Question

Correct mathematical equations for the Raoult's law are (a) \(\frac{P^{o}-P_{S}}{P_{s}}=\) mole fraction of solvent (b) \(\frac{P^{O_{S}}-P_{S}}{P_{o}}=\) mole fraction of a nonvolatile solute (c) \(P_{s} \propto\) mole fraction of solvent (d) \(P_{s} \propto\) mole fraction of a non volatile solute

Short answer

Correct option is (c).

Step-by-step solution

Understanding Raoult's Law

Raoult's law states that the partial vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent present in the solution. It is mathematically expressed as \( P_s = X_s \cdot P_s^o \), where \( P_s \) is the partial vapor pressure of the solvent, \( X_s \) is the mole fraction of the solvent, and \( P_s^o \) is the vapor pressure of the pure solvent.

Analyzing Raoult's Law Equations

Based on the understanding of Raoult's law, look for equations in the option that show either direct proportionality of the vapor pressure to the mole fraction of the solvent or express the decrease in vapor pressure due to a solute. Specifically, Raoult's law focuses on relating \( P_s \) either to the mole fraction of the solvent or showing how much \( P_s \) decreases relative to \( P_s^o \).

Evaluating Option (a)

Option (a) states \( \frac{P^o - P_s}{P_s} \) is the mole fraction of the solvent. This is incorrect because the mole fraction of the solvent should be \( \frac{P_s}{P_s^o} \).

Evaluating Option (b)

Option (b) states \( \frac{P^{O_s} - P_s}{P_o} \) is the mole fraction of a nonvolatile solute. While it seems to deal with the decrease in vapor pressure, Raoult’s law primarily expresses \( P_s \) concerning the solvent, not implicitly defining solute fractions without considering volatile solutes.

Evaluating Option (c)

Option (c) states \( P_s \propto \) mole fraction of solvent. This is a direct statement of Raoult’s law and is correct. It correctly describes how the vapor pressure of the solvent depends on its mole fraction.

Evaluating Option (d)

Option (d) states \( P_s \propto \) mole fraction of a nonvolatile solute. This is incorrect under Raoult's law, as \( P_s \) is associated with the solvent's mole fraction, not the solute.

Key concepts

Understanding Partial Vapor Pressure

In a solution, partial vapor pressure is a key concept in understanding the behavior of solvents. It denotes the pressure that a solvent's vapor exerts when the solvent is in equilibrium with its liquid form in a mixture. According to Raoult’s Law, this pressure is directly linked to the concentration of the solvent in the solution.

To put it simply, the partial vapor pressure of the solvent (P_s) is determined by multiplying its mole fraction (X_s) by the vapor pressure of the pure solvent (P_s^o). The formula is given by:

• \( P_s = X_s \cdot P_s^o \)

This formula shows us that if you increase the mole fraction of the solvent, the partial vapor pressure also rises. Conversely, if the mole fraction decreases, so too does the partial vapor pressure. This relationship highlights how sensitive the vapor pressure of a mixture is to the amount of solvent present. It's crucial in predicting how solvents will behave when mixed with other substances.

Mole Fraction of Solvent

Mole fraction is a way to express concentrations. It is defined as the ratio of the number of moles of a component (in this case, the solvent) to the total number of moles in the solution. It places emphasis on the proportion of one component in a solution rather than its absolute quantity.

For the solvent, the mole fraction (X_s) is calculated using the formula:

• \( X_s = \frac{n_s}{n_s + n_{solute}} \)

Here, \(n_s\) is the number of moles of the solvent, and \(n_{solute}\) is the number of moles of all solutes present. This fraction is pivotal in Raoult’s Law since it determines the solvent's contribution to the total vapor pressure.

The mole fraction adds context to Raoult’s Law by showing the proportional contribution of each component, emphasizing how a greater mole fraction of the solvent results in a higher partial vapor pressure.

Nonvolatile Solute

When talking about Raoult's Law, a nonvolatile solute is one that doesn’t evaporate into gas under given conditions, unlike the solvent. Such solutes remain entirely in the liquid phase, and their main effect on the solution is due to their presence lowering the available space for solvent molecules, thus diminishing the solvent’s vapor pressure.

By dissolving a nonvolatile solute in a solvent, you can witness a lowering of the vapor pressure of the solvent. This reduction occurs because the presence of solute particles reduces the number of solvent molecules at the surface, inhibiting their ability to escape as vapor. This is referred to as the "vapor pressure lowering effect," rooted in colligative properties which depend on the number of particles in solution, not their identity.

In summary, nonvolatile solutes don't directly contribute to the vapor pressure but impact it by affecting the solvent's behavior, fundamentally altering the properties of the solution as per Raoult's Law.