Question

Under the condition of similar temperature, which of the following solution will have minimum vapour pressure? (a) \(0.1 \mathrm{M}-\) sugar (b) \(0.1 \mathrm{M}-\mathrm{NaCl}\) (c) \(0.1 \mathrm{M}-\mathrm{BaCl}_{2}\) (d) \(0.1 \mathrm{M}-\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\)

Short answer

The solution of 0.1M Al2(SO4)3 will have the minimum vapor pressure because it produces the maximum number of particles per mole of solute, thereby reducing the vapor pressure the most according to Raoult's Law.

Step-by-step solution

Understanding Raoult's Law

The first step in solving this problem is understanding Raoult's Law, which states that the vapor pressure of a solvent above a solution is equal to the product of the vapor pressure of the pure solvent and the mole fraction of the solvent in the solution. For solutions of non-volatile solutes (like sugar and salts), the vapor pressure is directly related to the mole fraction of the solvent. This implies that the more particles of solute that are present in the solution, the lower the vapor pressure of the solution will be, because the mole fraction of the solvent decreases.

Counting the number of ions produced

To determine which solution will have the minimum vapor pressure, we need to recognize that the dissociation of ionic compounds in water will affect the number of particles in solution. More particles result in a lower vapor pressure. For each compound, determine the number of particles it forms in solution. Sugar, being a non-electrolyte, will not dissociate and will therefore contribute 1 particle per molecule. NaCl dissociates into 2 ions: Na+ and Cl-. BaCl2 dissociates into 3 ions: Ba2+ and 2 Cl-. Al2(SO4)3 dissociates into 5 ions: 2 Al3+ and 3 SO4^2-. Thus, Al2(SO4)3 will produce the most particles.

Comparing the number of particles produced

Now that we know the number of particles produced by each compound in solution, we can compare them to determine which will have the lowest vapor pressure. Sugar produces 1 particle, NaCl produces 2 particles, BaCl2 produces 3 particles, and Al2(SO4)3 produces 5 particles per formula unit dissolved. Therefore, the solution with Al2(SO4)3 will have more particles and, consequently, the lower vapor pressure.

Key concepts

Raoult's Law

Understanding the relationship between vapor pressure and the composition of solutions is central to solving problems in physical chemistry, such as predicting which solution will have the lowest vapor pressure. Raoult's Law provides a crucial piece of this puzzle. It states that the vapor pressure of a pure solvent will decrease proportionally when a non-volatile solute is dissolved in it. The equation is expressed as:
\[ P_{solution} = P_{solvent} \times X_{solvent} \]
where \( P_{solution} \) is the vapor pressure of the solution, \( P_{solvent} \) is the vapor pressure of the pure solvent, and \( X_{solvent} \) is the mole fraction of the solvent in the solution. Put simply, the more solute particles present, the lower the solvent's mole fraction and, thus, its vapor pressure. This is because the solute particles occupy space at the surface of the solution, inhibiting the evaporation of the solvent molecules.

Dissociation of Ionic Compounds

When ionic compounds dissolve in water, they dissociate into their constituent ions, an essential consideration when calculating vapor pressures using Raoult's Law. The extent of dissociation significantly alters the number of solute particles in the solution, which in turn affects the solution’s vapor pressure. The resultant ions act as separate particles, each one contributing to the lowering of the solution's vapor pressure. For instance, sodium chloride (NaCl) dissociates completely in aqueous solution to give two ions: Na+ and Cl-, effectively doubling the number of particles that would have been predicted by just considering the formula units. In the case of compounds like barium chloride (BaCl2) and aluminum sulfate (Al2(SO4)3), the dissociation results in three and five particles respectively, which further lower the vapor pressure compared to undissociated or weakly dissociated substances.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles of all components. In the context of solutions, the mole fraction of the solvent is particularly important in applying Raoult's Law. It is calculated using the formula:
\[ X_{component} = \frac{n_{component}}{n_{total}} \]
Here, \( n_{component} \) is the number of moles of the component (solute or solvent) in question, and \( n_{total} \) is the sum of the moles of all components in the solution. The mole fraction of the solvent decreases as more solute particles are introduced into the solution. This is key to understanding why the vapor pressure drops upon the addition of a solute: as the solvent's mole fraction decreases, Raoult's Law predicts a reduction in the solution's vapor pressure.

Colligative Properties

Colligative properties are properties of solutions that depend only on the number of solute particles in the solution, regardless of their nature. These properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure. A key aspect of colligative properties is that they are directly related to the mole fraction of the solute in a given solution, as established by Raoult's Law. The principle behind each property is that the presence of solute particles affects the physical behavior of the solvent. For example, in vapor pressure lowering, the additional solute particles disrupt the solvent molecules’ ability to vaporize, which is why a solution has a lower vapor pressure compared to the pure solvent. The impact is magnified when a solute dissociates into multiple particles, as is the case with many ionic compounds, effectively amplifying the solution's colligative properties.