When 40 g of substance is dissolved in 1000 g g of water, its freezing point is depressed by `1.86^@C. `K_f` for water is `1.86^@C` `mol^-1` then find the molar mass of the solute.

Options

A. 4
B. 10
C. 40
D. 400

Answer

To find the molar mass of the solute, we can use the formula for freezing point depression:

\[ \Delta T_f = K_f \cdot m \]

Where:
- \(\Delta T_f\) is the depression in freezing point,
- \(K_f\) is the freezing point depression constant,
- \(m\) is the molality of the solution.

### Step 1: Identify the given values
- Mass of the solute (substance) = 40 g
- Mass of the solvent (water) = 1000 g
- Depression in freezing point (\(\Delta T_f\)) = 1.86 °C
- Freezing point depression constant for water (\(K_f\)) = 1.86 °C kg/mol

### Step 2: Calculate the molality (m)
Molality is defined as the number of moles of solute per kilogram of solvent.

First, we need to convert the mass of the solvent from grams to kilograms:
\[ \text{Mass of solvent in kg} = \frac{1000 \text{ g}}{1000} = 1 \text{ kg} \]

Now, we can rearrange the freezing point depression formula to find molality:
\[ m = \frac{\Delta T_f}{K_f} \]

Substituting the values:
\[ m = \frac{1.86 \text{ °C}}{1.86 \text{ °C kg/mol}} = 1 \text{ mol/kg} \]

### Step 3: Calculate the number of moles of solute
Using the definition of molality:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]

Rearranging gives us:
\[ \text{moles of solute} = m \cdot \text{mass of solvent in kg} \]
\[ \text{moles of solute} = 1 \text{ mol/kg} \cdot 1 \text{ kg} = 1 \text{ mol} \]

### Step 4: Calculate the molar mass of the solute
Molar mass is defined as the mass of the solute divided by the number of moles of solute:
\[ \text{Molar mass} = \frac{\text{mass of solute}}{\text{moles of solute}} \]
\[ \text{Molar mass} = \frac{40 \text{ g}}{1 \text{ mol}} = 40 \text{ g/mol} \]

### Final Answer
The molar mass of the solute is **40 g/mol**.

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