Question

Two beakers are placed in a sealed box at \(25^{\circ} \mathrm{C}\). One beaker contains \(30.0 \mathrm{~mL}\) of a \(0.050 \mathrm{M}\) aqueous solution of a nonvolatile nonelectrolyte. The other beaker contains \(30.0 \mathrm{~mL}\) of a \(0.035 \mathrm{M}\) aqueous solution of \(\mathrm{NaCl}\). The water vapor from the two solutions reaches equilibrium. (a) In which beaker does the solution level rise, and in which one does it fall? (b) What are the volumes in the two beakers when equilibrium is attained, assuming ideal behavior?

Short answer

The solution level rises in the beaker containing the nonvolatile nonelectrolyte and falls in the beaker containing the NaCl solution. The final volumes of the solutions in the two beakers when equilibrium is attained are 30.025 mL and 29.975 mL, respectively.

Step-by-step solution

Calculate mole fractions

To determine which beaker's solution level rises and which falls, we need to calculate the mole fraction of solute and solvent in each beaker. To do this, we must first find the moles of solute and solvent in each solution. For a nonvolatile nonelectrolyte: Concentration = 0.050 M Solvent: water, volume = 30 mL Number of moles of solute: \( 0.050 \mathrm{M} \times 0.030 \mathrm{L} = 0.0015 \mathrm{mol}\) Number of moles of solvent: \( \dfrac{0.030 \mathrm{L} }{18.015 \mathrm{\dfrac{g}{mol}} \times 1 \mathrm{\dfrac{g}{mL}} }\ = 1.666 \mathrm{mol}\) Mole fraction of solute: \( \dfrac{0.0015}{0.0015 + 1.666} = 0.0009\) For NaCl solution: Concentration = 0.035 M Solvent: water, volume = 30 mL Number of moles of solute: \( 0.035 \mathrm{M} \times 0.030 \mathrm{L} = 0.00105 \mathrm{mol}\) Number of moles of solvent: \( \dfrac{0.030 \mathrm{L} }{18.015 \mathrm{\dfrac{g}{mol}} \times 1 \mathrm{\dfrac{g}{mL}} }\ = 1.666 \mathrm{mol}\) Mole fraction of solute: \( \dfrac{0.00105}{0.00105 + 1.666} = 0.00063\) Now that we have the mole fraction of solute and solvent in each beaker, we can proceed to calculate the vapor pressure of each solution.

Calculate vapor pressure

Using the mole fraction of solvent in each solution, we will now determine the vapor pressure of each solution using Raoult's Law: \( P_{solvent} = X_{solvent} \times P^*_\mathrm{solvent}\) Where \(P_{solvent}\) is the vapor pressure of the solvent in the solution, \(X_{solvent}\) is the mole fraction of the solvent, and \(P^*_\mathrm{solvent}\) is the vapor pressure of the pure solvent. The vapor pressure of pure water at \(25^{\circ} \mathrm{C}\) is 23.76 mmHg. For the nonvolatile nonelectrolyte solution: \( P_1 = X_{solvent\mathrm{(Nonelectrolyte)}} \times P^*_\mathrm{water}\) \( P_1 = (1 - 0.0009) \times 23.76 \mathrm{mmHg} = 23.58 \mathrm{mmHg}\) For the NaCl solution: \( P_2 = X_{solvent\mathrm{(NaCl)}} \times P^*_\mathrm{water}\) \( P_2 = (1 - 0.00063) \times 23.76 \mathrm{mmHg} = 23.65 \mathrm{mmHg}\) Now that we have the vapor pressure of each solution, we can determine which beaker's solution level rises and which falls.

Determine solution level change

Since the vapor pressure of the nonvolatile nonelectrolyte solution is lower than that of the NaCl solution, water vapor will move from the beaker with the NaCl solution to the beaker with the nonvolatile nonelectrolyte solution. As a result, the solution level in the beaker containing the nonvolatile nonelectrolyte will rise, and the solution level in the beaker containing the NaCl solution will fall.

