For the redox reaction given, what is the value of `(x)/(z)` ?
`xNO_(3)^(-)+yAs_(2)S_(3)+zH_(2)O to AsO_(4)^(3-)+ NO+ SO_(4)^(2-)+ H^(+)`.

Answer

To solve the problem of finding the value of \( \frac{x}{z} \) in the given redox reaction, we will follow these steps:

### Step 1: Write the balanced equation
The reaction is given as:
\[ x \text{NO}_3^{-} + y \text{As}_2\text{S}_3 + z \text{H}_2\text{O} \rightarrow \text{AsO}_4^{3-} + \text{NO} + \text{SO}_4^{2-} + \text{H}^{+} \]

### Step 2: Determine oxidation states
- For \( \text{NO}_3^{-} \), nitrogen has an oxidation state of +5, and in \( \text{NO} \), it is +2. This indicates that nitrogen is being reduced.
- For \( \text{As}_2\text{S}_3 \), arsenic has an oxidation state of +3 and changes to +5 in \( \text{AsO}_4^{3-} \). This indicates that arsenic is being oxidized.
- For sulfur in \( \text{As}_2\text{S}_3 \) (which is -2) changing to +6 in \( \text{SO}_4^{2-} \), sulfur is also being oxidized.

### Step 3: Write half-reactions
**Reduction half-reaction:**
\[ \text{NO}_3^{-} + 3e^{-} \rightarrow \text{NO} \]

**Oxidation half-reaction:**
1. For arsenic:
\[ \text{As}_2\text{S}_3 \rightarrow 2 \text{AsO}_4^{3-} + 4e^{-} \]
2. For sulfur:
\[ 3 \text{S}^{-2} \rightarrow 3 \text{SO}_4^{2-} + 24e^{-} \]

### Step 4: Combine half-reactions
To balance the electrons, we need to find the least common multiple of the electrons lost and gained. The total electrons lost from sulfur and arsenic is 28 electrons, while nitrogen gains 3 electrons.

We multiply the reduction half-reaction by 28 and the oxidation half-reaction by 3 to balance the electrons:
\[ 28 \text{NO}_3^{-} + 3 \text{As}_2\text{S}_3 + z \text{H}_2\text{O} \rightarrow 6 \text{AsO}_4^{3-} + 9 \text{SO}_4^{2-} + 28 \text{NO} + 8 \text{H}^{+} \]

### Step 5: Balance the equation
Now, we need to balance the number of hydrogen and oxygen atoms. After balancing, we find:
- \( x = 28 \)
- \( y = 3 \)
- \( z = 4 \)

### Step 6: Calculate \( \frac{x}{z} \)
Now we can calculate:
\[ \frac{x}{z} = \frac{28}{4} = 7 \]

### Final Answer
Thus, the value of \( \frac{x}{z} \) is \( 7 \).

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