Problem 122 - A Complete Resource Book in Chemistry for JEE Main

Question

When 2 moles each of Hexane and heptane are mixed and if \(\mathrm{P}_{\text {Hexane }}^{\circ}\) and \(\mathrm{P}_{\text {Heptane }}^{\circ}\) are \(50 \mathrm{~mm} \mathrm{Hg}\) and \(60 \mathrm{~mm} \mathrm{Hg}\) respectively than? (a) \(\mathrm{P}_{\text {Total }}=55 \mathrm{~mm} \mathrm{Hg}\) (b) \(\mathrm{P}_{\text {Hexane }}=25 \mathrm{~mm} \mathrm{Hg}\) (c) \(\mathrm{P}_{\text {Heptane }}=30 \mathrm{~mm} \mathrm{Hg}\) (d) None-Ideal solution

Step-by-Step Solution

Answer

The total pressure is 55 mm Hg, matching option (a).

Step 1: Calculate Mole Fraction of Hexane

Since the solution consists of 2 moles each of hexane and heptane, the total number of moles is 4. The mole fraction of hexane, \(X_{\text{Hexane}}\), is calculated as follows: \(X_{\text{Hexane}} = \frac{\text{moles of hexane}}{\text{total moles}} = \frac{2}{4} = 0.5\).

Step 2: Calculate Mole Fraction of Heptane

Similarly, the mole fraction of heptane, \(X_{\text{Heptane}}\), is: \(X_{\text{Heptane}} = \frac{\text{moles of heptane}}{\text{total moles}} = \frac{2}{4} = 0.5\). Since both components are in equal molar amounts, they have the same mole fraction.

Step 3: Apply Raoult's Law to Calculate Partial Pressure of Hexane

According to Raoult's Law, the partial pressure of hexane, \(P_{\text{Hexane}}\), is given by: \(P_{\text{Hexane}} = X_{\text{Hexane}} \times P_{\text{Hexane}}^\circ = 0.5 \times 50\, \text{mm Hg} = 25\, \text{mm Hg}\).

Step 4: Apply Raoult's Law to Calculate Partial Pressure of Heptane

Similarly, the partial pressure of heptane, \(P_{\text{Heptane}}\), is: \(P_{\text{Heptane}} = X_{\text{Heptane}} \times P_{\text{Heptane}}^\circ = 0.5 \times 60\, \text{mm Hg} = 30\, \text{mm Hg}\).

Step 5: Calculate Total Pressure

The total pressure, \(P_{\text{Total}}\), of the solution is the sum of the partial pressures: \(P_{\text{Total}} = P_{\text{Hexane}} + P_{\text{Heptane}} = 25\, \text{mm Hg} + 30\, \text{mm Hg} = 55\, \text{mm Hg}\). This matches option (a).

Key Concepts

Partial Pressure Calculation in Solutions

In a mixture of gases, the partial pressure is the pressure that a gas would exert if it alone occupied the entire volume of the mixture. When dealing with solutions, Raoult's Law comes into play to calculate these pressures. For a component in an ideal solution, the partial pressure is calculated by multiplying the component's mole fraction by the component's pure vapor pressure. In our exercise, we considered a mixture of hexane and heptane. Let's understand how the partial pressures were calculated:

• For hexane: Its mole fraction is calculated as 0.5 (since there are 2 moles out of a total of 4 moles). The pure vapor pressure of hexane is given as 50 mm Hg. Using Raoult's Law, the partial pressure of hexane is computed as:\[ P_{\text{Hexane}} = X_{\text{Hexane}} \times P_{\text{Hexane}}^\circ = 0.5 \times 50 \text{ mm Hg} = 25 \text{ mm Hg} \]
• For heptane: Similarly, with a mole fraction of 0.5 and a pure vapor pressure of 60 mm Hg, the partial pressure of heptane is:\[ P_{\text{Heptane}} = X_{\text{Heptane}} \times P_{\text{Heptane}}^\circ = 0.5 \times 60 \text{ mm Hg} = 30 \text{ mm Hg} \]

The total pressure of the solution is simply the sum of these two partial pressures, giving us 55 mm Hg.

Understanding Mole Fraction

The concept of mole fraction is a way of expressing the composition of a component in a mixture. It's essentially the ratio of the number of moles of a particular substance to the total number of moles in the solution. For calculating mole fractions, the formula is simple and intuitive:\[ X_{\text{compound}} = \frac{\text{moles of compound}}{\text{total moles}} \]In our situation, there are equal moles of hexane and heptane mixed together.

• Hexane Mole Fraction: With 2 moles of hexane in total 4 moles of solution, the mole fraction \( X_{\text{Hexane}} \) becomes: \( \frac{2}{4} = 0.5 \).
• Heptane Mole Fraction: Similarly, it is \( \frac{2}{4} = 0.5 \) for heptane.

Having equal mole fractions indicates a balanced composition, which often simplifies calculations and leads to useful properties in the solution, like in this example concerning partial pressures.

Exploring Ideal Solution Dynamics

An ideal solution follows Raoult's Law perfectly where each component's vapor pressure is directly proportional to its mole fraction. This law assumes no interaction forces between dissimilar molecules, which means the behavior of each molecule is unaffected by its neighbors. In the ideal solution of hexane and heptane, both substances exhibit behaviors that align with Raoult's Law, allowing predictable calculations of pressure contributions:

• Proportionality holds: The partial pressures of each substance (hexane and heptane) became exactly half of their respective pure vapor pressures because their mole fractions were 0.5.
• Additivity principle applies: The total vapor pressure of the solution is simply the sum of the partial pressures, validating its ideal nature.

It's important to acknowledge that not all mixtures behave ideally, but when they do, calculations and predictions become straightforward, as demonstrated in this mixture scenario.