Question

Which of the following will have the lowest total vapor pressure at \(25^{\circ} \mathrm{C} ?\) a. pure water (vapor pressure \(=23.8\) torr at \(25^{\circ} \mathrm{C}\) ) b. a solution of glucose in water with \(\chi_{\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}}=0.01\) c. a solution of sodium chloride in water with \(\chi_{\mathrm{NaCl}}=0.01\) d. a solution of methanol in water with \(\chi_{\mathrm{CH}_{3} \mathrm{OH}}=0.2\) (Consider the vapor pressure of both methanol \(\left[143\right.\) torr at \(\left.25^{\circ} \mathrm{C}\right]\) and water.)

Short answer

The solutions with the lowest total vapor pressure at 25°C are b) a solution of glucose in water with \(\chi_{C6H12O6}=0.01\) and c) a solution of sodium chloride in water with \(\chi_{NaCl}=0.01\), both having a vapor pressure of approximately 23.56 torr.

Step-by-step solution

Calculate the vapor pressures for each solution using Raoult's Law

Raoult's Law states that the partial vapor pressure of a component in a solution is equal to the product of the mole fraction of that component and its vapor pressure in the pure state. When considering volatile solutes, we need to take into account the vapor pressure of both the solute and the solvent. a. The vapor pressure of pure water at 25°C is given as 23.8 torr. b. For the glucose-water solution, glucose is a nonvolatile solute, so its vapor pressure contribution will be negligible. Therefore, we only need to calculate the vapor pressure of water in the solution. \(P_{water} = \chi_{C6 H12 O6} \times P^0_{water} = (1 - 0.01) \times 23.8 \approx 23.56 \) torr. c. Similarly, for the NaCl-water solution, NaCl is a nonvolatile solute. The vapor pressure of water is going to be \(P_{water} = (1 - 0.01)\times 23.8 \approx 23.56 \) torr. d. For the methanol-water solution, both the methanol and water are volatile. We need to calculate the partial vapor pressures of both methanol and water, then sum them to find the total vapor pressure. \(P_{water} = \chi_{CH3OH} \times P^0_{water} = (1 - 0.2) \times 23.8 = 19.04 \) torr. \(P_{methanol} = \chi_{CH3OH} \times P^0_{methanol} = 0.2 \times 143 = 28.6 \) torr. The total vapor pressure for the methanol-water solution is \(P_{total} = P_{water} + P_{methanol} = 19.04 + 28.6 = 47.64 \) torr.

Compare the vapor pressures and determine the answer

Now that we have calculated the vapor pressures for each of the solutions, we can compare their values: a. Pure water: 23.8 torr b. Glucose-water solution: 23.56 torr c. NaCl-water solution: 23.56 torr d. Methanol-water solution: 47.64 torr Based on the calculated vapor pressures, the solution with the lowest total vapor pressure at 25°C is the one of glucose in water with \(\chi_{C6 H12 O6} = 0.01\), as well as the one of sodium chloride in water with \(\chi_{NaCl} = 0.01\). Therefore, the answer is b) and c).

Key concepts

Raoult's Law

Understanding Raoult's Law is fundamental when studying vapor pressure in the context of solutions. It's a straightforward yet powerful principle that links the composition of a solution to its vapor pressure. Simply put, Raoult's Law asserts that the partial vapor pressure of any component of an ideal solution is directly proportional to its mole fraction in the solution.

This relationship is pertinent for calculating the vapor pressures of solutions, especially when determining the boiling or freezing points of a solution or when predicting the behavior of a component in a mixture. In mathematical terms, Raoult's Law can be articulated as:
\[ P = \(\chi \) \times P^{\circ} \]
where \( P \) denotes the partial vapor pressure of the component in the solution, \( \chi \) represents the mole fraction of the component, and \( P^{\circ} \) is the vapor pressure of the pure component at the same temperature.

When dealing with a solution that contains a nonvolatile solute, the total vapor pressure over the solution diminishes relative to that of the pure solvent. This phenomenon is because the nonvolatile solute occupies space that would otherwise be available for the solvent molecules to evaporate, effectively lowering the mole fraction of the solvent in the vapor phase.

Mole Fraction

Quantifying the composition of a solution is crucial for a myriad of chemical calculations, and the mole fraction is a key unit of measure for this purpose. The mole fraction is an expression of the ratio of the number of moles of a particular component to the total number of moles of all components in the solution.

Considering a binary mixture, if we call one component 'A' and the other 'B', the mole fraction of 'A', denoted as \( \chi_{A} \), would be calculated using the formula:
\[ \chi_{A} = \frac{n_{A}}{n_{A} + n_{B}} \]
where \( n_{A} \) and \( n_{B} \) are the moles of 'A' and 'B', respectively. Notably, the sum of the mole fractions in a solution always equals one.

In the context of the vapor pressure exercise outlined above, the mole fraction directly affects the resultant vapor pressure of the solution through Raoult's Law. A higher mole fraction of a volatile solvent in a solution will generally lead to a higher vapor pressure, assuming ideal behavior. On the other hand, the presence of a nonvolatile solute will always decrease the mole fraction of the solvent, leading to a reduction in vapor pressure, as reflected by the negligible vapor pressure contribution of glucose in the provided exercise.

Nonvolatile Solute

Diving into the realm of solutions, it is pivotal to distinguish between volatile and nonvolatile solutes due to their impact on the vapor pressure of the solution. A nonvolatile solute is a substance that has negligible vapor pressure at a given temperature and does not readily evaporate. Because nonvolatile solutes do not contribute significantly to the vapor pressure of a solution, they are primarily responsible for lowering the vapor pressure of the solvent by decreasing its mole fraction.

As seen in the exercise, when a nonvolatile solute such as glucose or sodium chloride is dissolved in water, the water's vapor pressure is slightly lower than it would be in its pure state. This is a direct application of Raoult's Law: since the mole fraction of water is reduced, the vapor pressure it contributes to the solution is also reduced. Solutions containing nonvolatile solutes are also subject to colligative properties, which depend solely on the number of solute particles and not their identity, leading to phenomena such as boiling point elevation and freezing point depression.

In contrast, with volatile solutes, like the methanol in example d, both components of the mixture contribute to the total vapor pressure. The calculation of the total vapor pressure involves considering the contributions of each volatile component in the solution, translating to an additive effect rather than just a simple reduction due to a nonvolatile component.