A solution of 1 g phenol in 50 ml of diethly ehter bolied at a temperature elevated by `0.632^(@)C`. The molal elevation constant for diethyl either is 2.12 kg/ mol and density is 0.714 g/ml. Calculate molecular weight of phenol.

Answer

To calculate the molecular weight of phenol from the given data, we will follow these steps:

### Step 1: Understand the Formula for Boiling Point Elevation
The formula for boiling point elevation is given by:
\[ \Delta T_b = K_b \times m \]
where:
- \(\Delta T_b\) = elevation in boiling point
- \(K_b\) = molal elevation constant
- \(m\) = molality of the solution

### Step 2: Calculate Molality
Molality (\(m\)) is defined as:
\[ m = \frac{\text{Number of moles of solute}}{\text{Mass of solvent in kg}} \]
The number of moles of solute can be expressed as:
\[ \text{Number of moles} = \frac{w_2}{M} \]
where:
- \(w_2\) = mass of solute (phenol) in grams
- \(M\) = molecular weight of the solute (phenol)

Thus, we can rewrite the molality as:
\[ m = \frac{w_2}{M \times w_1} \]
where \(w_1\) is the mass of the solvent in kg.

### Step 3: Find the Mass of Solvent
Given that the volume of diethyl ether is 50 mL and its density is 0.714 g/mL, we can calculate the mass of the solvent:
\[ w_1 = \text{Density} \times \text{Volume} = 0.714 \, \text{g/mL} \times 50 \, \text{mL} = 35.7 \, \text{g} \]
Now convert grams to kilograms:
\[ w_1 = \frac{35.7}{1000} = 0.0357 \, \text{kg} \]

### Step 4: Substitute Values into the Boiling Point Elevation Formula
Now we can substitute the values into the boiling point elevation formula:
\[ \Delta T_b = K_b \times m \]
Substituting for \(m\):
\[ \Delta T_b = K_b \times \frac{w_2}{M \times w_1} \]
Rearranging for \(M\):
\[ M = \frac{K_b \times w_2}{\Delta T_b \times w_1} \]

### Step 5: Substitute Known Values
Now substitute the known values:
- \(K_b = 2.12 \, \text{kg/mol}\)
- \(w_2 = 1 \, \text{g}\)
- \(\Delta T_b = 0.632 \, \text{°C}\)
- \(w_1 = 0.0357 \, \text{kg}\)

Thus:
\[ M = \frac{2.12 \times 1}{0.632 \times 0.0357} \]

### Step 6: Calculate the Molecular Weight
Calculating the denominator:
\[ 0.632 \times 0.0357 \approx 0.0226 \]
Now calculate \(M\):
\[ M = \frac{2.12}{0.0226} \approx 93.96 \, \text{g/mol} \]

### Step 7: Final Result
The molecular weight of phenol is approximately:
\[ M \approx 94 \, \text{g/mol} \]