THE VIBRATIONS OF MOLECULES I

THE SIMPLE HARMONIC OSCILLATOR

STUDENT INSTRUCTIONS

These instructions accompany the MATHCAD diskfile horun.mcd.

This program illustrates the classical harmonic oscillator. First we will derive the differential equation of motion for a mass m connected to a heavy wall with a spring of force constant k. The coordinate system is chosen so that when the mass is at rest with no forces acting on it the value of x = 0. When the spring is stretched x > 0, and when the spring is compressed x < 0. The potential energy is given below:

The kinetic energy T is given by the expression

where is the time derivative of the distance x or the velocity. The potential energy V is given by where k is the force constant and x the displacement from the equilibrium position. To obtain the differential equation of motion we use the Lagrange function L

which is then substituted into Lagrange's equation of motion

gives the differential equation or

EXERCISE 1

Substitute L in equation (3) into Lagrange's equation (4) to obtain the differential equation (5). Remember that when you take a partial derivative you assume that all other variables are constants.

EXERCISE 2

In the computational document set k=3, m=1, and in the initial values matrix set x=.2 and dx/dt = 0. Run the program (press F9) and note the graph on page 2. The red line represents x as a function of time and the green dash line represents dx/dt. When x=0 what is the velocity dx/dt? When the dx/dt=0 what is the value of x?

EXERCISE 3

Now change k by a factor of 10. How has the frequency changed? Return k to its original value and raise m by a factor of 10. What happens to the frequency?

EXERCISE 4

Change k to 300 and the mass m to 1. Now observe the graph. Is this the expected behavior? Is there an explanation for this graph?

THE DIATOMIC MOLECULE

The same type of differential equation applies to the diatomic molecule with modest adjustments of the above equations. We describe the motions in terms of an internal coordinate q which is the amount that the distance between the atoms has been distorted from the equilibrium bond distance. For the mass we must use the reduced mass which is calculated from the individual atom masses as follows:

where m1 and m2 are the masses in kg of the individual atoms. This quantity can be calculated using gram molecular weights M1 and M2

where Nav is Avagadro's number. The kinetic energy

The potential energy q is written

The differential equation of motion then becomes

where .

A graph suitable for animation is given at the end of the computational document. Place the cursor just to the right of this diagram and then click on 'Window' then 'animation', the 'create' and a dialog box should appear. Move the cursor to the upper-left side of the diagram and hold down the mouse button to mark off the region you wish to animate.

Now press the light bulb button which establishes automatic mode calculation. Now click on animate and the computer will prepare the frames of the atoms at different positions. A small playback picture appears at the left. Click on the arrow at the left bottom and the animation will play back.