Osomotic pressure [in atm] of a 0.1 M solution of `K_4[Fe(CN)_6]`, which undergoes 50% dissociation, wil be ___ at `27^(@)C` :

Options

A. 7.38
B. 3.69
C. 405.9
D. none of these

Explanation

To find the osmotic pressure of a 0.1 M solution of \( K_4[Fe(CN)_6] \) that undergoes 50% dissociation at \( 27^{\circ}C \), we can follow these steps:

### Step 1: Understand the formula for osmotic pressure
The osmotic pressure (\( \pi \)) can be calculated using the formula:
\[ \pi = I \cdot C \cdot R \cdot T \]
where:
- \( I \) = Van't Hoff factor
- \( C \) = concentration of the solution (in mol/L)
- \( R \) = universal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin

### Step 2: Convert temperature to Kelvin
Given the temperature is \( 27^{\circ}C \):
\[ T = 27 + 273 = 300 \, K \]

### Step 3: Determine the dissociation and Van't Hoff factor
The compound \( K_4[Fe(CN)_6] \) dissociates as follows:
\[ K_4[Fe(CN)_6] \rightarrow 4K^+ + [Fe(CN)_6]^{4-} \]
This means that for every mole of \( K_4[Fe(CN)_6] \), it produces 5 moles of ions (4 from \( K^+ \) and 1 from \( [Fe(CN)_6]^{4-} \)).

Since the solution undergoes 50% dissociation, we can denote the degree of dissociation as \( \alpha = 0.5 \).

### Step 4: Calculate the Van't Hoff factor (\( I \))
The Van't Hoff factor \( I \) can be calculated using the formula:
\[ I = 1 + n \cdot \alpha \]
where \( n \) is the number of particles produced per formula unit. Here, \( n = 4 + 1 = 5 \) (4 \( K^+ \) ions and 1 \( [Fe(CN)_6]^{4-} \) ion).

Substituting the values:
\[ I = 1 + 4 \cdot 0.5 = 1 + 2 = 3 \]

### Step 5: Substitute values into the osmotic pressure formula
Now we can substitute \( I \), \( C \), \( R \), and \( T \) into the osmotic pressure formula:
\[ \pi = I \cdot C \cdot R \cdot T \]
\[ \pi = 3 \cdot 0.1 \cdot 0.0821 \cdot 300 \]

### Step 6: Calculate the osmotic pressure
Calculating the above expression:
\[ \pi = 3 \cdot 0.1 \cdot 0.0821 \cdot 300 = 7.389 \]
Rounding it off, we get:
\[ \pi \approx 7.39 \, \text{atm} \]

### Final Answer
The osmotic pressure of the solution is approximately \( 7.39 \, \text{atm} \).

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