Problem 130 - A Complete Resource Book in Chemistry for JEE Main

Question

Correct mathematical equations for the Raoult's law are (a) \(\frac{P^{O}-P_{S}}{P_{s}}=\) mole fraction of solvent (b) \(\frac{P^{o} s_{s} P_{S}}{P_{o}}=\) 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

Step-by-Step Solution

Answer

Option (c) is correct: \(P_s \propto\) mole fraction of solvent.

Step 1: Understand Raoult's Law

According to Raoult's Law, the partial vapor pressure of a solvent in a solution (P_s) is directly proportional to the mole fraction of the solvent (\(x_{solvent}\)) in the solution. This can be expressed as:\[P_s = x_{solvent} \times P^o\]where \(P^o\) is the vapor pressure of the pure solvent.

Step 2: Analyze Option (a)

Option (a) suggests the equation:\[\frac{P^O-P_s}{P_s} = \text{mole fraction of solvent}\]This equation is incorrect, as it implies a ratio involving changes in pressure rather than directly relating the actual partial pressure to the mole fraction of the solvent.

Step 3: Analyze Option (b)

Option (b) suggests:\[\frac{P^o s_s P_s}{P_o} = \text{mole fraction of a nonvolatile solute}\]This equation does not correctly represent the concepts described in Raoult's Law since the law primarily deals with the vapor pressure of the solvent and its proportional relationship to its mole fraction.

Step 4: Analyze Option (c)

Option (c) proposes that:\[P_s \propto \text{mole fraction of solvent}\]This directly aligns with Raoult's Law, which states that the partial vapor pressure of the solvent is proportional to its mole fraction in the solution (\(P_s \propto x_{solvent}\)).

Step 5: Analyze Option (d)

Option (d) suggests:\[P_s \propto \text{mole fraction of a nonvolatile solute}\]This is incorrect according to Raoult's Law, as the law does not relate the mole fraction of the nonvolatile solute to the partial vapor pressure of the solvent.

Key Concepts

Partial Vapor Pressure

Partial vapor pressure is a key concept in understanding Raoult's Law. It refers to the pressure exerted by a vapor in equilibrium with its liquid at a given temperature. When a solvent is part of a solution, its partial vapor pressure can be influenced by the presence of solutes.
In the case of Raoult's Law, the partial vapor pressure of a solvent, denoted as \( P_s \), hinges on the mole fraction of that solvent. Essentially, it's the amount of pressure the solvent's vapor contributes to the total pressure exerted by the solution's vapor phase. This relationship is pivotal in predicting how the presence of solutes affects the volatility of solvents in mixtures.
For instance, when you mix a solvent with other substances, the solvent's tendency to escape into the vapor diminishes, thereby lowering its partial vapor pressure. This concept helps to understand solution behavior in various chemical and physical processes.

Mole Fraction

The mole fraction is a crucial term when discussing solutions and it plays a leading role in Raoult's Law. It's a way to express the composition of a mixture and is defined as the ratio of the number of moles of a component to the total number of moles in the mixture.
The mole fraction, \( x \), of a component in a solution is calculated as:

• \( x_{component} = \frac{n_{component}}{n_{total}} \)

Where \( n_{component} \) is the number of moles of the specific component and \( n_{total} \) is the total number of moles of all components in the solution.
This measure is dimensionless and is crucial for calculating the partial vapor pressure exerted by a solvent, as Raoult's Law expresses that \( P_s = x_{solvent} \times P^o \). Understanding the mole fraction aids in assessing how changing the amount of a solvent or solute in a solution can impact its vapor pressure and other properties.

Non-Volatile Solute

When dealing with solutions, distinguishing between volatile and non-volatile solutes is essential. A non-volatile solute is a substance with very low vapor pressure and doesn't easily vaporize at the given temperature.
In the context of Raoult's Law, non-volatile solutes don't directly contribute to the vapor phase above the solution. This means they don't affect the partial vapor pressure of solvents through direct vapor pressure addition.
However, non-volatile solutes influence a solution's overall properties by lowering the overall vapor pressure of the solvent. They do this by reducing the solvent's mole fraction in the solution. Therefore, adding a non-volatile solute to a solvent decreases the solvent's partial vapor pressure without the solute itself contributing to vapor pressure.

Proportional Relationship

Raoult's Law brilliantly illustrates the proportional relationship inherent in solutions involving volatile components, often denoted through the partial vapor pressure equation \( P_s = x_{solvent} \times P^o \).
This law states that the partial vapor pressure of a solvent in a solution is directly proportional to its mole fraction. In simpler terms, as the mole fraction of the solvent increases, the partial vapor pressure of that solvent also rises, assuming the vapor pressure of the pure solvent remains constant.

• This relationship becomes fundamental when predicting how a solution will behave when the composition changes.
• It clarifies how diluting a solution (by decreasing the mole fraction of the solvent) leads to a reduction in vapor pressure, and vice versa.

Understanding this relationship aids in predicting and controlling the volatility of a solution as well as comprehending various absorption, distillation, and boiling phenomena. Such insights are invaluable across chemical, industrial, and environmental applications.