Problem 83 - Chemistry The Central Science

Question

The osmotic pressure of a \(0.010 \mathrm{M}\) aqueous solution of \(\mathrm{CaCl}_{2}\) is found to be 0.674 atm at \(25^{\circ}\) C. (a) Calculate the van't Hoff factor, \(i\), for the solution. (b) How would you expect the value of \(i\) to change as the solution becomes more concentrated? Explain.

Step-by-Step Solution

Answer

The van't Hoff factor, i, for a 0.010 M CaCl2 solution is 0.92, calculated using the formula \( i = \frac{\Pi_{experimental}}{\Pi_{ideal}} =\frac{0.674 \, \text{atm}}{0.733 \, \text{atm}} \). As the concentration of the solution increases, the van't Hoff factor is expected to deviate further from the ideal value of 3, due to increased ionic interactions.

Step 1: Calculate the theoretical osmotic pressure for a 0.010 M CaCl2 solution

Assuming that CaCl2 dissociates ideally into one Ca2+ ion and two Cl- ions (i = 3), we can calculate the theoretical osmotic pressure using the formula: \( \Pi_{ideal} = i_{ideal} \times c \times R \times T \) Where \(i_{ideal}\) is the van't Hoff factor for ideal dissociation, which is 3 in our case. We will plug in the values to obtain the theoretical osmotic pressure: \( \Pi_{ideal} = 3 \times 0.010 \, \text{M} \times 0.0821 \, \frac{\text{L atm}}{\text{mol K}} \times 298.15 \, \text{K} \)

Step 2: Calculate the van't Hoff factor, i

Before calculating the van't Hoff factor, first compute the value of \( \Pi_{ideal} \): \( \Pi_{ideal} = 0.733 \, \text{atm} \) Now we can calculate the van't Hoff factor by comparing the experimental osmotic pressure to the theoretical osmotic pressure: \( i = \frac{\Pi_{experimental}}{\Pi_{ideal}} \) \( i = \frac{0.674 \, \text{atm}}{0.733 \, \text{atm}} \)

Step 3: Compute the van't Hoff factor

Divide the experimental osmotic pressure by the theoretical osmotic pressure: \( i = 0.92 \) Thus, the van't Hoff factor for this 0.010 M CaCl2 solution is 0.92.

Step 4: Discuss the change in the van't Hoff factor as the concentration increases

As the concentration of the CaCl2 solution increases, the ions will be more crowded in the solution, leading to a higher chance of the ions interacting with one another. These interactions may decrease the extent to which the ions are separated from one another and, consequently, the van't Hoff factor could deviate further from the ideal value of 3. Thus, as the concentration increases, we would expect the van't Hoff factor to decrease from the ideal value.

Key Concepts

The van't Hoff Factor

The van't Hoff factor, often represented as \( i \), is a measure used to account for the effect of solute particles on colligative properties such as osmotic pressure, boiling point elevation, and freezing point depression. It is particularly useful for solutions containing ionic compounds, which dissociate in water to produce multiple ions. Here are some key points about the van't Hoff factor:

• For a completely dissociated ionic compound like \( \text{CaCl}_2 \), which ideally dissociates into three ions (one \( \text{Ca}^{2+} \) and two \( \text{Cl}^- \) ions), the theoretical van't Hoff factor would be 3.
• In practice, the van't Hoff factor may differ from the theoretical value due to ion interactions and other real-world factors. For example, in this exercise, \( i \) is calculated to be 0.92, lower than the ideal value of 3.
• The experimental van't Hoff factor is determined by comparing the experimental osmotic pressure with the theoretical osmotic pressure calculated assuming ideal dissociation.

Understanding \( i \) helps explain discrepancies between expected and observed colligative properties.

Ion Interactions

Ion interactions play a crucial role in determining the behavior of ions in a solution, particularly as concentration increases. These interactions can cause deviations from the ideal behavior expected based on the theoretical van't Hoff factor.

• As the concentration of ions in a solution rises, ions are closer together, increasing the likelihood of interactions between them.
• Such interactions can include attractions or repulsions that influence the extent of dissociation of the ionic compound.
• The more significant these interactions, the farther the behavior of the solution may deviate from ideal, impacting the van't Hoff factor and causing it to be lower than the expected value.

Therefore, ion interactions are a vital factor to consider when predicting and understanding the properties of an ionic solution.

Solution Concentration Effects

Solution concentration has a profound effect on the physical properties of a solution, and it is particularly evident in the context of colligative properties like osmotic pressure.Increased Concentration Effects
As the concentration of the solute increases:

• The chance of ion interactions increases, which can reduce the number of freely moving ions, affecting properties like osmotic pressure.
• This often results in the real van't Hoff factor being lower than the ideal one, as seen with \( \text{CaCl}_2 \) in the exercise.
• The solution may behave less ideally. Factors such as ion pairing and clustering can occur, further affecting the observed properties.

By knowing these effects, we can better predict and explain how changes in concentration influence the physical properties and behaviors of solutions.