Calculate the molal elevation constant, `"K"_("b")` for water and the boiling point of `0.1` molal urea solution. Latent heat of vaporisation of water is `9.72` kcal `"mol"^(-1)` at `373.15` K.

Answer

To solve the problem, we will follow these steps:

### Step 1: Note down the given data
- Molality of urea solution (m) = 0.1 molal
- Latent heat of vaporization of water (ΔH_vaporization) = 9.72 kcal/mol
- Boiling point of water (T_b) = 373.15 K

### Step 2: Convert latent heat from kcal to joules
1. Convert kcal to calories:
\[ 9.72 \text{ kcal} = 9.72 \times 1000 \text{ cal} = 9720 \text{ cal} \]
2. Convert calories to joules:
\[ 9720 \text{ cal} \times 4.184 \text{ J/cal} = 40668.48 \text{ J} \]

### Step 3: Use the formula to calculate the molal elevation constant (K_b)
The formula relating K_b to the latent heat of vaporization is:
\[ K_b = \frac{R \cdot M_1 \cdot T_b^2}{1000 \cdot \Delta H_{vaporization}} \]
Where:
- \( R \) = 8.314 J/(mol·K) (universal gas constant)
- \( M_1 \) = 18 g/mol (molar mass of water)
- \( T_b \) = 373.15 K
- \( \Delta H_{vaporization} = 40668.48 \text{ J} \)

Substituting the values into the formula:
\[ K_b = \frac{8.314 \cdot 18 \cdot (373.15)^2}{1000 \cdot 40668.48} \]

### Step 4: Calculate K_b
1. Calculate \( T_b^2 \):
\[ (373.15)^2 = 139,507.6225 \text{ K}^2 \]
2. Substitute and calculate:
\[ K_b = \frac{8.314 \cdot 18 \cdot 139507.6225}{1000 \cdot 40668.48} \]
\[ K_b = \frac{8.314 \cdot 18 \cdot 139507.6225}{40668480} \]
\[ K_b = \frac{0.512 \text{ K kg/mol}} \]

### Step 5: Calculate the boiling point elevation (ΔT_b)
Using the formula for boiling point elevation:
\[ \Delta T_b = i \cdot K_b \cdot m \]
Where:
- \( i \) = van 't Hoff factor for urea = 1 (since urea is a non-electrolyte)
- \( m \) = 0.1 molal

Substituting the values:
\[ \Delta T_b = 1 \cdot 0.512 \cdot 0.1 = 0.0512 \text{ K} \]

### Step 6: Calculate the new boiling point
The new boiling point of the urea solution is:
\[ T_{boiling} = T_b + \Delta T_b = 373.15 + 0.0512 = 373.20 \text{ K} \]

### Final Answers
- The molal elevation constant \( K_b \) for water is **0.512 K kg/mol**.
- The boiling point of the 0.1 molal urea solution is **373.20 K**.