Problem 131 - A Complete Resource Book in Chemistry for JEE Main

Question

For the binary ideal solution having 1 mole of \(\mathrm{A}\) and 2 moles of \(\mathrm{B}\) at \(298 \mathrm{~K}\) which one is correct? (a) \(\Delta \mathrm{S}_{\text {mix }}=-\mathrm{R} \sum \mathrm{x}_{\mathrm{i}} \ln \mathrm{X}_{\mathrm{i}}\) (b) \(\Delta \mathrm{H}_{\operatorname{mix}}=0\) (c) \(\Delta \mathrm{G}_{\operatorname{mix}}=\mathrm{RT} \sum \mathrm{x}_{\mathrm{i}} \ln \mathrm{x}_{\mathrm{i}}\) (d) \(\Delta \mathrm{H}_{\operatorname{mix}}=(+)\) ve

Step-by-Step Solution

Answer

Options (a), (b), and (c) are correct.

Step 1: Determine Mole Fractions

To find the mole fractions of A and B, we use the given moles: 1 mole of A and 2 moles of B. The total moles is 1 + 2 = 3 moles.- Mole fraction of A, \(x_A = \frac{1}{3}\).- Mole fraction of B, \(x_B = \frac{2}{3}\).

Step 2: Evaluate each Option

We will evaluate each option to check which are correct.(a) \( \Delta S_{\text{mix}} = -R \sum x_i \ln X_i \) is correct for ideal solutions: \(-R(x_A \ln x_A + x_B \ln x_B)\). (b) \( \Delta H_{\operatorname{mix}} = 0 \) is correct for an ideal solution, as there is no enthalpy change during mixing.(c) \( \Delta G_{\operatorname{mix}} = RT \sum x_i \ln x_i \) is correct and negative for ideal solutions: \(RT(x_A \ln x_A + x_B \ln x_B)\).(d) \( \Delta H_{\operatorname{mix}} = (+) \) ve is incorrect as ideal solutions have zero enthalpy change during mixing.

Step 3: Conclude Correct Options

Based on the evaluations: - Option (a) is correct because it reflects the entropy change for the ideal mixing process. - Option (b) is correct because the enthalpy of mixing for an ideal solution is zero. - Option (c) is correct as it uses the correct expression for Gibbs energy change. - Option (d) is incorrect since the enthalpy change for ideal mixing is zero.

Key Concepts

Mole Fraction

Understanding mole fraction is crucial in the study of solutions, especially when dealing with ideal solutions. \( \text{Mole fraction} \) is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the moles of one component to the total number of moles in the solution. For a binary solution with components A and B, the mole fraction of component A \( x_A \) is given by:

• \( x_A = \frac{\text{moles of A}}{\text{total moles in solution}} \)

In our example, with 1 mole of A and 2 moles of B, the total amount is 3 moles:

• Mole fraction of A, \( x_A = \frac{1}{3} \).
• Mole fraction of B, \( x_B = \frac{2}{3} \).

Mole fractions are dimensionless quantities and always add up to 1. They help in simplifying the calculations for other thermodynamic properties in solution chemistry.

Enthalpy of Mixing

Enthalpy of mixing is an essential part of understanding how substances interact in a solution. For an **ideal solution**, this concept simplifies significantly because such solutions assume no interaction forces between different molecules change upon mixing. Therefore, the **enthalpy of mixing** \( \Delta H_{\text{mix}} \) is zero.

• In mathematical terms, \( \Delta H_{\operatorname{mix}} = 0 \).

When substances mix ideally, their overall energy neither increases nor decreases—basically, the idea is that the solution feels the same way energetically as the pure components. In a practical sense, this means an ideal solution requires no heat to form, nor does it release heat.

Gibbs Energy of Mixing

The Gibbs energy of mixing is a vital concept when analyzing solution behavior it indicates the spontaneity of the mixing process. For **ideal solutions**, the equation for the change in Gibbs energy upon mixing is:

• \( \Delta G_{\operatorname{mix}} = RT \sum x_i \ln x_i \)

Here, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( x_i \) is the mole fraction of each component.For our mixture:

• \( \Delta G_{\operatorname{mix}} = RT \left( x_A \ln x_A + x_B \ln x_B \right) \).

This expression simplifies the understanding that mixing is a process driven by entropy rather than enthalpy changes. In essence, the Gibbs energy change is negative, showing that mixing in an ideal solution is a spontaneous process.

Entropy of Mixing

Entropy, often referred to as the measure of disorder, increases when substances are mixed, especially in **ideal solutions**. The **entropy of mixing** \( \Delta S_{\operatorname{mix}} \) can be expressed as:

• \( \Delta S_{\text{mix}} = -R \sum x_i \ln x_i \)

Where \( R \) is the universal gas constant, and \( x_i \) is the mole fraction of each component.In a binary system:

• \( \Delta S_{\text{mix}} = -R \left( x_A \ln x_A + x_B \ln x_B \right) \).

This formula highlights that entropy increases as components mix due to the increased randomness and dispersal of molecules. The increase in entropy is a fundamental driving force for the mixing process in ideal solutions, contributing to the overall spontaneity and Gibbs energy reduction.