At `25^(@)C`, a solution containing `0.2g` of polyisobutylene in `100mL` of benzene develpoed a rise of `2.4mm` at osmotic equilibrium. Calculate the molecular weight of polyisobutylene if the density of solution is `0.88 g// mL`

Options

A. `2.39 xx10^(5)g`
B. `33.9xx10^(5)g`
C. `43.8xx10^(5)g`
D. `78.6xx10^(5)g`

Explanation

To calculate the molecular weight of polyisobutylene from the given data, we can follow these steps:

### Step 1: Understand the given data
- Mass of polyisobutylene (solute), \( m = 0.2 \, \text{g} \)
- Volume of benzene (solvent), \( V = 100 \, \text{mL} = 0.1 \, \text{L} \)
- Rise in osmotic pressure, \( \Delta h = 2.4 \, \text{mm} \)
- Density of the solution, \( \rho = 0.88 \, \text{g/mL} \)

### Step 2: Convert the rise in height to osmotic pressure
To convert the height of the liquid column (in mm) to osmotic pressure (in atm), we can use the formula:

\[ \pi = \frac{\Delta h \cdot \rho \cdot g}{760} \]

Where:
- \( \Delta h \) = 2.4 mm
- \( \rho \) = density of the solution = 0.88 g/mL
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- 760 mm Hg = 1 atm

First, convert the height to meters and then calculate the osmotic pressure:

\[ \pi = \frac{2.4 \, \text{mm} \cdot 0.88 \, \text{g/mL} \cdot 9.81 \, \text{m/s}^2}{760 \, \text{mm Hg}} \]

Calculating this gives:

\[ \pi = \frac{2.4 \times 0.88 \times 9.81}{760} \approx 0.024 \, \text{atm} \]

### Step 3: Use the osmotic pressure equation
The osmotic pressure equation is given by:

\[ \pi = \frac{n}{V}RT \]

Where:
- \( n \) = number of moles of solute
- \( V \) = volume of solution in liters
- \( R \) = ideal gas constant = 0.0821 L·atm/(K·mol)
- \( T \) = temperature in Kelvin = 25°C = 298 K

Rearranging for \( n \):

\[ n = \pi V / RT \]

Substituting the known values:

\[ n = \frac{0.024 \, \text{atm} \cdot 0.1 \, \text{L}}{0.0821 \, \text{L·atm/(K·mol)} \cdot 298 \, \text{K}} \approx 9.8 \times 10^{-4} \, \text{mol} \]

### Step 4: Calculate the molar mass
The molar mass \( M \) can be calculated using the formula:

\[ M = \frac{m}{n} \]

Where:
- \( m \) = mass of solute = 0.2 g
- \( n \) = number of moles calculated above

Substituting the values:

\[ M = \frac{0.2 \, \text{g}}{9.8 \times 10^{-4} \, \text{mol}} \approx 204.08 \, \text{g/mol} \]

### Final Answer
The molecular weight of polyisobutylene is approximately **204.08 g/mol**.

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