1.4275 g sample of `[Cr(NH_3)_6]SO_4Cl` (mol. Wt. =285.5 ) is dissolved to prepare 250 mL solution showing an osmotic pressure of 1.478 atm at `27^@C`. Which of the following statements are correct about this solution ?
(I)Each molecule furnishes three ions in solution
(ii)The van't Hoff factor is 3.
(iii)Equilibrium molarity of `[Cr(NH_3)_6]SO_4Cl`=0
(iv)Equilibrium molarity of `[Cr(NH_3)_6]^(3+)` =0.02 M

Options

A. I and III only
B. II and IV only
C. I,II and IV only
D. all of these

Explanation

To solve the problem step by step, we will analyze the provided information and apply relevant formulas.

### Step 1: Calculate the number of moles of the complex
We start by calculating the number of moles of the complex `[Cr(NH_3)_6]SO_4Cl` using the formula:

\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} \]

Given:
- Mass = 1.4275 g
- Molar mass = 285.5 g/mol

Calculating the number of moles:

\[ \text{Number of moles} = \frac{1.4275 \, \text{g}}{285.5 \, \text{g/mol}} = 0.005 \, \text{moles} \]

**Hint:** Remember to always use the correct units when calculating moles.

### Step 2: Calculate the concentration of the solution
Next, we calculate the concentration (C) of the solution in moles per liter (M):

\[ C = \frac{\text{Number of moles}}{\text{Volume in liters}} \]

Given:
- Volume = 250 mL = 0.250 L

Calculating the concentration:

\[ C = \frac{0.005 \, \text{moles}}{0.250 \, \text{L}} = 0.02 \, \text{M} \]

**Hint:** Always convert mL to L when calculating concentration.

### Step 3: Use the osmotic pressure formula
The osmotic pressure (π) is given by the formula:

\[ \pi = i \cdot C \cdot R \cdot T \]

Where:
- \( \pi = 1.478 \, \text{atm} \)
- \( R = 0.0821 \, \text{L atm/(K mol)} \)
- \( T = 27^\circ C = 300 \, \text{K} \) (convert Celsius to Kelvin)

Rearranging the formula to solve for the van't Hoff factor (i):

\[ i = \frac{\pi}{C \cdot R \cdot T} \]

Substituting the values:

\[ i = \frac{1.478}{0.02 \cdot 0.0821 \cdot 300} \]

Calculating:

\[ i = \frac{1.478}{4.926} \approx 3 \]

**Hint:** Check your calculations carefully to ensure accuracy.

### Step 4: Analyze the dissociation of the complex
Since the van't Hoff factor \( i = 3 \), this indicates that each molecule of the complex dissociates into three ions in solution. The dissociation can be represented as:

\[ [Cr(NH_3)_6]SO_4Cl \rightarrow [Cr(NH_3)_6]^{3+} + SO_4^{2-} + Cl^{-} \]

### Step 5: Determine equilibrium molarities
1. **Equilibrium molarity of `[Cr(NH_3)_6]SO_4Cl`**: Since the complex dissociates completely, its equilibrium concentration is 0 M.

2. **Equilibrium molarity of `[Cr(NH_3)_6]^{3+}`**: The concentration of the cation after dissociation is equal to the initial concentration of the complex, which is 0.02 M.

### Conclusion
Based on the analysis:
- (I) Each molecule furnishes three ions in solution: **True**
- (II) The van't Hoff factor is 3: **True**
- (III) Equilibrium molarity of `[Cr(NH_3)_6]SO_4Cl` = 0: **True**
- (IV) Equilibrium molarity of `[Cr(NH_3)_6]^{3+}` = 0.02 M: **True**

**Final Answer:** All statements (I, II, III, IV) are correct.