Problem 11 - Chemistry The Central Science

Question

Suppose you had a balloon made of some highly flexible semipermeable membrane. The balloon is filled completely with a 0.2\(M\) solution of some solute and is submerged in a 0.1 \(\mathrm{M}\) solution of the same solute: Initially, the volume of solution in the balloon is 0.25 L. Assuming the volume outside the semipermeable membrane is large, as the illustration shows, what would you expect for the solution volume inside the balloon once the system has come to equilibrium through osmosis? [Section 13.5\(]\)

Step-by-Step Solution

Answer

The final volume of the solution inside the balloon when the system reaches equilibrium through osmosis is \(0.5 L\).

Step 1: Identify the initial conditions

We are given the following information: - The initial volume of the solution inside the balloon is \(0.25 L\). - The initial concentration (molarity) of the solution inside the balloon is \(0.2 M\). - The initial concentration of the solution outside the balloon is \(0.1 M\). Since both solutions contain the same solute, we can assume their van't Hoff factors (i) are equal.

Step 2: Calculate osmotic pressures

Using the formula \(Π = iMRT\), we'll calculate the osmotic pressures for both the inside and outside solutions. Let's assume the temperature is constant and equal for both solutions, and the van't Hoff factor (i) is equal for both solutions. Hence the difference in osmotic pressure inside and outside the balloon depends only on the difference in molarity. Diff_Π = iRT(M_inside - M_outside)

Step 3: Calculate the equilibrium concentration

When the system reaches equilibrium through osmosis, the osmotic pressures of both the inside and outside solutions will be equal. Thus, the final concentration of both solutions will also be equal. Let's denote the equilibrium concentration as M_eq.

Step 4: Calculate the moles of solute inside the balloon

Before osmosis occurs, we can calculate the moles of solute inside the balloon. moles_solute_inside = M_inside × V_inside = 0.2 M × 0.25 L = 0.05 moles

Step 5: Find the final volume of the solution inside the balloon

As the system reaches equilibrium, the moles of solute inside the balloon stay constant. We can use the equilibrium concentration to find the final volume. moles_solute_inside = M_eq × V_eq We already know M_eq = M_inside = M_outside from Step 3. Therefore, we can solve for the final volume: V_eq = moles_solute_inside / M_eq = 0.05 moles / 0.1 M = 0.5 L The final volume of the solution inside the balloon when the system reaches equilibrium through osmosis is 0.5 L.

Key Concepts

Semipermeable Membrane

A semipermeable membrane is a crucial concept in understanding osmosis. This type of membrane allows certain molecules or ions to pass through it while blocking others. In this exercise, the balloon is made of a semipermeable membrane which permits the passage of water molecules but prevents the solute from crossing.
This selective permeability is essential for osmosis:

• Water molecules can move freely across the membrane.
• Solute particles are too large or incompatible to pass through.

The semipermeable membrane thus acts as a barrier that helps establish concentration differences, which are critical for the movement of water during osmosis.
This movement occurs until the solution inside and outside the membrane reaches equilibrium.

Osmotic Pressure

Osmotic pressure is a measure of the tendency of water to move across a semipermeable membrane due to concentration differences. In simple terms, it is the pressure needed to stop the fluid movement caused by osmosis. The exercise uses the formula \(\Pi = iMRT\) to explain how osmotic pressure is calculated, where:

• \(i\) is the van't Hoff factor (usually 1 for non-dissociating solutes).
• \(M\) is the molarity of the solution.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

The difference in osmotic pressure between the inside and outside of the balloon dictates the direction of water movement. Higher osmotic pressure inside a solution means water will move into that solution to balance the concentration.

Equilibrium Concentration

Equilibrium concentration is reached when the system is balanced, meaning the concentration of solute on either side of the semipermeable membrane is equal and there's no net movement of water. In our scenario, initially, the balloon had a solute concentration of \(0.2 M\) while the external solution was \(0.1 M\).
Over time, water moved into the balloon due to higher osmotic pressure inside it. This movement continued until the solute concentration inside the balloon equaled that outside. At equilibrium, the concentration, \(M_{eq}\), becomes identical on both sides of the membrane. The concept of equilibrium ensures that no further net osmosis occurs.

• The system reaches equilibrium, achieving balance.
• Water movement halts, reaching a steady state.

This equilibrium state leads to a final balloon volume which can be calculated using the initial moles of solute and the new equilibrium concentration.