Question

The osmotic pressure of \(0.010-M\) solutions of \(\mathrm{CaCl}_{2}\) and urea at \(25^{\circ} \mathrm{C}\) are 0.605 and 0.245 atm, respectively. Calculate the van't Hoff factor for the \(\mathrm{CaCl}_{2}\) solution.

Short answer

The van't Hoff factor for \( \mathrm{CaCl}_{2} \) is approximately 2.5.

Step-by-step solution

Understand the Formula for Osmotic Pressure

Osmotic pressure \( \Pi \) is calculated using the formula \( \Pi = i \, M \, R \, T \), where \( i \) is the van't Hoff factor, \( M \) is the molarity, \( R \) is the ideal gas constant (0.0821 \( L \, atm \, K^{-1} \, mol^{-1} \)), and \( T \) is the temperature in Kelvin. For this problem, \( T = 25^{\circ} \mathrm{C} = 298 \, K \). This formula relates the concentration of the solute to the osmotic pressure of the solution.

Calculate the Osmotic Pressure for Ideal Solution

Calculate the expected osmotic pressure for an ideal (non-dissociating) solution, like urea. Given that the osmotic pressure of the 0.010-M urea solution is 0.245 atm, we can assume \( i = 1 \) for urea, which is a non-electrolyte.

Use the Osmotic Pressure Formula for \( \mathrm{CaCl}_{2} \)

For \( \mathrm{CaCl}_{2} \), the measured osmotic pressure \( \Pi \) is 0.605 atm. Using \( \Pi = i \, M \, R \, T \), we can rearrange to solve for \( i \): \[ i = \frac{\Pi}{M \, R \, T} = \frac{0.605}{0.010 \times 0.0821 \times 298} \].

Calculate \( i \) for \( \mathrm{CaCl}_{2} \)

Substitute the known values into the rearranged formula: \[ i = \frac{0.605}{0.010 \times 0.0821 \times 298} \approx 2.5 \]. Hence, the van't Hoff factor for \( \mathrm{CaCl}_{2} \) is approximately 2.5.

Key concepts

Osmotic Pressure

Osmotic pressure is a fundamental concept in chemistry, often used to understand how solutes in a solution influence the movement of water across a semipermeable membrane. Osmotic pressure, denoted by the Greek letter \( \Pi \), refers to the pressure required to stop this water movement. The concept is particularly important in biological processes like the maintenance of cell turgor pressure.
Osmotic pressure is calculated using the formula: \( \Pi = i \cdot M \cdot R \cdot T \). Here:

• \( i \) is the van’t Hoff factor, accounting for the degree of ionization of the solute.
• \( M \) is the molarity of the solution, representing concentration.
• \( R \) is the ideal gas constant (0.0821 \( L \, atm \, K^{-1} \, mol^{-1} \)).
• \( T \) is the temperature in Kelvin.

The concept of osmotic pressure is pivotal in understanding phenomena like plant root absorption and kidney filtration.

Molarity

Molarity is a key concept in chemistry that quantifies the concentration of a solution. It's defined as the number of moles of solute per liter of solution. In formulaic terms, molarity \( M \) is expressed as \( M = \frac{n}{V} \), where:

• \( n \) = number of moles of solute,
• \( V \) = volume of the solution in liters.

Molarity helps chemists understand the proportion of molecules or ions in a given solution, which is crucial for reactions and processes such as titrations and stoichiometric calculations.
In the context of osmotic pressure, molarity is used to determine how much solute is present, affecting how the solution behaves when a semipermeable membrane is involved.

Ideal Gas Constant

The ideal gas constant \( R \) is an essential constant in the equations governing the behavior of gases. It appears in the equation of state for an ideal gas, been a part of various calculations like osmotic pressure and even in thermodynamics equations.
The value of \( R \) is 0.0821 \( L \, atm \, K^{-1} \, mol^{-1} \), which is derived from the properties of gases under standard conditions. The constant's universality makes it useful across different scenarios, from explaining gas behavior to predicting changes in gas properties with temperature or volume.
It's crucial in the osmotic pressure equation, linking chemical processes with broader physical principles, and allowing the interaction between gas laws and solutions.

Electrolyte Dissociation

Electrolyte dissociation is the process where compounds split into ions when dissolved in water. Electrolytes, such as salts, acids, and bases, dissociate to form positively and negatively charged ions in a solution. This property is essential for numerous biochemical and physical processes.
CaCl₂ is a common example of an electrolyte. It dissociates into Ca²⁺ and 2 Cl^- ions in solution. The extent of dissociation impacts the van’t Hoff factor, which influences osmotic pressure by increasing the number of particles in a solution.

• For a compound like urea, which does not dissociate, \( i \) is typically 1.
• In contrast, dissociation increases \( i \) for an electrolyte, affecting both the boiling point elevation and freezing point depression of solutions.

Understanding electrolyte dissociation helps chemists predict how substances interact in solutions and affects the calculation of osmotic pressure through the van’t Hoff factor.