A solution containing 12.5g of a non electrolyte substance in 175g of water gave a boiling point elevation of 0.70 K. Calculate the molar mass of the substance. (Elevation constant for water, `K_(b)` = 0.52 K kg `"mol"^(-1)`)

Explanation

To calculate the molar mass of the non-electrolyte substance, we can follow these steps:

### Step 1: Write the formula for boiling point elevation
The boiling point elevation (\( \Delta T_b \)) is given by the formula:
\[ \Delta T_b = K_b \cdot m \]
where:
- \( \Delta T_b \) = boiling point elevation (in K)
- \( K_b \) = ebullioscopic constant of the solvent (water in this case)
- \( m \) = molality of the solution

### Step 2: Calculate the molality of the solution
Molality (\( m \)) is defined as:
\[ m = \frac{\text{mass of solute (in kg)}}{\text{mass of solvent (in kg)}} \]
Given:
- Mass of solute = 12.5 g = 0.0125 kg
- Mass of solvent (water) = 175 g = 0.175 kg

Substituting the values:
\[ m = \frac{0.0125 \, \text{kg}}{0.175 \, \text{kg}} = 0.07143 \, \text{mol/kg} \]

### Step 3: Substitute values into the boiling point elevation formula
Now, substituting the values of \( \Delta T_b \) and \( K_b \) into the boiling point elevation formula:
\[ 0.70 = 0.52 \cdot m \]
Substituting \( m \):
\[ 0.70 = 0.52 \cdot 0.07143 \]

### Step 4: Solve for molality
To find \( m \):
\[ m = \frac{0.70}{0.52} \approx 1.346 \, \text{mol/kg} \]

### Step 5: Relate molality to the number of moles and calculate molar mass
Molality can also be expressed in terms of moles of solute and mass of solvent:
\[ m = \frac{n}{\text{mass of solvent (in kg)}} \]
where \( n \) is the number of moles of solute. Rearranging gives:
\[ n = m \cdot \text{mass of solvent (in kg)} = 1.346 \cdot 0.175 \approx 0.2356 \, \text{mol} \]

### Step 6: Calculate the molar mass
Molar mass (\( M \)) is given by:
\[ M = \frac{\text{mass of solute (in g)}}{n} \]
Substituting the values:
\[ M = \frac{12.5 \, \text{g}}{0.2356 \, \text{mol}} \approx 53.06 \, \text{g/mol} \]

### Final Answer
The molar mass of the non-electrolyte substance is approximately **53.06 g/mol**.

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