The Lennard-Jones potential is a model that describes the interaction between two non-bonded molecules. It accounts for both attractive and repulsive forces:

• Attractive Forces: At larger distances, molecules attract each other due to van der Waals forces.
• Repulsive Forces: At very short distances, molecules repel each other due to the overlap of electron clouds.

The formula for the Lennard-Jones potential is:

$$ U(r)=4 \varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right] $$

Where:

• $U(r)$ is the potential energy as a function of the distance $r$ between the molecules.
• $\varepsilon$ (epsilon) is the depth of the potential well, representing the strength of the attraction.
• $\sigma$ (sigma) is the distance at which the potential is zero.
• $r$ is the distance between the molecules.

Calculation

Assume the Lennard-Jones potential model with $\sigma=0.36 nm$ and $\varepsilon=1.38 E -21 J$ :

(a) Compute the potential energy for two molecules located at positions ($0,0$) and ($0,0.4 nm$). (b) Suppose a third molecule was located at ($0.5,0$). Compute the potential energy for the system. (c) To develop a very crude insight on the methods of averaging, we can think of the average potential energy as defining an average distance between the molecules. As the volume expands, the average distance between molecules increases and the magnitude of the average potential energy decreases in accordance with the Lennard-Jones model. For the potential energy from (b), compute the average distance, $\langle r\rangle$, that corresponds to the average potential energy for this system of molecules. (d) Suppose the volume of the system in (c) expands by a factor of two. How would that affect the average distance, $\langle r\rangle$, and what would you estimate as the new intermolecular energy? (e) Assume approximately four molecules can fit around a central molecule in a liquid before it is too crowded and another layer starts to build up. Assuming the Lennard-Jones energy is practically zero beyond the first layer (i.e. ignore all but the first layer), and the average distance between the central molecule and its four neighbors is $\langle r\rangle=0.55 nm$, estimate the intermolecular energy around one single molecule and that for one mole of similar molecules.

🧠Python snippets

https://gist.github.com/viadean/d4ef317a8a48ae8e17fb4350cc1d7fd8

Explanations:

• Clear Function Definitions: The code is organized into functions (lennard_jones_potential, distance, and find_average_distance) for better readability and reusability.
• Accurate Distance Calculation: The distance function correctly calculates the Euclidean distance between two points.
• Lennard-Jones Implementation: The lennard_jones_potential function accurately implements the Lennard-Jones potential formula.
• Average Distance Function: added the find_average_distance function. This function uses a binary search to find the distance that results in the given potential.