Question

The molecular bonding in a diatomic molecule such as the nitrogen \(\left(\mathrm{N}_{2}\right)\) molecule can be modeled by the Lennard Jones potential, which has the form $$ U(x)=4 U_{0}\left(\left(\frac{x_{0}}{x}\right)^{12}-\left(\frac{x_{0}}{x}\right)^{6}\right) $$ where \(x\) is the separation distance between the two nuclei and \(x_{0}\), and \(U_{0}\) are constants. Determine, in terms of these constants, the following: a) the corresponding force function; b) the equilibrium separation \(x_{0}\), which is the value of \(x\) for which the two atoms experience zero force from each other; and c) the nature of the interaction (repulsive or attractive) for separations larger and smaller than \(x_{0}\).

Short answer

a) The force function is: $$ F(x) = 4U_{0}\left(12\left(\frac{x_{0}^{13}}{x^{14}}\right) - 6\left(\frac{x_{0}^{7}}{x^8}\right)\right) $$ b) The equilibrium separation is: $$ x_0 $$ c) The nature of the interaction: - Repulsive for separations smaller than \(x_0\). - Attractive for separations larger than \(x_0\).

Step-by-step solution

Calculate the Force Function

In physics, the force between two particles is the negative derivative of the potential energy function with respect to the separation distance. Thus, we can find the force function by differentiating the given potential energy function \(U(x)\) with respect to \(x\), as follows: $$ F(x) = -\frac{dU(x)}{dx} $$ Differentiating the given potential energy function and negating the result, we have: $$ F(x) = -\frac{d(4U_{0}((\frac{x_{0}}{x})^{12} - (\frac{x_{0}}{x})^{6}))}{dx} $$

Differentiate U(x)

Using the chain rule, let's differentiate \(U(x)\): $$ \frac{dU(x)}{dx} = 4U_{0}\left(-12\left(\frac{x_{0}}{x}\right)^{12}\left(\frac{-x_{0}}{x^2}\right) - 6\left(\frac{x_{0}}{x}\right)^6\left(\frac{-x_{0}}{x^2}\right)\right) $$

Simplify the Force Function

Now we can simplify the force function by removing the negative sign and grouping the terms: $$ F(x) = 4U_{0}\left(12\left(\frac{x_{0}^{13}}{x^{14}}\right) - 6\left(\frac{x_{0}^{7}}{x^8}\right)\right) $$

Equilibrium Separation

The equilibrium separation \(x_{0}\) is the value of \(x\) for which the atoms experience zero force from each other. So, we have to find when the force function \(F(x)\) is equal to zero: $$ F(x) = 0 $$ Setting \(F(x)\) to zero and substituting \(x=x_0\), we get $$ 0 = 4U_{0}\left(12\left(\frac{x_{0}^{13}}{x_{0}^{14}}\right) - 6\left(\frac{x_{0}^7}{x_{0}^8}\right)\right) $$ Canceling \(x_{0}\) in both terms, we get $$ 0 = 12 - 6 $$ This equation is true, so there is an equilibrium point at \(x=x_0\).

Analyze Interactions

To find the nature of the interaction for separations larger and smaller than \(x_{0}\), we can analyze the force function: For \(x>x_0\): Notice that when \(x > x_0\), the terms \(\frac{x_{0}^{13}}{x^{14}}\) and \(\frac{x_{0}^7}{x^8}\) becomes smaller, making the force function negative. So, the interaction is attractive. For \(x<x_0\): For cases when \(x < x_0\), the terms \(\frac{x_{0}^{13}}{x^{14}}\) and \(\frac{x_{0}^7}{x^8}\) becomes larger, making the force function positive. So, the interaction is repulsive. In conclusion, we have the following: a) The force function is: $$ F(x) = 4U_{0}\left(12\left(\frac{x_{0}^{13}}{x^{14}}\right) - 6\left(\frac{x_{0}^{7}}{x^8}\right)\right) $$ b) The equilibrium separation is: $$ x_0 $$ c) The nature of the interaction: - Repulsive for separations smaller than \(x_0\). - Attractive for separations larger than \(x_0\).

Key concepts

Molecular Bonding

Molecular bonding is the force that holds together atoms in a molecule. When we model these forces in a molecule, like a nitrogen diatomic molecule \(\mathrm{N}_2\), we often use potential energy functions to describe how the energy of the system changes with the distance between atoms. The Lennard-Jones potential is a common model that describes a balance of attractive and repulsive interactions at varying distances.

In this potential, two key factors—attraction due to van der Waals forces and repulsion due to Pauli repulsion between electrons—determine how strongly the two atoms are drawn to each other or pushed apart. This interplay of forces dictates the stability and properties of the molecule, including how it behaves under different physical conditions.

Equilibrium Separation

The concept of equilibrium separation is particularly important in understanding molecular structure. It represents the ideal distance between two atomic nuclei where the potential energy is at its minimum, and consequently, the forces between the atoms are balanced. In the context of the Lennard-Jones potential, the equilibrium separation, denoted as \(x_0\), is the point at which the attractive and repulsive forces cancel out to produce zero net force.

Equilibrium separation is not only a theoretical concept; it has practical implications for properties such as bond length. In our exercise, with the nitrogen diatomic molecule, understanding and calculating \(x_0\) allows us to predict the optimal distance that leads to a stable molecular bond under normal conditions. This balanced state is crucial for the molecule's existence and the formation of matter as we know it.

Force Function

The force function quantitatively describes the force experienced by atoms as a function of their separation distance. It's derived from the potential energy function by taking its negative derivative with respect to distance. This relationship captures how the force changes when atoms move closer or further apart from each other.

In our nitrogen diatomic molecule example, the force function helps us understand the nuances of molecular interaction. When the separation is greater than \(x_0\), atoms are attracted towards each other; conversely, when it's less, the atoms repel. The force function thus defines the behavior of the atoms and is used to predict the dynamics of the molecular system under different conditions, like changes in temperature or pressure. Evaluating the force function at equilibrium gives us insight into the stability of the molecular structure; it shows the balance point at which the molecule would be at rest if undisturbed.