0.067 molar aqueous solution of a binary electrolyte `A^(+)B^(-)` shows 2.46 atm osmotic pressure at `27^(@)C` . What fraction of `A^(+)B^(-)` remains unionised ?

Options

A. 0.1
B. 0.15
C. 0.5
D. Zero

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
We have a 0.067 M solution of a binary electrolyte \( A^+B^- \) and the osmotic pressure \( \pi \) is given as 2.46 atm at a temperature of 27°C.

### Step 2: Convert temperature to Kelvin
To use the ideal gas law in the osmotic pressure formula, we need to convert the temperature from Celsius to Kelvin:
\[ T(K) = 27 + 273 = 300 \, K \]

### Step 3: Use the osmotic pressure formula
The formula for osmotic pressure is given by:
\[ \pi = iCRT \]
Where:
- \( \pi \) = osmotic pressure (2.46 atm)
- \( i \) = van 't Hoff factor (number of particles the solute breaks into)
- \( C \) = molarity of the solution (0.067 mol/L)
- \( R \) = ideal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin (300 K)

### Step 4: Rearranging the formula to find \( i \)
We can rearrange the formula to solve for \( i \):
\[ i = \frac{\pi}{CRT} \]

### Step 5: Substitute the values into the equation
Now we can substitute the values we have:
\[ i = \frac{2.46}{0.067 \times 0.0821 \times 300} \]

### Step 6: Calculate \( i \)
Calculating the denominator:
\[ 0.067 \times 0.0821 \times 300 = 1.649 \]
Now substituting this back into the equation for \( i \):
\[ i = \frac{2.46}{1.649} \approx 1.49 \]

### Step 7: Relate \( i \) to the degree of ionization \( \alpha \)
For a binary electrolyte \( A^+B^- \), the van 't Hoff factor \( i \) can be expressed as:
\[ i = 1 + \alpha \]
Where \( \alpha \) is the degree of ionization (the fraction of the solute that ionizes).

### Step 8: Solve for \( \alpha \)
From our previous calculation:
\[ 1 + \alpha = 1.49 \]
Thus,
\[ \alpha = 1.49 - 1 = 0.49 \]

### Step 9: Calculate the fraction that remains unionized
The fraction of the electrolyte that remains unionized is given by:
\[ \text{Fraction unionized} = 1 - \alpha \]
Substituting the value of \( \alpha \):
\[ \text{Fraction unionized} = 1 - 0.49 = 0.51 \]

### Final Answer
The fraction of \( A^+B^- \) that remains unionized is approximately 0.5.

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