Question

Lattice model mixture. You have a solution with molecules of type \(A\) mixed with molecules of type \(B\) at \(T=300 \mathrm{~K}\). Suppose that every two \(A B\) molecular contacts are less stable than one \(A . A\) plus one \(B B\) contact, by an energy \(k T\), and that every molecule has an average of six neighbor contacts. For \(x_{A}=x_{B}=1 / 2\), to what temperature do you have to heat the system to get \(A\) to dissolve fully in \(B\) ?

Short answer

450 K

Step-by-step solution

- Understand the Problem

A solution contains molecules of type A and type B at a temperature of 300 K. It's given that two AB contacts are less stable than one AA contact plus one BB contact by an energy of \(kT\). Each molecule has an average of six neighbor contacts. We need to determine the temperature at which A will fully dissolve in B when the mole fractions of A and B are both 0.5.

- Define the Intermolecular Energy Difference

Given that every two AB molecular contacts are less stable than one AA plus one BB contact by \(kT\), the energy difference for each AB contact is \(\frac{kT}{2}\).

- Calculate the Interaction Terms

For a molecule in a lattice with six neighbors, if the mole fractions \(x_A = x_B = \frac{1}{2}\), the number of AB contacts per molecule will be \(\frac{6}{2} = 3\). Thus, the total energy difference due to AB interactions is \(3 \times \frac{kT}{2} = \frac{3kT}{2}\).

- Set Up the Equation for Dissolution

To find the temperature at which A fully dissolves in B, we need the system to reach a state where the entropy and the intermolecular energy lead to full miscibility. Given that the energetic penalty must be balanced by the thermal energy, the condition is that the thermal energy \(kT'\) overcomes the energy difference \(\frac{3kT}{2}\). Therefore, \(kT' = \frac{3kT}{2}\).

- Solve for the Temperature

From \(kT' = \frac{3kT}{2}\), we solve for \ T'\ to get \(T' = \frac{3}{2} \times 300K = 450K\).

Key concepts

Intermolecular Energy

Intermolecular energy refers to the potential energy interactions between molecules in a substance. These interactions occur due to forces such as hydrogen bonding, van der Waals forces, and ionic bonds. In a lattice model mixture, the energy of molecular interactions depends on the types of molecules involved and their relative positions.

In the exercise, it's given that the intermolecular energy for AB contacts is less stable by an energy of \(kT\) compared to AA and BB contacts. This energy difference is crucial to determining how molecules A and B interact within the mixture. To solve the problem, we assume each molecule has six neighboring contacts. When calculating the energy difference due to AB contacts, knowing this intermolecular energy differential helps us understand the factors affecting the miscibility of the compounds.

Understanding intermolecular energy provides insight into the stability and dissolution behavior of mixtures. If one type of molecule prefers to stay with similar molecules (less unfavorable interactions), it's likely due to favorable intermolecular energies.

Molecular Interactions

Molecular interactions are fundamental to understanding how substances mix and behave in different environments. These interactions can be attractive or repulsive and are driven by the intrinsic properties of the molecules, such as polarity, size, and charge.

In the context of the lattice model mixture in the exercise, A and B molecules interact with each other and their neighbors. The specific interaction between different types of molecules (AA, BB, and AB) determines the overall behavior of the mixture. The energy difference calculated, where AB contacts are less stable by \(kT\), helps us understand why there might be a tendency not to mix fully unless enough thermal energy is added.

Molecular interactions involve a balance of forces. Attractive forces help substances mix, while repulsive forces act to keep them apart. The balance of these interactions dictates whether a substance will be miscible or immiscible under given conditions.

Miscibility

Miscibility is the ability of two substances to mix and form a homogeneous solution. It is influenced by factors such as temperature, pressure, and the nature of the molecular interactions between the substances involved.

In the provided exercise, we aim to determine the temperature at which molecule A fully dissolves in molecule B. For molecules to be fully miscible, the energy penalty from unfavorable interactions needs to be offset by thermal energy. This means achieving a state where the entropy (disorder) of the system outweighs the energetic costs associated with mixing.

As calculated, the temperature where full miscibility occurs is 450K. This is because the thermal energy at this temperature is sufficient to overcome the unfavorable energy difference caused by AB molecular interactions. By heating the system to 450K, the molecules gain enough kinetic energy to surpass the intermolecular energy barriers, leading to a fully mixed state.

Understanding miscibility is crucial in many fields, including chemistry, materials science, and pharmaceuticals. It helps predict how different substances will interact when combined and under various conditions.