Step 1: Understanding the Concept:
Freezing point depression is a colligative property that describes the decrease in the freezing point of a liquid when a solute is added.
In a pure solvent, molecules are in a dynamic equilibrium between the liquid and solid phases at the freezing point.
Solute particles disrupt the orderly arrangement of solvent molecules into a solid lattice.
As a result, a lower temperature is required to freeze the solution compared to the pure solvent.
The amount by which the freezing point drops depends on the concentration of the solute particles.
Step 2: Key Formula or Approach:
The change in freezing point (\(\Delta T_{f}\)) is given by the formula:
\[ \Delta T_{f} = i \cdot K_{f} \cdot m \]
Where:
\(\Delta T_{f}\) is the depression in freezing point ($T_{f}^{\circ} - T_{f}$).
\(i\) is the van't Hoff factor.
\(K_{f}\) is the molal freezing point depression constant (cryoscopic constant).
\(m\) is the molality of the solution.
The freezing point of the solution (\(T_{f}\)) is then calculated as:
\[ T_{f} = T_{f}^{\circ} - \Delta T_{f} \]
For water, the standard freezing point (\(T_{f}^{\circ}\)) is \(0^{\circ}C\).
Step 3: Detailed Explanation:
Step-by-step calculation:
1. Determine the van't Hoff factor (i):
The solute is \(NaCl\) (Sodium chloride). It is a strong electrolyte that dissociates fully in aqueous solution:
\(NaCl \rightarrow Na^{+} + Cl^{-}\).
Since it produces 2 ions, \(i = 2\).
2. Identify given values:
Molality (\(m\)) = 0.1 m.
Cryoscopic constant (\(K_{f}\)) = \(1.86^{\circ}C \cdot m^{-1}\).
3. Calculate the depression (\(\Delta T_{f}\)):
\[ \Delta T_{f} = 2 \times 1.86 \times 0.1 \]
\[ \Delta T_{f} = 0.372^{\circ}C \]
4. Determine the freezing point of the solution:
Pure water freezes at \(0^{\circ}C\).
\[ T_{f} = 0^{\circ}C - 0.372^{\circ}C \]
\[ T_{f} = -0.372^{\circ}C \].
This means that instead of freezing at $0^{\circ}C$, the solution will remain liquid until it reaches $-0.372^{\circ}C$.
Step 4: Final Answer:
The freezing point of the 0.1 m NaCl solution is \(-0.372^{\circ}C\).