At `27^(@)C`,3.92 gm `H_(2)SO_(4)` is present in 250 ml solution. The osmotic pressure of this solution is 1.5 atm. If the osmotic pressure of solution of NaOH is 2 atm at same temperature, then concentration of NaOH solution is

Options

A. 0.32 M
B. 12.183 M
C. `72.3 gm//lit`
D. 1 M `Na_(2)SO_(4)`

Answer

To solve the problem, we need to find the concentration of the NaOH solution using the information given about the H₂SO₄ solution and the principles of osmotic pressure. Here’s a step-by-step breakdown:

### Step 1: Understand the Formula for Osmotic Pressure
The osmotic pressure (π) of a solution is given by the formula:
\[ \pi = iCRT \]
Where:
- \( \pi \) = osmotic pressure
- \( i \) = Van 't Hoff factor (number of particles the solute dissociates into)
- \( C \) = molarity (concentration in moles per liter)
- \( R \) = universal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin

### Step 2: Calculate the Molarity of H₂SO₄
Given:
- Mass of H₂SO₄ = 3.92 g
- Volume of solution = 250 mL = 0.250 L
- Molar mass of H₂SO₄ = 98 g/mol

First, calculate the number of moles of H₂SO₄:
\[ \text{Moles of H₂SO₄} = \frac{\text{mass}}{\text{molar mass}} = \frac{3.92 \, \text{g}}{98 \, \text{g/mol}} \approx 0.04 \, \text{mol} \]

Now, calculate the molarity (C) of the H₂SO₄ solution:
\[ C = \frac{\text{moles}}{\text{volume in L}} = \frac{0.04 \, \text{mol}}{0.250 \, \text{L}} = 0.16 \, \text{M} \]

### Step 3: Determine the Van 't Hoff Factor for H₂SO₄
H₂SO₄ dissociates into 3 ions:
\[ \text{H₂SO₄} \rightarrow 2\text{H}^+ + \text{SO₄}^{2-} \]
Thus, the Van 't Hoff factor \( i \) for H₂SO₄ is 3.

### Step 4: Calculate the Osmotic Pressure for H₂SO₄
Using the osmotic pressure formula:
\[ \pi = iCRT \]
For H₂SO₄:
\[ 1.5 = 3 \times 0.16 \times R \times T \]
Since R and T are constants, we can express this as:
\[ 1.5 = 3 \times 0.16 \times k \quad \text{(where \( k = R \times T \))} \]
From this, we can find \( k \):
\[ k = \frac{1.5}{3 \times 0.16} = \frac{1.5}{0.48} \approx 3.125 \]

### Step 5: Use the Same k to Find Concentration of NaOH
For NaOH, we know:
- \( \pi = 2 \, \text{atm} \)
- Van 't Hoff factor \( i \) for NaOH is 2 (as it dissociates into Na⁺ and OH⁻).

Using the osmotic pressure formula for NaOH:
\[ 2 = 2 \times C \times k \]
Substituting \( k \):
\[ 2 = 2 \times C \times 3.125 \]
Solving for \( C \):
\[ C = \frac{2}{2 \times 3.125} = \frac{2}{6.25} = 0.32 \, \text{M} \]

### Final Answer
The concentration of the NaOH solution is:
\[ \boxed{0.32 \, \text{M}} \]