Calculate final volumes

The solution levels rise and fall until equilibrium is reached, i.e., when both solutions have the same vapor pressure. We can calculate the final volumes of the solutions in the two beakers using the mole fractions and vapor pressures at equilibrium. Let ΔV be the volume change in the NaCl solution beaker. \( X_{solvent_\mathrm{NaCl \mathrm{,final}}} = \dfrac{30-\Delta V}{30} \) Since the vapor pressures are equal at equilibrium, we can write: \( (1-X_{solvent_\mathrm{NaCl\mathrm{,final}}})P_{water}^* = (1-X_{solvent_\mathrm{Nonelectrolyte}})P_{water}^*\) Substituting the values and simplifying: \( (1 - \dfrac{30-\Delta V}{30})23.76 = (1 - 0.0009)23.76\) Solving for ΔV gives us: \( \Delta V = 0.025 \mathrm{L}\) So the final volume of the nonvolatile nonelectrolyte solution is 30 mL + 0.025 L = 30.025 mL, and the final volume of the NaCl solution is 30 mL - 0.025 L = 29.975 mL. In summary, the solution level rises in the beaker containing the nonvolatile nonelectrolyte, and it falls in the beaker containing the NaCl solution. The final volumes of the solutions in the two beakers when equilibrium is attained are 30.025 mL and 29.975 mL, respectively.

Key concepts

Raoult's Law

Raoult's Law is crucial in understanding how the vapor pressure of a solution is affected when a solute is introduced. This law states that the partial vapor pressure of the solvent in a solution is equal to the mole fraction of the solvent multiplied by the vapor pressure of the pure solvent. The equation is:\[ P_{solvent} = X_{solvent} \times P_{solvent}^* \]Where \( P_{solvent} \) is the vapor pressure of the solvent above the solution, \( X_{solvent} \) is the mole fraction of the solvent, and \( P_{solvent}^* \) is the vapor pressure of the pure solvent.When you add a solute to a solvent, it generally causes the vapor pressure to decrease because the solvent molecules are less likely to escape into the vapor phase. This is referred to as the "lowering of vapor pressure." It’s important to note that Raoult's Law applies ideally to solutions where the solute doesn't alter the nature of the solvent or vice versa significantly.

Mole Fraction

The mole fraction is a way to express concentration by representing the ratio of moles of one component to the total moles in the solution. For a solution, you need to calculate both the moles of solute and moles of solvent to find the mole fractions. For example, in the problem above:

• Calculate the moles of solute: \( ext{Number of moles of solute} = ext{Molarity} \times ext{Volume in Liters} \)
• Calculate the moles of solvent: \( ext{Number of moles of solvent} = \frac{ ext{Volume of solvent}}{ ext{Molar mass of solvent}} \)
• Apply the equation for mole fraction: \( X_{component} = \frac{ ext{Moles of component}}{ ext{Total moles in solution}} \)

The mole fraction gives insight into the proportion of solvent present, which directly influences the solution's vapor pressure using Raoult’s Law.

Equilibrium

In the context of solutions, equilibrium refers to the state where the rate of evaporation equals the rate of condensation for each solution. At equilibrium in the sealed box scenario, the vapor pressures of the solvents in both beakers equalize. This balance occurs because molecules move back and forth between the liquid and vapor phases until the two rates are the same. Key points about equilibrium in such systems include:

• Equilibrium is dynamic, meaning molecules continuously change phase, but the overall amounts remain stable.
• Achieving equilibrium ensures that the vapor pressures in each system become equal.
• In the example provided, the NaCl solution's higher initial vapor pressure causes water vapor to move, raising the vapor pressure of the nonelectrolyte solution until both reach the same level.

When equilibrium is reached, the physical properties of the solutions stabilize, indicating no further net exchange of water vapor between the beakers, thus giving us the final volumes of the solutions in each beaker.