Van't Hoff factor for K2SO4 at complete dissociation and its effect on boiling p

Van't Hoff Factor and Boiling Point Elevation for $K_2SO_4$ This question tests the understanding of the Van't Hoff factor and its application in colligative properties, specifically boiling point elevation. It requires students to determine the theoretical Van't Hoff factor for an ionic compound like potassium sulfate ($K_2SO_4$) assuming complete dissociation and then use this factor to calculate the boiling point elevation. The core concepts are dissociation of electrolytes and the relationship between boiling point elevation, molality, and the Van't Hoff factor.

Concept Overview

The Van't Hoff factor ($i$) quantifies the extent to which a solute dissociates or associates in a solution. For ionic compounds that dissociate completely, $i$ is equal to the number of ions produced per formula unit. Boiling point elevation ($Delta T_b$) is a colligative property that depends on the concentration of solute particles, and it is directly proportional to the molality ($m$) of the solution and the Van't Hoff factor ($i$).

Step 1: Determine the dissociation of $K_2SO_4$ in water. Potassium sulfate ($K_2SO_4$) is a strong electrolyte. When it dissolves in water, it dissociates into potassium ions ($K^+$) and sulfate ions ($SO_4^{2-}$). Assuming complete dissociation, the dissociation equation is: $K_2SO_4(s) \rightarrow 2K^+(aq) + SO_4^{2-}(aq)$ This equation shows that one formula unit of $K_2SO_4$ produces two potassium ions and one sulfate ion.

Step 2: Calculate the theoretical Van't Hoff factor ($i$) for $K_2SO_4$. The Van't Hoff factor ($i$) is defined as the ratio of the observed colligative property to the colligative property calculated assuming no dissociation. For complete dissociation, $i$ is equal to the total number of ions formed from one formula unit of the solute. From the dissociation equation in Step 1, we have 2 $K^+$ ions and 1 $SO_4^{2-}$ ion, totaling $2 + 1 = 3$ ions. Therefore, the theoretical Van't Hoff factor for $K_2SO_4$ at complete dissociation is: $i = 3$

Step 3: Understand the formula for boiling point elevation. The elevation in boiling point ($Delta T_b$) of a solvent upon the addition of a non-volatile solute is given by the formula: $\Delta T_b = i \cdot K_b \cdot m$ where:

• $Delta T_b$ is the boiling point elevation (in $^\circ C$ or $K$).
• $i$ is the Van't Hoff factor of the solute.
• $K_b$ is the ebullioscopic constant of the solvent (for water, $K_b = 0.52 \, ^\circ C \cdot kg/mol$).
• $m$ is the molality of the solution (in mol/kg).

Step 4: Apply the Van't Hoff factor to calculate boiling point elevation. The question asks for the effect of complete dissociation on the boiling point elevation. This means we need to use the calculated Van't Hoff factor ($i=3$) in the boiling point elevation formula. If $K_2SO_4$ were a non-electrolyte ($i=1$), the boiling point elevation would be $Delta T_b = K_b \cdot m$. However, because it dissociates into 3 ions, the effective concentration of particles in the solution is tripled, leading to a higher boiling point elevation. Thus, the boiling point elevation for a solution of $K_2SO_4$ at complete dissociation will be: $\Delta T_b = 3 \cdot K_b \cdot m$ This shows that the boiling point elevation is three times greater than it would be if $K_2SO_4$ did not dissociate.

Key Takeaways:

• The Van't Hoff factor ($i$) accounts for the dissociation or association of solutes in solution.
• For strong electrolytes like $K_2SO_4$, assuming complete dissociation, $i$ equals the number of ions produced per formula unit.
• Boiling point elevation is a colligative property that increases with the number of solute particles, as described by $Delta T_b = i \cdot K_b \cdot m$.
• A higher Van't Hoff factor leads to a greater boiling point elevation for the same molality.

Answer: The Van't Hoff factor for $K_2SO_4$ at complete dissociation is 3. This means the boiling point elevation for a $K_2SO_4$ solution will be three times what it would be for a non-electrolyte of the same molality, i.e., $Delta T_b = 3 \cdot K_b \cdot m$.