Which relations are not correct for an aqueous dilute solution of `K_(3)PO_(4)` if its degree of dissociation is `alpha`?

Options

A. `DeltaP/P^(@)=(Molalityxx18xx(1+3alpha))/1000`
B. `DeltaP/P^(@)=(pi_(obs)xx18xx(1+3alpha))/(RT xx 1000)`
C. `DeltaP/P^(@)=(DeltaT_(f)_(obs)xx18)/(K_(f) xx 1000)`
D. Mw of `K_(3)PO_(4))=Mw_(obs)xx(1+3alpha)`

Explanation

To solve the question regarding the incorrect relations for an aqueous dilute solution of \( K_3PO_4 \) with a degree of dissociation \( \alpha \), we will follow these steps:

### Step 1: Understand the Dissociation of \( K_3PO_4 \)
When potassium phosphate (\( K_3PO_4 \)) dissociates in water, it produces:
- 3 potassium ions (\( 3K^+ \))
- 1 phosphate ion (\( PO_4^{3-} \))

The dissociation can be represented as:
\[ K_3PO_4 \rightarrow 3K^+ + PO_4^{3-} \]

### Step 2: Determine the Degree of Dissociation
Let the initial concentration of \( K_3PO_4 \) be 1 mole. After dissociation, if \( \alpha \) is the degree of dissociation, the concentrations will be:
- \( K_3PO_4 \) remaining: \( 1 - \alpha \)
- \( K^+ \) produced: \( 3\alpha \)
- \( PO_4^{3-} \) produced: \( \alpha \)

### Step 3: Calculate the Van't Hoff Factor \( i \)
The Van't Hoff factor \( i \) is given by the total number of particles in solution after dissociation divided by the number of formula units initially present. Thus:
\[ i = \frac{(1 - \alpha) + 3\alpha + \alpha}{1} = 1 + 3\alpha \]

### Step 4: Analyze the Relations
Now we will analyze the relations given in the options to identify which one is incorrect.

1. **Relative Lowering of Vapor Pressure**:
\[ \frac{\Delta P}{P_0} = i \cdot m \cdot \frac{M_w}{1000} \]
This relation is correct as it incorporates the Van't Hoff factor.

2. **Osmotic Pressure**:
\[ \Pi = i \cdot C \cdot RT \]
This relation is also correct, where \( C \) is the molarity of the solution.

3. **Freezing Point Depression**:
\[ \Delta T_f = i \cdot K_f \cdot m \]
This relation is also correct.

4. **Molecular Weight Relation**:
\[ i = \frac{M_{calculated}}{M_{observed}} \]
This relation is correct as well.

### Step 5: Identify the Incorrect Relation
From the analysis, we find that all the relations derived from the principles of colligative properties are correct. However, the second option regarding osmotic pressure appears to have been misinterpreted in the context of the question.

### Conclusion
The incorrect relation for an aqueous dilute solution of \( K_3PO_4 \) with degree of dissociation \( \alpha \) is:
\[ \text{Option 2: } \Pi \neq \text{(incorrect form)} \]