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 \(x_{\mathrm{E}}\), 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 \(x_{\mathrm{E}}\).

Short answer

Answer: The nature of interaction is repulsive for separations larger than the equilibrium separation and attractive for separations smaller than the equilibrium separation.

Step-by-step solution

Find the force function

To calculate the force, we need to take the negative derivative of the potential energy function with respect to the separation distance \(x\). Our given potential energy formula is: $$ U(x)=4U_{0}\left(\left(\frac{x_{0}}{x}\right)^{12}-\left(\frac{x_{0}}{x}\right)^{6}\right) $$ Differentiate \(U(x)\) with respect to \(x\) and multiply by -1: $$ F(x) = -\frac{dU}{dx} = -4U_0\left(-12\frac{x_0^{12}}{x^{13}} + 6\frac{x_0^6}{x^7}\right) $$ Simplify to obtain the force function: $$ F(x) = 4U_0\left(12\frac{x_0^{12}}{x^{13}} - 6\frac{x_0^6}{x^7}\right) $$

Find the equilibrium separation \(x_{\mathrm{E}}\)

To find the equilibrium separation, we want to find the value \(x_{\mathrm{E}}\) for which the force between the two atoms is zero. So we set our force function to zero and solve for \(x\): $$ 0 = 4U_0\left(12\frac{x_0^{12}}{x^{13}} - 6\frac{x_0^6}{x^7}\right) $$ Divide by \(4U_0\): $$ 0 = 12\frac{x_0^{12}}{x^{13}} - 6\frac{x_0^6}{x^7} $$ Move the second term to the other side: $$ 6\frac{x_0^6}{x^7} = 12\frac{x_0^{12}}{x^{13}} $$ Multiplying both sides with \(x^{13}\) and \(x^7\) to get rid of the denominators: $$ 6x^{13}x_0^6 = 12x^{7}x_0^{12} $$ Divide by \(6x_0^6\): $$ x^6 = 2x_0^6 $$ Take the sixth root of both sides: $$ x_{\mathrm{E}} = x_0\sqrt[6]{2} $$

Determine the nature of interaction

Now, we will analyze the force function, \(F(x)\), to determine the nature of interaction (repulsive or attractive) for separations larger and smaller than \(x_{\mathrm{E}}\). For \(x > x_{\mathrm{E}}\): We will check the sign of the force function: $$ F(x) = 4U_0\left(12\frac{x_0^{12}}{x^{13}} - 6\frac{x_0^6}{x^7}\right) > 0 $$ Since the force is positive for \(x > x_{\mathrm{E}}\), the interaction is repulsive. For \(x < x_{\mathrm{E}}\): We will check the sign of the force function: $$ F(x) = 4U_0\left(12\frac{x_0^{12}}{x^{13}} - 6\frac{x_0^6}{x^7}\right) < 0 $$ Since the force is negative for \(x < x_{\mathrm{E}}\), the interaction is attractive. In summary: a) The corresponding force function is given by: $$ F(x) = 4U_0\left(12\frac{x_0^{12}}{x^{13}} - 6\frac{x_0^6}{x^7}\right) $$ b) The equilibrium separation \(x_{\mathrm{E}}\) is given by: $$ x_{\mathrm{E}} = x_0\sqrt[6]{2} $$ c) The nature of interaction is: - Repulsive for separations larger than \(x_{\mathrm{E}}\) - Attractive for separations smaller than \(x_{\mathrm{E}}\)

Key concepts

Molecular Bonding

Molecular bonding is the force that holds atoms together in molecules, creating stable structures from individual atoms. This force arises due to the interactions between the electrons of one atom with the nucleus of another, leading to a balance that can be energetically favorable. In the context of Lennard-Jones potential, molecular bonding is described through a potential energy function that takes into account both attractive and repulsive forces.
Attraction occurs due to forces like van der Waals interactions or hydrogen bonding, while repulsion is caused when electron clouds or nuclei get too close, resulting in an increase in energy. At a certain point, the energy reaches a minimum, indicating the most stable configuration of the molecule, also known as the equilibrium separation. Understanding the Lennard-Jones potential helps explain how atoms combine and stay together to form molecules.

Equilibrium Separation

The term equilibrium separation, denoted as \( x_{\mathrm{E}} \), refers to the particular distance between two atoms or molecules at which the net force between them is zero. This distance signifies the point where the attractive and repulsive forces balance each other out, leading to a state of equilibrium. At the equilibrium separation, the potential energy of the system is at its minimum, meaning the atoms or molecules are at their most stable configuration.
In the Lennard-Jones potential, as shown in the exercise, the equilibrium separation \( x_{\mathrm{E}} \) can be algebraically derived by setting the force function to zero and solving for \( x \). For a diatomic molecule like nitrogen (\(\mathrm{N}_{2}\)), using the given potential function, the equilibrium separation is determined to be \( x_{\mathrm{E}} = x_0\sqrt[6]{2} \). Grasping this concept is crucial for understanding molecular structures and behaviors at different distances.

Force Function

The force function is derived from the potential energy function of a system by taking the negative gradient of the potential energy with respect to position. In a physical sense, the force function tells us how the force between two particles varies with their separation. Generally, a force function includes terms that account for attractive forces (which are negative and promote bonding) and repulsive forces (which are positive and prevent atoms from collapsing into each other).
For the Lennard-Jones potential, as seen in the exercise, the force function is found by taking the negative derivative of the potential energy function with respect to the separation distance \( x \). The resulting function is capable of describing the behavior of the forces as atoms move closer or farther away from each other. Specifically, this force function will predict an attractive force when atoms are closer than the equilibrium separation and a repulsive force when they are farther away, which is key to modeling the behavior of molecular bonds in different situations.