Question

The relative motion of two atoms in a molecule can be described as the motion of a single body of mass \(m\) moving in one dimension, with potcntial encrgy \(U(r)=A / r^{12}-B / r^{6}\) where \(r\) is the separation between the atoms and \(A\) and \(B\) are positive constants. a) Find the equilibrium separation, \(r_{0}\) of the atoms, in terms of the constants \(A\) and \(B\) b) If moved slightly, the atoms will oscillate about their equilibrium separation. Find the angular frequency of this oscillation, in terms of \(A, B,\) and \(m\).

Short answer

Question: Calculate the equilibrium separation and angular frequency of oscillations for a diatomic molecule with the potential energy function given by \(U(r) = -Ar^{-12} + Br^{-6}\), where \(A\), \(B\), and \(r\) are constants and \(r\) is the separation between the atoms. Answer: The equilibrium separation \(r_0\) is given by the expression: \(r_0 = \sqrt[6]{2\frac{A}{B}}\). The angular frequency of oscillations \(\omega\) is given by the expression: \(\omega = \sqrt{\frac{1}{m}\left(-78 A^{1/3}B^{19/3}+ 42 A^{7/3}B^{5/6}\right)}\).

Step-by-step solution

(Step 1: Calculate the equilibrium separation \(r_0\))

To find the equilibrium separation, we will look for the point where the force between the atoms is zero, which corresponds to the minimum of the potential energy function. To find it, we need to differentiate the potential energy function \(U(r)\) with respect to \(r\) and set it equal to zero: \(\frac{dU(r)}{dr} = -12\frac{A}{r^{13}} + 6\frac{B}{r^7} = 0\) Now, we will solve this equation for the equilibrium separation \(r_0\).

(Step 2: Solve the equation for the equilibrium separation \(r_0\))

We have the equation: \(-12\frac{A}{r^{13}} + 6\frac{B}{r^7} = 0\) To solve it, we will first isolate \(A\): \(A = \frac{1}{2}r^{13}\frac{B}{r^7} = \frac{1}{2}r^6B\) Now, let's denote the equilibrium separation by \(r_0\): \(r_0^6 = 2\frac{A}{B}\) Finally, we have the equilibrium separation \(r_0\) in terms of the constants \(A\) and \(B\): \(r_0 = \sqrt[6]{2\frac{A}{B}}\)

(Step 3: Calculate the second derivative of the potential energy function)

To find the angular frequency of oscillations around the equilibrium, we will consider the second derivative of the potential energy function. It corresponds to the effective spring constant in the Hooke's law relating the force to the displacement. The second derivative of the potential energy function is given by: \(\frac{d^2 U(r)}{dr^2} = -12 \cdot 13 \frac{A}{r^{14}} + 6 \cdot 7 \frac{B}{r^8}\)

(Step 4: Evaluate the second derivative at the equilibrium separation \(r_0\))

Now, we need to evaluate the second derivative at the equilibrium point \(r_0\): \(\frac{d^2 U(r_0)}{dr^2} = -12 \cdot 13 \frac{A}{r_0^{14}} + 6 \cdot 7 \frac{B}{r_0^8}\)

(Step 5: Calculate the angular frequency of oscillation)

According to Hooke's law, the angular frequency of oscillation is given by: \(\omega = \sqrt{\frac{k}{m}}\) In our case, \(k = \frac{d^2 U(r_0)}{dr^2}\), so we get: \(\omega = \sqrt{\frac{1}{m}\left(-12 \cdot 13 \frac{A}{r_0^{14}} + 6 \cdot 7 \frac{B}{r_0^8}\right)}\) Now, we need to use the expression for the equilibrium separation \(r_0=\sqrt[6]{2\frac{A}{B}}\): \(\omega = \sqrt{\frac{1}{m}\left(-12 \cdot 13 \frac{A}{\left(\sqrt[6]{2\frac{A}{B}}\right)^{14}} + 6 \cdot 7 \frac{B}{\left(\sqrt[6]{2\frac{A}{B}}\right)^8}\right)}\) By simplifying the expression, we arrive at the final expression for the angular frequency of oscillations in terms of \(A, B\) and \(m\): \(\omega = \sqrt{\frac{1}{m}\left(-78 A^{1/3}B^{19/3}+ 42 A^{7/3}B^{5/6}\right)}\)

Key concepts

Equilibrium Separation

Understanding the concept of equilibrium separation is crucial in molecular physics, especially when analyzing the behavior of atoms within a molecule. Equilibrium separation, denoted as \( r_0 \), refers to the distance between two atoms in a molecule where the potential energy is at its minimum, and the system is in a stable state.

During the equilibrium state, the forces between atoms are balanced, resulting in no net force to cause movement. For the potential energy function given by \( U(r) = A / r^{12} - B / r^{6} \), we can find \( r_0 \) by setting the derivative of \( U(r) \) with respect to \( r \) equal to zero, symbolizing the force balance. This step leads to the critical equation for \( r_0 \) where \( r_0^6 = 2A/B \), and subsequently \( r_0 = \sqrt[6]{2A/B} \), linking the equilibrium separation directly to the constants of the potential energy function.

Angular Frequency of Oscillation

When a molecule experiences a slight displacement from its equilibrium position, the atoms oscillate about this point. The angular frequency of oscillation, \( \omega \), describes the speed of these oscillations and is a key concept in vibrational molecular dynamics.

The angular frequency can be calculated using Hooke's law, which suggests that the oscillatory motion of atoms can be considered similar to that of a mass attached to a spring. To obtain \( \omega \), we first determine the effective spring constant by calculating the second derivative of the potential energy function at the equilibrium separation. The angular frequency is then found using the formula \( \omega = \sqrt{k/m} \), where \( k \) is the effective spring constant and \( m \) is the mass of the oscillating system. The final expression for \( \omega \) involves the constants from the potential energy function as well as the mass of the body, capturing the intricacies of molecular vibration.

Potential Energy Function

The potential energy function, \( U(r) \), is a fundamental concept in the study of molecular physics, offering insight into the interaction and energy between atoms in a molecule. It describes how the potential energy of a system changes with the separation between atoms.

In our case, the potential energy function is proportional to \( 1/r^{12} \) and \( 1/r^{6} \), where \( r \) represents the separation between two atoms. The values of \( A \) and \( B \) are constants that dictate the depth and shape of the potential curve. The analysis of this function enables the determination of equilibrium points and the study of how atoms oscillate when displaced from this point.

Hooke's Law

Hooke's law is an essential principle in classical physics, and it has significant implications in molecular physics when describing the oscillatory motion of atoms. It states that the force needed to extend or compress a spring by some distance is proportional to that distance. It is commonly written as \( F = -kx \), with \( k \) being the spring constant, and \( x \) the displacement from the spring's rest position.

In the context of molecular physics, Hooke's law helps us approximate the behavior of atomic bonds as springs, where the displacement is the change in the separation from the equilibrium position. By analyzing the potential energy function and utilizing Hooke's law, we can deduce the angular frequency of oscillations of atoms, which is indispensable for understanding molecular vibrations and dynamics.