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 B at \(298 \mathrm{~K}\) which one is correct? (a) \(\Delta \mathrm{S}_{\operatorname{mix}}=-\mathrm{R} \sum \mathrm{x}_{1} \ln \mathrm{x}_{\mathrm{i}}\) (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

Correct options: (a) and (b).

Step 1: Define Ideal Solution Characteristics

An ideal solution is characterized by \(\Delta H_{\text{mix}} = 0\) and \(\Delta G_{\text{mix}} < 0\). This is because there are no interactions between the components beyond those in the pure solvents.

Step 2: Calculate Total Moles

Calculate the total number of moles in the solution. Given 1 mole of A and 2 moles of B, the total moles is \(1 + 2 = 3\).

Step 3: Find Mole Fractions

Find the mole fractions of A and B:- Mole fraction of A, \(x_A = \frac{1}{3}\).- Mole fraction of B, \(x_B = \frac{2}{3}\).

Step 4: Verify Entropy of Mixing

For an ideal solution, the entropy change of mixing is\[\Delta S_{\text{mix}} = -R(x_A \ln x_A + x_B \ln x_B)\]This is consistent with option (a), which is a correct form for \(\Delta S_{\text{mix}}\).

Step 5: Verify Enthalpy of Mixing

Since it is an ideal solution, \(\Delta H_{\text{mix}} = 0\) as there are no heat exchanges in mixing; hence, option (b) is correct.

Step 6: Verify Gibbs Free Energy of Mixing

The Gibbs free energy change for an ideal solution is given by\(\Delta G_{\text{mix}} = RT(x_A \ln x_A + x_B \ln x_B)\)Since option (c) represents this, it is incorrect as it does not consider the negative sign for the free energy change.

Step 7: Check Endothermic Mixing

Option (d) suggests \(\Delta H_{\text{mix}} > 0\), which is incorrect for ideal solutions since for them \(\Delta H_{\text{mix}} = 0\) as highlighted previously.

Key Concepts

Entropy of Mixing

When two substances are mixed, their molecules spread out over a larger volume, leading to an increase in disorder. This randomness is a classic characteristic when dealing with the entropy of mixing. For ideal solutions, this can be quantitatively understood using the formula:

• \[ \Delta S_{\text{mix}} = -R(x_A \ln x_A + x_B \ln x_B) \]

where \( R \) is the gas constant, and \( x_A \) and \( x_B \) are the mole fractions of the substances \( A \) and \( B \).
The negative sign indicates that the random distribution of molecules results in an increase in entropy, enhancing the disorder of the system.
This formula perfectly aligns with the basic thermodynamic principle that mixing increases entropy because the components become more evenly distributed.

Enthalpy of Mixing

Ideal solutions follow specific rules one of which is related to enthalpy. Enthalpy describes the heat exchange during a process. For ideal solutions, the enthalpy of mixing is defined as

• \( \Delta H_{\text{mix}} = 0 \)

This indicates no heat is absorbed or evolved during the mixing process as there are no interactions between the different molecules that are different than those in each pure substance.
Why is this zero? In real substances, intermolecular forces have to be overcome or formed when they mix, but in ideal solutions, these forces are presumed to be unchanged.
This characteristic of an ideal solution tells us the energy of the system remains unchanged when the substances are mixed.

Gibbs Free Energy of Mixing

Gibbs free energy is a key concept for understanding spontaneity in chemical processes. For mixing processes, it allows us to predict whether mixing will occur spontaneously. The Gibbs free energy change is calculated as:

• \[ \Delta G_{\text{mix}} = RT(x_A \ln x_A + x_B \ln x_B) \]

where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin.
This equation states, at any given temperature, the free energy usually decreases when the components mix.
A negative value of \( \Delta G_{\text{mix}} \) suggests that the mixing process is spontaneous as the energy of the system is minimized. For ideal solutions, this fundamental theorem reinforces the notion that when no additional interactions exist beyond the ideal assumptions, substances will mix naturally due to entropy considerations.