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

Question

When 2 moles each of Hexane and heptane are mixed and if \(\mathrm{P}_{\text {Hexane }}^{\mathrm{t}}\) 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) \(\mathrm{P}_{\text {Herane }}=25 \mathrm{~mm} \mathrm{Hg}\) (c) \(\mathrm{P}_{\text {Heptane }}=30 \mathrm{~mm} \mathrm{Hg}\) (d) None-Ideal solution

Step-by-Step Solution

Answer

Options (a), (b), and (c) are correct; (d) is incorrect.

Step 1: Determine the Mole Fractions

When hexane and heptane are mixed, we have two moles of each. The total number of moles in the solution is 2 (Hexane) + 2 (Heptane) = 4 moles. The mole fraction of hexane, \(X_{\text{Hexane}}\), is \(\frac{2}{4} = 0.5\). Similarly, the mole fraction of heptane, \(X_{\text{Heptane}}\), is also \(\frac{2}{4} = 0.5\).

Step 2: Calculate Partial Pressures

Using Raoult's Law, calculate the partial pressure exerted by each component. For hexane: \(\mathrm{P}_{\text{Hexane}} = X_{\text{Hexane}} \times \mathrm{P}_{\text{Hexane}}^{\circ} = 0.5 \times 50 = 25\) mm Hg. For heptane: \(\mathrm{P}_{\text{Heptane}} = X_{\text{Heptane}} \times \mathrm{P}_{\text{Heptane}}^{\circ} = 0.5 \times 60 = 30\) mm Hg.

Step 3: Calculate the Total Pressure

The total pressure of the solution is the sum of the partial pressures of hexane and heptane. \(\mathrm{P}_{\text{Total}} = \mathrm{P}_{\text{Hexane}} + \mathrm{P}_{\text{Heptane}} = 25 + 30 = 55\) mm Hg.

Step 4: Verify against the Options

From the calculations: \(\mathrm{P}_{\text{Total}} = 55\) mm Hg, \(\mathrm{P}_{\text{Hexane}} = 25\) mm Hg, and \(\mathrm{P}_{\text{Heptane}} = 30\) mm Hg. Compare these results with the options given. The calculated total pressure matches option (a), partial pressure of hexane matches option (b), and partial pressure of heptane matches option (c). Therefore, options (a), (b), and (c) are correct.

Key Concepts

Understanding Mole Fraction

The mole fraction is a simple concept that expresses the ratio of moles of a component to the total number of moles in a solution. In the case of a solution made by mixing two substances, like hexane and heptane, the mole fraction helps in determining how much of each substance is present relative to the total mixture.

For example, when 2 moles of hexane and 2 moles of heptane are mixed, the total number of moles becomes 4. The mole fraction, which is denoted by the symbol \(X\), can then be calculated for each component:

• The mole fraction of hexane, \(X_{\text{Hexane}}\), is \(\frac{2}{4} = 0.5\).
• Similarly, the mole fraction of heptane, \(X_{\text{Heptane}}\), is also \(\frac{2}{4} = 0.5\).

With mole fractions, we can now apply Raoult's Law to determine how each component contributes to the total pressure in the solution.

Partial Pressure Calculation

The partial pressure of a gas component in a mixture is a measure of that gas's contribution to the total pressure. Raoult's Law plays a crucial role here, as it allows us to predict the partial pressures in ideal solutions.

According to Raoult's Law, the partial pressure of a component is given by the product of its mole fraction and its pure component vapor pressure:

• For hexane, the partial pressure \(P_{\text{Hexane}}\) is calculated as \(X_{\text{Hexane}} \times P_{\text{Hexane}}^{\circ}\), where \(P_{\text{Hexane}}^{\circ}\) is the vapor pressure of pure hexane. This gives \(0.5 \times 50 = 25\, \, \text{mm Hg}\).
• Similarly, for heptane, the partial pressure \(P_{\text{Heptane}}\) is \(X_{\text{Heptane}} \times P_{\text{Heptane}}^{\circ}\). This comes out as \(0.5 \times 60 = 30\, \, \text{mm Hg}\).

To find the total pressure of the mixture, you simply add the calculated partial pressures: \(P_{\text{Total}} = 25 + 30 = 55\, \, \text{mm Hg}\). This total pressure is what you would anticipate from the system under ideal conditions.

Ideal and Non-Ideal Solutions

To better understand Raoult's Law, it's important to distinguish between ideal and non-ideal solutions. An ideal solution perfectly follows Raoult's Law, meaning the interactions between molecules of different substances are similar to the interactions within each pure component.

In an ideal solution, the calculated total and partial pressures match perfectly with experimental results. This was shown by our earlier calculations, where the calculated pressure values correctly matched the exercise answers.

• Ideal solutions: Obey Raoult's Law, showing linear relationships between composition and properties.
• Non-ideal solutions: Deviate from Raoult's predictions due to differing intermolecular forces, requiring activity coefficients for accurate description.

The exercise assumes ideal behavior, as evidenced by the correct totals derived using Raoult’s Law, but in real-world scenarios, many solutions exhibit non-ideal behavior that requires a more comprehensive analysis.