The reaction `S_(2)O_(8)^(2-) + 3I^(ɵ) rarr 2SO_(4)^(2-) + I_(3)^(ɵ)` is of first order both with respect to persulphate and iofide ions. Taking the initial concentration as `a` and `b`, respectively, and taking `x` as the concentration of the triofide at time `t`, a differential rate equation can be written.
Two suggested mechanism for the reaction are:
I. `S_(2)O_(8)^(2-)+I^(ɵ) hArr SO_(4)I^(ɵ)+SO_(4)^(2-) ("fast")`
`I^(ɵ)+SO_(4)I^(ɵ) overset(k_(1))rarrI_(2) + SO_(4)^(2-)` (show)
`I^(ɵ) + I_(2) overset(k_(2))rarr I_(3)^(ɵ) ("fast")`
II. `S_(2)O_(8)^(2-) + I^(ɵ) overset(k_(1))rarr S_(2)O_(8) I^(2-) (slow)`
`S_(2)O_(8)I^(3-) overset(k_(2))rarr2SO_(4)^(2-)+I^(o+) ("fast")`
`I^(o+) + I^(ɵ) overset(k_(3)) rarr I_(2) ("fast")`
`I_(2) + I^(o+) overset(k_(4))rarr I_(3)^(ɵ) ("fast")`
For the reaction `I_(2)+2S_(2)O_(3)^(2-) rarr S_(4)O_(6)^(2-) + 2I^(ɵ)`
I. `(-d[I_(2)])/(dt) = -(1)/(2) (d[S_(2)O_(3)^(2-)])/(dt)`
II. `(-d[I_(2)])/(dt) = -2 (d[S_(2)O_(3)^(2-)])/(dt)`
III. `(-d[I_(2)])/(dt) = -2 (d[I^(ɵ)])/(dt) xx (d[S_(2)O_(3)^(2-)])/(dt)`
IV. `(d[S_(4)O_(6)^(2-)])/(dt) = (1)/(2)(d[I^(ɵ)])/(dt)`
The correct option is

Options

A. Only I
B. I and IV
C. II and IV
D. Only III

Explanation

To solve the problem, we need to analyze the given reaction and the mechanisms proposed. The reaction in question is:

\[ \text{I}_2 + 2 \text{S}_2\text{O}_3^{2-} \rightarrow \text{S}_4\text{O}_6^{2-} + 2 \text{I}^- \]

### Step 1: Write the Rate Law Expression

For the reaction, we can express the rate of disappearance of reactants and the rate of appearance of products. The rate law can be determined based on the stoichiometry of the reaction.

1. The rate of disappearance of \( \text{I}_2 \) is given by:
\[ -\frac{d[\text{I}_2]}{dt} \]

2. The rate of disappearance of \( \text{S}_2\text{O}_3^{2-} \) is:
\[ -\frac{1}{2} \frac{d[\text{S}_2\text{O}_3^{2-}]}{dt} \]

3. The rate of appearance of \( \text{S}_4\text{O}_6^{2-} \) is:
\[ \frac{1}{1} \frac{d[\text{S}_4\text{O}_6^{2-}]}{dt} \]

4. The rate of appearance of \( \text{I}^- \) is:
\[ -\frac{1}{2} \frac{d[\text{I}^-]}{dt} \]

### Step 2: Setting Up the Rate Expressions

From the stoichiometry of the reaction, we can equate the rates:

\[ -\frac{d[\text{I}_2]}{dt} = -\frac{1}{2}\frac{d[\text{S}_2\text{O}_3^{2-}]}{dt} = \frac{1}{1}\frac{d[\text{S}_4\text{O}_6^{2-}]}{dt} = -\frac{1}{2}\frac{d[\text{I}^-]}{dt} \]

### Step 3: Analyze the Given Options

Now we will analyze the options provided in the question:

1. **Option I**:
\[ -\frac{d[\text{I}_2]}{dt} = -\frac{1}{2}\frac{d[\text{S}_2\text{O}_3^{2-}]}{dt} \]
This is correct based on our derivation.

2. **Option II**:
\[ -\frac{d[\text{I}_2]}{dt} = -2\frac{d[\text{S}_2\text{O}_3^{2-}]}{dt} \]
This is incorrect because the stoichiometric coefficient for \( \text{S}_2\text{O}_3^{2-} \) is 2, thus it should be \( -\frac{1}{2} \).

3. **Option III**:
\[ -\frac{d[\text{I}_2]}{dt} = -2\frac{d[\text{I}^-]}{dt} \cdot \frac{d[\text{S}_2\text{O}_3^{2-}]}{dt} \]
This is incorrect as it incorrectly combines the rates.

4. **Option IV**:
\[ \frac{d[\text{S}_4\text{O}_6^{2-}]}{dt} = \frac{1}{2}\frac{d[\text{I}^-]}{dt} \]
This is also incorrect as the stoichiometry does not support this relationship.

### Conclusion

The correct options are:
- **Option I** is correct.
- **Option II** is incorrect.
- **Option III** is incorrect.
- **Option IV** is incorrect.

Thus, the only correct statement is from **Option I**.