Question

Solutions \(\mathrm{A}\) and \(\mathrm{B}\) have osmotic pressures of 2.4 and 4.6 atm, respectively, at a certain temperature. What is the osmotic pressure of a solution prepared by mixing equal volumes of \(\mathrm{A}\) and \(\mathrm{B}\) at the same temperature?

Short answer

The osmotic pressure of the mixture is 3.5 atm.

Step-by-step solution

Understanding Osmotic Pressure

Osmotic pressure (π) is a colligative property, meaning it depends on the number of solute particles in a solution rather than their identity. The total osmotic pressure of a mixture of solutions can be found by considering the contributions of each separate solution.

Osmotic Pressure Formula

When solutions are mixed, the osmotic pressure can be averaged if they are mixed in equal volumes. The formula for the osmotic pressure of a mixture is: \[ \pi_{mix} = \frac{V_A \pi_A + V_B \pi_B}{V_A + V_B} \] where \( V_A \) and \( V_B \) are the volumes of solutions A and B, respectively, and \( \pi_A \) and \( \pi_B \) are their osmotic pressures.

Setting Up the Problem

Since the volumes of solutions A and B are equal, we can let \( V_A = V_B = V \). Therefore, the formula simplifies to: \[ \pi_{mix} = \frac{V \pi_A + V \pi_B}{V + V} = \frac{\pi_A + \pi_B}{2} \]

Calculate the Osmotic Pressure of the Mixture

Substitute the given values into the formula: \[ \pi_{mix} = \frac{2.4 \, \text{atm} + 4.6 \, \text{atm}}{2} \]Calculate the numerical value:\[ \pi_{mix} = \frac{7.0 \, \text{atm}}{2} = 3.5 \, \text{atm} \]

Conclusion

The osmotic pressure of the solution prepared by mixing equal volumes of solutions A and B is 3.5 atm.

Key concepts

Colligative Properties

Colligative properties are fascinating because they depend not on the type of particles present in a solution, but rather on their quantity. This includes traits like vapor pressure lowering, boiling point elevation, freezing point depression, and indeed, osmotic pressure. When a solute is added to a solvent, these properties change in predictable ways. In the case of osmotic pressure, when solute particles are added, they affect how solvents pass through a semipermeable membrane.
As a result, the more particles present in a solution, regardless of their individual characteristics, the greater the osmotic pressure. It's like focusing entirely on the number of people in a room, without caring about who they are. Understanding this makes dealing with solutions much easier and highlights the importance of considering particle count in solution chemistry.

Solution Mixing

Mixing solutions is a common practice in chemistry and involves combining two or more solutions to form a new one. In our problem, we mix solutions with different osmotic pressures to create a new solution with a combined osmotic pressure. This process often involves understanding and calculating how each component contributes to the final mixture.
When solutions of equal volumes are mixed, as in the original exercise, the combined properties, such as osmotic pressure, are averaged. This means you take equal parts of each solution, contributing equally to the final mixture. It's like making a smoothie by blending equal amounts of banana and strawberry, where each flavor blends evenly into the final drink.

• The original properties of each solution (such as their osmotic pressures) are significant in determining the characteristics of the blend.
• Equal mixing can result in averaging the properties.

This helps simplify calculations and predictions about the final solution.

Pressure Calculation

Calculating pressure, especially osmotic pressure, involves understanding both the nature of the contributing solutions and the overall process of mixing. In our context, osmotic pressure is determined by using a specific formula that considers the pressures of individual solutions and their respective volumes. When we have solutions A and B with osmotic pressures of 2.4 atm and 4.6 atm, we use the formula:\[\pi_{mix} = \frac{V_A \pi_A + V_B \pi_B}{V_A + V_B}\]
Here, the given solution volumes are equal, simplifying the formula to:\[\pi_{mix} = \frac{\pi_A + \pi_B}{2}\] This equation reflects the averaging of osmotic pressures due to equal amounts of mixing.

Equal Volumes of Solutions

Equal volumes of solutions being mixed result in a straightforward averaging of their individual properties, such as osmotic pressure. When volumes are equal, it aligns perfectly with formulas where simplification to an average value can be applied, as shown in the solution formula.
The key takeaway is that knowing the volumes of the solutions can significantly streamline the calculation process. In the case where solution A and B have the same volume, the average means each contributes 50% to the final osmotic pressure. This principle can apply to other properties that depend on concentration or the number of solute particles. Mixing in this manner ensures the influences of each original solution are evenly represented in the final mix.