You take on a research assistantship with a molecular...

You take on a research assistantship with a molecular physicist. She is studying the vibrations of diatomic molecules. In these vibrations, the two atoms in the molecule move back and forth along the line connecting them (see Figure $20.5 \mathrm{c}$ ). As an introduction to her research, she asks you to familiarize yourself with the Lennard-Jones potential (see Example 7.9 ), which describes the potential energy function for a diatomic molecule. She asks you to determine the effective spring constant, in terms of the parameters $\sigma$ and $\epsilon,$ for the bond holding the atoms together in the molecule for small vibrations around the equilibrium separation $r_{\mathrm{cq}} .$ After being stumped for a while, you ask her for a hint. She responds, "Example 7.9 provides the derivative of the potential energy function. Compare that to Equation 7.29 to find the force between the atoms. You want to show that $F$ is of the form $-k x$, and find $k .$ Let the separation distance $r=r_{\mathrm{eq}}+x,$ where $x$ is small and take advantage of the series approximations in Appendix Section B.5." Wow, that's several hints! You sit down and get to work.

Key Concepts

Lennard-Jones Potential

The Lennard-Jones potential is a mathematical model used to describe the interaction between a pair of neutral atoms or molecules. It captures both the attractive and repulsive forces at play and is characterized by parameters that control the depth of the potential well and the equilibrium separation. This potential is fundamental in molecular physics and chemistry as it provides insight into molecular binding and interactions.

Harmonic (Small Oscillation) Approximation

When a system exhibits small vibrations around an equilibrium point, the potential energy function can be approximated by a quadratic function. This approximation simplifies the analysis by treating the motion as that of a simple harmonic oscillator, where the force is linearly proportional to the displacement from equilibrium. This concept is vital for analyzing molecular vibrations and extracting effective parameters like the spring constant.

Taylor Series Expansion

The Taylor series expansion is a mathematical tool used to approximate functions using polynomials. In the context of molecular vibrations, it is applied to the potential energy function near the equilibrium separation. Retaining terms up to the second order, the expansion yields a quadratic function whose curvature directly relates to the effective spring constant, enabling a simplified description of the system’s dynamics.

Effective Spring Constant

The effective spring constant quantifies the stiffness of the bond in a molecule under small displacements from its equilibrium position. It is derived from the curvature of the potential energy function at the minimum, obtained by taking the second derivative with respect to the displacement. This constant is crucial for modeling molecular vibrations using the harmonic oscillator model.

Force Derivation from Potential Energy

The relationship between force and potential energy is given by F = -dU/dr, indicating that the force is the negative gradient of the potential energy. In the context of small vibrations, this derivative is used to derive a linear force law (Hooke's law), where the force is proportional to the displacement, and the constant of proportionality is the effective spring constant.

Transcript

00:01 In this problem, we want to find the effective spring constant of a diatomic molecule.

00:07 So diatomic molecule is just a molecule with two atoms.

00:11 These atoms are going to be kind of jiggling back and forth, and we want to know, you know, if we model this as a spring, what is that spring constant? to do this, we're going to be using the equation on screen right now.

00:23 This is the leonard jones potential.

00:27 And what we'll be using is the fact that b -force is equal to the derivative of the potential with respect to distance and it's actually negative of that.

00:43 So let's take this derivative.

00:47 What we'll get, if we take this derivative, so we'll have sigma to the 12 up top, and then r to the negative 12, that gives us the negative 12, and then it's going to be r to the 13.

01:00 And now for this one, it's going to be quite similar.

01:04 We're going to get a negative 6, so that becomes positive.

01:07 6 divided by r to the 7, and we get sigma to the 6 on top.

01:12 Okay, so this is the force on the particles.

01:16 Now we're also given another hint here, which is to assume that these two atoms are near their equilibrium length, right? if you have a molecule, you would assume that they're not going to be far away from the equilibrium since they're staying together, essentially.

01:34 That's what we're going to assume that r is equal to r equilibrium plus x, where x is very small.

01:45 Okay, so this allows us to do an approximation.

01:53 Specifically it's a taylor series approximation.

01:57 And the equation for this is this.

02:00 If you have 1 plus delta where delta here is small and you raise that to the power of, we'll call it n.

02:08 This will be approximately 1 plus n delta.

02:12 So that is the approximation.

02:14 So let's apply that to this.

02:16 So let's just plug in what r is here.

02:19 And negative 12, sigma to the 12.

02:23 So the req plus x to the 13 plus 6 sigma to the 6.

02:33 R .e .q.

02:34 Plus x to the 7.

02:36 Okay.

02:38 Now to make this look like this, we're going to factor out req from this.

02:48 So this will be negative 12 sigma to the 12 divided by req to the 13.

02:56 1 plus x over the req term.

03:02 And then the same thing for this other term.

03:06 So req to the 7, it's factored out, and we get 1 plus x over req again.

03:14 Okay.

03:15 And now, all we have to do here is apply that approximation.

03:19 So this will be approximately equal to 4 epsilon, negative 12 sigma 12 over req to the 13, and this is going to be 1 plus 13 x over req.

03:40 All right.

03:45 And then plus 6 sigma to 6th, divided by req to 7.

03:53 1 plus 7xr .e .q.

04:04 Now, do notice i accidentally made this a plus.

04:08 That should be a minus since the powers of these things is negative.

04:14 Okay, that is what we get though from this expansion.

04:20 And now we notice that we do have these terms here and here that don't contain an x.

04:27 So let's think about how we can get rid of these.

04:31 And how we can get rid of these is the fact that f is equal to zero when x is equal to zero, right? when x is equal to zero, we're in the equilibrium position, which means that the force should be zero...

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