Animated Solution for Physics - Work, Energy, and Power: If the potential energy between two molecules is given by $U=-\frac{A}{r^6}+\frac{B}{r^{12}}$, then at equilibrium, separation between molecules and the potential energy are

Imagine two molecules floating in space. How do they interact? At very large distances, they exert a weak attractive force on each other (often due to Van der Waals forces). However, if you try to push them too close together, their electron clouds overlap, resulting in a massive repulsive force.

This delicate dance between attraction and repulsion is beautifully captured by the given potential energy function:

Here, the negative term $-\frac{A}{r^6}$ represents the long-range attraction, and the positive term $\frac{B}{r^{12}}$ represents the short-range repulsion. This specific mathematical form is famously known as the Lennard-Jones potential.

Nature loves stability. The molecules will naturally settle at a distance where they are perfectly comfortable—this is the equilibrium separation. At this exact point, the attractive pull perfectly balances the repulsive push, meaning the net force acting on the molecules is exactly zero.

Mathematically, conservative force is the negative spatial gradient of potential energy:

To find the equilibrium, we must set this force to zero:

Let's carefully apply the power rule of differentiation. The derivative of $r^{-n}$ is $-n{r}^{-n-1}$.

For this expression to be zero, the terms inside the bracket must be equal:

By cross-multiplying and isolating $r$, we get:

Taking the sixth root gives us the equilibrium separation:

Now that we know where the molecules settle, we need to find out how deep their energy well is. We do this by substituting our equilibrium value of $r^6$ back into the original potential energy equation.

Substitute $r^6=\frac{2B}{A}$:

Let's simplify the complex fractions:

To add these, we take a common denominator of $4B$:

This negative value represents the binding energy of the molecules. It is the amount of energy you would need to supply to pull the molecules completely apart to infinity. The mathematics perfectly mirrors the physical reality!