Problem 82 - General Chemistry: Principles and Modern Applications

Question

Calculate the van't Hoff factors of the following weak electrolyte solutions: (a) \(0.050 \mathrm{m}\) HCHO \(_{2}\), which begins to freeze at \(-0.0986^{\circ} \mathrm{C}\).(b) \(0.100 \mathrm{M} \mathrm{HNO}_{2},\) which has a hydrogen ion (and nitrite ion) concentration of \(6.91 \times 10^{-3} \mathrm{M}\).

Step-by-Step Solution

Answer

The calculated van't Hoff factor for a 0.050m solution of HCHO_2 is computed and extracted from the result in Step 1. And for a 0.100M solution of HNO_2, the calculated van't Hoff factor is computed and extracted from the result in Step 2.

Step 1: Calculation of van't Hoff factor for part (a)

The freezing point depression of the solution is \(\Delta T_{f} = 0.0986^{\circ}C\) and the molality (m) of the HCHO_2 solution is 0.050m. The cryoscopic constant for water \(K_{f}\) is 1.86 K·kg/mol. Now we rearrange the formula \(\Delta T_{f} = i \cdot K_{f} \cdot m\) for 'i': \(i = \frac{\Delta T_{f}}{K_{f} \cdot m}\). Now substitute the given values into the formula to calculate the van't Hoff factor.

Step 2: Calculation of van't Hoff factor for part (b)

For HNO_2, the molar concentration is 0.100M and the ion concentration is \(6.91 \times 10^{-3} M\). The ion concentration indicates the concentration of both H+ ions and NO_2- ions, which totals to \(2 \times 6.91 \times 10^{-3} = 1.382 \times 10^{-2} M\). Van’t Hoff factor 'i' is calculated by dividing the molar ion concentration by the molar concentration of the compound: \(i = \frac{1.382 \times 10^{-2} M}{0.100 M}\). Calculate this ratio to obtain the van't Hoff factor.

Key Concepts

Freezing Point Depression

Freezing Point Depression is a colligative property of solutions. This means it depends on the number of solute particles in the solution rather than their identity. When a solute is added to a solvent, the freezing point of the resulting solution is lower than that of the pure solvent. This phenomenon occurs because the solute particles disrupt the formation of the solid structure of the solvent, needing lower temperatures to achieve freezing.
For example, when you add salt to icy roads, it helps to melt the ice by lowering its freezing point. The formula used to calculate the change in freezing point is \[ \Delta T_f = i \cdot K_f \cdot m \] where \( \Delta T_f \) is the freezing point depression, \( i \) is the van’t Hoff factor, \( K_f \) is the cryoscopic constant of the solvent, and \( m \) is the molality of the solution. The van't Hoff factor reflects the number of particles the solute breaks into in the solution, which is crucial in determining the extent of the freezing point depression.

Weak Electrolytes

Weak electrolytes are substances that partially dissociate into ions in solution. Unlike strong electrolytes, which dissociate completely, weak electrolytes exist in equilibrium between the undissociated molecules and the ions in the solution.
Common examples include acetic acid and ammonia. Because they do not fully dissociate, weak electrolytes create fewer ions in solution, which affects properties like conductivity and colligative properties such as freezing point depression.

• Acetic Acid: Only partially ionizes into acetate ions and hydrogen ions.
• Ammonia: Ionizes partly to form ammonium ions and hydroxide ions.

In the context of a calculation involving the Van't Hoff factor, a weak electrolyte's van't Hoff factor is less than its stoichiometric value due to incomplete ionization. This partial ionization implies that fewer particles are contributing to colligative properties, making understanding of the van’t Hoff factor essential to accurately calculate changes such as freezing point depression.

Ion Concentration

Ion Concentration is a measure of how many ions are present in a solution. It is a key concept in understanding the behavior of both strong and weak electrolytes. The concentration of ions determines many physical and chemical properties of solutions, such as conductivity and pH.
In the case of weak electrolytes, the ion concentration can be calculated using known acid or base dissociation constants, as these substances only partially ionize:

• For a weak acid, the equilibrium concentration of ions like \( H^+ \) can be deduced from the acid dissociation constant \( K_a \).
• For a weak base, the \( OH^- \) ion concentration can be derived from the base dissociation constant \( K_b \).

Ion concentration also plays a critical role in calculating the van't Hoff factor, as it is the number of ions that directly influences the degree of ionization. By understanding ion concentration, we can grasp how many particles are in the solution, which helps in determining colligative properties like freezing point depression.