The unbalanced equation for the reaction of `P_(4)S_(3)` with nitrate in aqueous acidic medium is given below.
`P_(4)S_(3)+NO_(3)^(-)rarr H_(3)PO_(4)+SO_(4)^(2-)+NO`
The number of moles of water required per mol of `P_(4)S_(3)` is `(x)/(3)`, the value of x is

Answer

To solve the problem, we need to balance the chemical equation for the reaction of \( P_4S_3 \) with nitrate ions in an aqueous acidic medium. The unbalanced equation is:

\[ P_4S_3 + NO_3^- \rightarrow H_3PO_4 + SO_4^{2-} + NO \]

### Step 1: Write the unbalanced equation
We start with the unbalanced equation provided:

\[ P_4S_3 + NO_3^- \rightarrow H_3PO_4 + SO_4^{2-} + NO \]

### Step 2: Identify the products and reactants
In this reaction:
- Reactants: \( P_4S_3 \) and \( NO_3^- \)
- Products: \( H_3PO_4 \), \( SO_4^{2-} \), and \( NO \)

### Step 3: Balance phosphorus (P)
There are 4 phosphorus atoms in \( P_4S_3 \) and 1 phosphorus atom in each \( H_3PO_4 \). Therefore, we need 4 moles of \( H_3PO_4 \):

\[ P_4S_3 + NO_3^- \rightarrow 4H_3PO_4 + SO_4^{2-} + NO \]

### Step 4: Balance sulfur (S)
There are 3 sulfur atoms in \( P_4S_3 \). To balance sulfur, we need 3 moles of \( SO_4^{2-} \):

\[ P_4S_3 + NO_3^- \rightarrow 4H_3PO_4 + 3SO_4^{2-} + NO \]

### Step 5: Balance nitrogen (N)
The nitrogen comes from the \( NO_3^- \) and produces \( NO \). If we assume \( x \) moles of \( NO \) are produced, we need to balance the nitrogen. The number of \( NO \) produced will depend on the number of \( NO_3^- \) consumed.

### Step 6: Balance oxygen (O)
Now we need to balance the oxygen atoms. Count the total number of oxygen atoms on both sides of the equation.

- Left side:
- From \( NO_3^- \): 3 oxygen atoms per nitrate.
- Right side:
- From \( 4H_3PO_4 \): \( 4 \times 4 = 16 \) oxygen atoms.
- From \( 3SO_4^{2-} \): \( 3 \times 4 = 12 \) oxygen atoms.
- From \( NO \): 1 oxygen atom.

Total on the right side = \( 16 + 12 + 1 = 29 \) oxygen atoms.

### Step 7: Add water to balance hydrogen (H)
To balance the hydrogen, we will need to add water molecules. Each \( H_3PO_4 \) contributes 3 hydrogen atoms, and we have 4 of them, contributing a total of \( 4 \times 3 = 12 \) hydrogen atoms.

To balance the hydrogen, we need to add water molecules to the left side. If we add \( y \) moles of water, we will have:

- Left side: \( 2y \) hydrogen atoms from water.
- Right side: \( 12 \) hydrogen atoms from \( H_3PO_4 \).

### Step 8: Set up the equation for hydrogen balance
We can set up the equation:

\[ 2y = 12 \]

Solving for \( y \):

\[ y = 6 \]

### Step 9: Calculate water per mole of \( P_4S_3 \)
Since we are looking for the number of moles of water required per mole of \( P_4S_3 \), we need to find the ratio. The equation shows that for every 1 mole of \( P_4S_3 \), we need 6 moles of water.

### Step 10: Express in the form \( \frac{x}{3} \)
The problem states that the number of moles of water required per mole of \( P_4S_3 \) is \( \frac{x}{3} \). We found that we need 6 moles of water:

\[ \frac{x}{3} = 6 \]

To find \( x \):

\[ x = 6 \times 3 = 18 \]

### Final Answer
The value of \( x \) is \( 18 \).