The observed osmotic pressure for a `0.10M` solution of `Fe(NH_(4))_(2)(SO_(4))_(2)` at `25^(@)C` is `10.8 atm`. The expected and experimental (observed) values of Van't Hoff factor (i) will be respectively: `(R = 0.082 L atm k^(-) mol^(-1))`

Options

A. 3 and `5.42`
B. 5 and `3.42`
C. 4 and `4.00`
D. 5 and `4.42`

Explanation

To solve the problem, we need to determine the expected and observed values of the Van't Hoff factor (i) for a 0.10 M solution of \( \text{Fe(NH}_4\text{)}_2\text{(SO}_4\text{)}_2 \) at 25°C, given the observed osmotic pressure of 10.8 atm.

### Step-by-Step Solution:

**Step 1: Convert Temperature to Kelvin**
- The temperature given is 25°C. To convert it to Kelvin, we add 273.
\[ T = 25 + 273 = 298 \, \text{K} \]

**Step 2: Use the Osmotic Pressure Formula**
- The osmotic pressure (\( \Pi \)) is given by the formula:
\[ \Pi = iCRT \]
where:
- \( C \) = concentration of the solution (0.10 M)
- \( R \) = ideal gas constant (0.082 L atm K\(^{-1}\) mol\(^{-1}\))
- \( T \) = temperature in Kelvin (298 K)

**Step 3: Calculate Theoretical Osmotic Pressure**
- Plugging in the values into the formula:
\[ \Pi_{\text{theoretical}} = i \cdot C \cdot R \cdot T \]
Assuming \( i = 1 \) for the theoretical calculation (as a starting point):
\[ \Pi_{\text{theoretical}} = 1 \cdot 0.10 \cdot 0.082 \cdot 298 \]
\[ \Pi_{\text{theoretical}} = 2.466 \, \text{atm} \]

**Step 4: Calculate the Observed Van't Hoff Factor (i)**
- The observed osmotic pressure is given as 10.8 atm. Using the formula for Van't Hoff factor:
\[ i = \frac{\Pi_{\text{observed}}}{\Pi_{\text{theoretical}}} \]
Substituting the values:
\[ i = \frac{10.8}{2.466} \approx 4.38 \]

**Step 5: Determine the Expected Van't Hoff Factor**
- The expected dissociation of \( \text{Fe(NH}_4\text{)}_2\text{(SO}_4\text{)}_2 \) is:
- 1 \( \text{Fe}^{2+} \)
- 2 \( \text{NH}_4^{+} \)
- 2 \( \text{SO}_4^{2-} \)

Thus, the expected total number of particles is:
\[ 1 + 2 + 2 = 5 \]
Therefore, the expected Van't Hoff factor \( i_{\text{expected}} = 5 \).

### Final Results:
- Expected Van't Hoff factor \( i_{\text{expected}} = 5 \)
- Observed Van't Hoff factor \( i_{\text{observed}} \approx 4.38 \)

### Summary:
- The expected and experimental values of Van't Hoff factor are \( 5 \) and \( 4.38 \) respectively.