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 \(25^{\circ} \mathrm{C}\). (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 the given \(0.010\: M\) CaCl2 solution is calculated to be \(2.76\). As the solution becomes more concentrated, the value of \(i\) is expected to decrease due to stronger ion-ion interactions and the formation of ion pairs, leading to incomplete dissociation.

Step 1: Find the theoretical osmotic pressure for i=1

From the ideal gas law, the molar concentration of a substance in solution is directly related to its osmotic pressure: \[Osmotic\: Pressure = OSP_{Theological} = (M)(R)(T)\] Here, M is the molarity of the solution, which is \(0.010\: M\) R is the ideal gas constant, \(0.0821\frac{L \cdot atm}{mol \cdot K}\) T is the temperature in Kelvin, \(25^{\circ}C +(273.15\) K \(=298.15\) K Now, plug in these values to find the theoretical osmotic pressure, \(OSP_{Theoretical}\). \[ OSP_{Theoretical} = (0.010\: M)(0.0821\frac{L \cdot atm}{mol \cdot K})(298.15\: K) = 0.244\: atm\]

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

We know that the van't Hoff factor, \(i\), is the ratio of the measured osmotic pressure to the theoretical osmotic pressure when \(i=1\): \[i = \frac{OSP_{Measured}}{OSP_{Theoretical}}\] Given the osmotic pressure, \(OSP_{Measured} = 0.674\: atm\), we can find the value of \(i\): \[i = \frac{0.674\: atm}{0.244\: atm} = 2.76\]

Step 3: Part (b) - Expected change in the value of i with increased concentration

When an ionic solute dissolves in water, it dissociates into its constituent ions. The degree of dissociation affects the van't Hoff factor. In highly concentrated solutions, the ions are closely packed and experience stronger ion-ion interactions, which can lead to incomplete dissociation. In such cases, the van't Hoff factor would be lower than in a more dilute solution. Moreover, as the concentration increases, the ions are more likely to interact with each other and form ion pairs rather than existing as individual ions. This will also result in decreased values of the van't Hoff factor at higher concentrations.

Key Concepts

Van't Hoff Factor

The van't Hoff factor, symbolized by the letter i, plays a crucial role in determining the properties of solutions, particularly for electrolytes, which are substances that dissolve to form ions in a solution. It represents the number of particles into which a compound dissociates in solution.

When a substance that does not ionize or dissociate in solution is dissolved, such as glucose in water, the van't Hoff factor is 1. For substances that do dissociate, like salts, i can be greater than 1. For example, NaCl would theoretically have an i of 2, as it dissociates into one sodium ion and one chloride ion.

However, the actual van't Hoff factor observed experimentally can differ due to a variety of reasons, such as ionic interactions at higher concentrations, which impede complete dissociation, as well as ion pairing—where the positive and negative ions attract each other and act as a single unit instead of separate entities. This concept becomes particularly useful for calculating related colligative properties such as osmotic pressure, boiling point elevation, and freezing point depression for ionic compounds.

Ideal Gas Law

The Ideal Gas Law is fundamental to understanding the relationship between the physical properties of gases. It establishes a correlation between pressure (P), volume (V), temperature (T), and the amount of substance in moles (n) using the equation: \[PV = nRT\]

Where R is the universal gas constant, which has a value of 0.0821 L atm mol^-1 K^-1. This equation assumes that the gas particles occupy no volume and that there are no intermolecular forces between them—conditions that are approximated by real gases at low pressure and high temperature.

In the context of osmosis, the osmotic pressure can be related to the ideal gas law by treating the solute particles in the solution as a 'gas'. This conceptual translation allows us to use the ideal gas law to predict osmotic pressure as if the dissolved particles were in a gaseous state, exerting pressure on the walls of the container.

Ionic Dissociation

Ionic dissociation refers to the process in which solid ionic compounds dissolve in a solvent and separate into their constituent ions, typically within an aqueous solution. The degree of dissociation is a measure of how completely the ionic compound separates into ions.

Ionic dissociation is influenced by factors such as the nature of the solute, the solvent, and the concentration of the solution. In dilute solutions, ionic compounds generally dissociate completely. However, at higher concentrations, various forces like electrostatic interactions between ions become more significant, resulting in incomplete dissociation, and the formation of ion pairs.

Understanding ionic dissociation is crucial when calculating properties like conductivity, osmotic pressure, and the van't Hoff factor. Indeed, in the exercise provided, the distinction between the theoretical and observed osmotic pressure points to the reality that CaCl₂ does not fully dissociate into three separate ions (Ca²⁺ and two Cl^- ions) in solution. This phenomenon is what drove the observed van't Hoff factor to be different from the expected theoretical value, illustrating the importance of such interactions in real-world scenarios.