify peroxide oxygens.
4. Recalculate using the correct oxygen values.
Step 3 : Detailed Explanation:
Standard algebraic method: $2(+1) + 2(x) + 8(-2) = 0 \Rightarrow 2 + 2x - 16 = 0 \Rightarrow 2x = 14 \Rightarrow x = +7$.
Since $+7$ exceeds sulfur's maximum valence, we must look at the structure: $HO-SO_2-O-O-SO_2-OH$.
In this structure, two oxygen atoms form a peroxide linkage ($O-O$). These two oxygens have an oxidation state of $-1$ each.
The remaining six oxygen atoms are in standard double bonds or hydroxyl groups, having an oxidation state of $-2$ each.
Correct equation: $2(\text{for } H) + 2x(\text{for } S) + 6(-2)(\text{for normal } O) + 2(-1)(\text{for peroxide } O) = 0$.
Simplify: $2 + 2x - 12 - 2 = 0$.
This reduces to $2x - 12 = 0$, which gives $x = +6$.
Thus, sulfur exists in its maximum $+6$ oxidation state.
Step 4 : Final Answer:
The oxidation state of sulfur in Marshall's acid is +6.