Question

An ideal solution is obtained by dissolving \(n\) moles of non-volatile, non- electrolyte solute in \(N\) moles of solvent. If the vapour pressure of solution is \(P\) and the vapour pressure of pure solvent is ' \(P^{a}\), then (a) \(\frac{P^{\circ}-P}{P}=\frac{n}{N}\) (b) \(\frac{P^{\circ}-P}{P^{\circ}}=\frac{n}{N}\) (c) \(\frac{P^{o}-P}{P^{\circ}}=\frac{N}{n}\) (d) \(\frac{P^{\circ}-P}{P}=\frac{N}{n}\)

Short answer

The correct relationship between the vapour pressures and number of moles of solute and solvent is given by option (b): \(\frac{P^{\circ}-P}{P^{\circ}}=\frac{n}{N}\).

Step-by-step solution

Understanding Raoult's Law

Raoult's Law states that the partial vapor pressure of a component in a solution is directly proportional to the mole fraction of the component present in the solution. Mathematically, it can be expressed as \( P = P^{\circ}x \), where \( P \) is the vapor pressure of the solvent in the solution, \( P^{\circ} \) is the vapor pressure of the pure solvent, and \( x \) is the mole fraction of the solvent in the solution.

Calculate Mole Fraction of Solvent

The mole fraction of the solvent, \( x \), is calculated by the formula \( x = \frac{N}{N + n} \), where \( N \) is the number of moles of solvent and \( n \) is the number of moles of solute.

Apply Raoult's Law to the Solution

Substitute the mole fraction of the solvent into Raoult's Law to get \( P = P^{\circ}\frac{N}{N + n} \). Then, rearrange the equation to isolate \( P \) on one side: \( P^{\circ} - P = P^{\circ} - P^{\circ}\frac{N}{N + n} = P^{\circ}\frac{n}{N + n} \).

Simplify to Find Correct Relation

Divide both sides by \( P^{\circ} \) to get \( \frac{P^{\circ}-P}{P^{\circ}} = \frac{n}{N + n} \). Since \( n \) is much smaller than \( N \), we can ignore it in the denominator, giving us \( \frac{P^{\circ}-P}{P^{\circ}} = \frac{n}{N} \), which corresponds to option (b).

Key concepts

Ideal Solution

When discussing solutions in chemistry, an 'ideal solution' is one where specific assumptions are held true to make calculations and predictions more straightforward. In an ideal solution, the intermolecular forces between solute and solvent molecules are almost similar to those between the molecules of the pure solvent or solute alone.

This means that there are no significant changes in enthalpy when a solute dissolves in a solvent – the process is essentially entropy-driven. The special thing about ideal solutions is that they follow Raoult's Law perfectly, without any deviations. Raoult's Law predicts how the vapor pressure of the solvent is affected upon adding a non-volatile solute, assuming no significant volume change and that the solute particles do not interact in a way that would affect the evaporation of the solvent.

Vapor Pressure

Vapor pressure is a critical concept when it comes to understanding physical properties of solutions. It is the pressure exerted by the vapor of a substance when it is in a closed system in dynamic equilibrium with its liquid phase. Simply put, it is a measure of a liquid's tendency to evaporate.

The vapor pressure of a pure solvent is higher than that of a solution containing a non-volatile solute. Why is this the case? Because the presence of a non-volatile solute reduces the number of solvent molecules on the surface, which in turn reduces the number of molecules that can escape into the vapor phase. This reduction in escaping tendency is quantitatively described by Raoult's Law and leads to a decrease in vapor pressure, which is proportional to the concentration of the solute in the solution.

Mole Fraction

The mole fraction is an expression of 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 mixture. This dimensionless quantity is vital in the application of Raoult's Law.

For a two-component system, such as a solute dissolved in a solvent, the mole fraction of the solvent (\( x_{solvent} \)) would be calculated as \( x_{solvent} = \frac{N}{N + n} \), where \( N \) is the number of moles of solvent and \( n \) is the number of moles of solute. The mole fraction plays a direct role in determining the vapor pressure of the solvent in the solution, as it is an essential variable within Raoult's Law.

Non-Volatile Solute

A non-volatile solute is a substance that has little to no tendency to transform into a gas under existing conditions. In other words, it does not vaporize. When a non-volatile solute is dissolved in a solvent to form a solution, its presence lowers the solution's vapor pressure compared to the pure solvent.

The reason behind this phenomenon is that the non-volatile solute particles occupy space at the liquid’s surface, where evaporation takes place. This reduces the number of solvent molecules at the surface, meaning less available to evaporate, which leads to a lower vapor pressure. This effect is quantifiable using Raoult's Law and plays a significant role in boiling point elevation and freezing point depression, two important colligative properties.