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Chapter 13: Problem 11

Suppose you had a balloon made of some highly flexible semipermeable membrane. The balloon is filled completely with a \(0.2 \mathrm{M}\) solution of some solute and is submerged in a 0.1 \(M\) solution of the same solute: Initially, the volume of solution in the balloon is \(0.25 \mathrm{~L}\). 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]

Short Answer

Expert verified
At equilibrium, the volume of the solution inside the balloon will be 0.5 L. This is based on calculating the osmotic pressure for both solutions before osmosis, finding the concentration inside the balloon at equilibrium, determining initial moles of solute, and using these values to find the final volume of the solution inside the balloon.

Step by step solution

01

Identify the osmotic pressure for both solutions before osmosis

On one side of the semipermeable membrane, the balloon is filled with a 0.2 M solution while outside the membrane lies a 0.1 M solution for the same solute. We'll use the formula, 螤 = iMRT, to find the osmotic pressures of both solutions. Assuming i = 1 (No degree of dissociation since the solute is unknown and not mentioned) and the temperature T remains constant. R is gas constant (0.0821 L atm K鈦宦� mol鈦宦�). 螤1_initial = (1)(0.2 M)(0.0821 L atm K鈦宦� mol鈦宦�)(T) = 0.01642T atm 螤2_initial = (1)(0.1 M)(0.0821 L atm K鈦宦� mol鈦宦�)(T) = 0.00821T atm
02

Find the concentration inside the balloon at equilibrium

At equilibrium, the osmotic pressures should be equal for both sides of the semipermeable membrane. Let's denote the final concentration inside the balloon as M_final. Since the molarity of the solution outside the membrane remains the same, we get: 螤1_final = 螤2_initial (1)(M_final)(0.0821 L atm K鈦宦� mol鈦宦�)(T) = 0.00821T atm Now, solving for M_final: M_final = 0.1 M
03

Calculate the initial moles of solute inside the balloon

Using the initial concentration and volume given, we can calculate the initial numbers of moles of solute present inside the balloon. initial moles = concentration 脳 volume initial moles = (0.2 mol/L) 脳 (0.25 L) = 0.05 mol
04

Find the final volume of the solution inside the balloon

At equilibrium, the number of moles of the solute inside the balloon remains the same. We'll use the final concentration (M_final) and the initial moles to calculate the final volume of the solution. final volume = initial moles 梅 M_final final volume = (0.05 mol) 梅 (0.1 mol/L) = 0.5 L So, at equilibrium, the volume of the solution inside the balloon is 0.5 L.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Semipermeable Membrane
Semipermeable membranes are special types of barriers that allow some substances to pass while blocking others. Specifically, they let solvent molecules move through but stop larger solute molecules. This characteristic plays a crucial role in biological processes and many scientific applications. In the described problem, the balloon acts as a semipermeable membrane, allowing water (solvent) to move in and out to balance solute concentrations. Without this selective permeability, osmosis wouldn't occur effectively.
Osmotic Pressure
Osmotic pressure is the force exerted by the concentration of solutes in a solution, driving the movement of solvent through a semipermeable membrane. When two solutions with different solute concentrations are separated by such a membrane, the solvent moves from the area of low solute concentration to high solute concentration. The formula for osmotic pressure is \( 螤 = iMRT \), where \( i \) is the van't Hoff factor, \( M \) is molarity, \( R \) is the gas constant, and \( T \) is temperature. In our example, osmotic pressure causes water to move into the balloon, raising its volume until equilibrium is reached.
Molarity
Molarity is a measure of the concentration of solute in a solution. It is expressed as moles of solute per liter of solution (mol/L or M). In this exercise, the balloon starts with a molarity of 0.2 M, while the surrounding solution is 0.1 M. This difference drives osmosis. By calculating the concentration initially and at equilibrium, we can predict how the system will adjust. This measurement is crucial for understanding how solute concentration impacts osmotic pressure and volume changes.
Equilibrium
Equilibrium in the context of osmosis refers to the point where the osmotic pressures on both sides of the semipermeable membrane are equal. At this stage, the net movement of solvent stops, even if the solutions may still have different concentrations. In our example, equilibrium is reached when the inside and outside pressures match, resulting in a balanced volume within and outside the balloon. Understanding equilibrium is key to predicting changes in solution behavior and ensuring systems naturally move toward balance.

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Most popular questions from this chapter

Breathing air that contains \(4.0 \%\) by volume \(\mathrm{CO}_{2}\) over time causes rapid breathing, throbbing headache, and nausea, among other symptoms. What is the concentration of \(\mathrm{CO}_{2}\) in such air in terms of (a) mol percentage, (b) molarity, assuming 101.3 kPa pressure and a body temperature of \(37^{\circ} \mathrm{C} ?\)

Indicate the principal type of solute-solvent interaction in each of the following solutions and rank the solutions from weakest to strongest solute- solvent interaction: (a) KCl in water, (b) \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) in benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right),(\mathbf{c})\) methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) in water.

Two nonpolar organic liquids, benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) and toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right),\) are mixed. (a) Do you expect \(\Delta H_{\text {soln }}\) to be a large positive number, a large negative number, or close to zero? Explain. (b) Benzene and toluene are miscible with each other in all proportions. In making a solution of them, is the entropy of the system increased, decreased, or close to zero, compared to the separate pure liquids?

Adrenaline is the hormone that triggers the release of extra glucose molecules in times of stress or emergency. A solution of \(0.64 \mathrm{~g}\) of adrenaline in \(36.0 \mathrm{~g}\) of \(\mathrm{CCl}_{4}\) elevates the boiling point by \(0.49^{\circ} \mathrm{C}\). Calculate the approximate molar mass of adrenaline from this data.

Which of the following in each pair is likely to be more soluble in water: (a) cyclohexane \(\left(\mathrm{C}_{6} \mathrm{H}_{12}\right)\) or glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\), (b) propionic acid \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COOH}\right)\) or sodium propionate \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COONa}\right),(\mathbf{c}) \mathrm{HCl}\) or ethyl chloride \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}\right) ?\) Explain in each case.
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