CHEM 110-Debye
MOLECULAR VIBRATIONS
11/16/2002

Molar heat capacity, the amount of heat required to raise the temperature of one mole of a substance by 1 ^oC, is found to increase with molecular complexity. Why? In this experiment you will see computer simulations of the internal vibrational motions of molecules of various complexities.

The overall motion of a molecule can be divided up into translational, rotational and vibrational motion. Each of these three basic types of motion can be used to store energy:


1. Translational Motion- subdivided into motions along the x-, y-, and z-axes with each of these individual motions contributing ½RT to the average molar translational energy of a substance. The average molar energy stored in translational motion for a mole of any substance at temperature T is then 
   E_trans = ½RT + ½RT + ½RT = (3)½RT= 3RT/2
    
2. Rotational Motion - subdivided into rotations around the x-, y-, and z-axes with each such rotation contributing ½RT to the average molar rotational energy. However, rotational motion stores energy only if the rotational motion involves a nucleus located off the axis of rotation. A monatomic substance has its mass located directly on all three axes and cannot store energy in rotational motion. A linear molecular has two such qualified rotations (rotation around the molecular axis does not qualify as the atoms are located directly on the rotation axis) and its average molar energy stored in rotational motion is  
   E_rot(linear) = ½RT + ½RT = (2)½RT = RT
   A non-linear molecule can store energy in rotations around all three axes so that its average molar rotational energy is  
   E_rot(non-linear) = ½RT + ½RT + ½RT == (3)½RT = (3/2)RT
    
3. Vibrational Motion - subdivided into so-called normal modes of vibration which rapidly increase with the number of atoms in the molecule. Each of these normal vibrational modes contributes RT to the average molar energy of the substance and is a primary reason why heat capacities increase with molecular complexity. If there are X_vib modes of vibration, then the vibrational energy contributes X_vib(RT) to the average molar energy of the substance. A monatomic substance can have no internal vibrations as such motion would be called translation.

You will be looking at the increasing numbers and features of the normal modes of vibration as molecules become larger in size and more complex in their structure. To a large extent, it is these internal molecular vibrations which are responsible for the increased heat capacity of substances composed of larger molecules. The number of vibrational modes can be calculated from the number of atoms (N) in a molecule once you know if the molecule is linear or non-linear. For linear molecules, the number of vibrational modes is 3N - 5; for non-linear molecules, the number of vibrational modes is 3N - 6.


• Monatomic Substance - A monatomic substance can store energy only in translational motion. There are neither rotational nor vibrational modes.
   
• Linear Molecule - A linear molecule can store translational energy three ways and rotational energy in two ways so that X_vib(linear) = 3N - 3 - 2 = 3N - 5 equals the number of vibrational modes. For example, the CO₂ molecule consists of three atoms so that X_vib = 3N - 5 = 9 - 5 = 4 vibrational modes. The diatomic HCl molecule has X_vib = 3N - 5 = 6 - 5 = 1 vibrational mode.
   
• Non-linear Molecule - these molecules can store translational energy three ways and also rotational energy three ways. The number of normal vibrational modes is given by X_vib(non-linear) = 3N - 3 - 3 = 3N - 6. For example, the non-linear H₂S molecule will have X_vib = 3N - 6 = 9 - 6 = 3 normal modes of vibration.


Particular vibrational modes often are found to be dominated by either bending motions or stretching motions and chemists label them as such. Some vibrational modes appear to involve combinations of significant amounts of bending and stretching motions and cannot easily be identified as being one or the other. Here some examples showing the calculation of the number of vibrational modes of some molecules:



Molecule or Atom	Geometry	Number of Vibrational Modes	Average Molar Energy	Molar Cv
Ar	point	3N - 3 = 0	3RT/2 + 0 + 0 = (3/2)RT	(3/2)R
H2	linear	3N - 5 = 1	3RT/2 + 2(½RT) + (1)RT = (7/2)RT	(7/2)R
CO2	linear	3N - 5 = 4	3RT/2 + 2(½RT) + (4)RT = (13/2)RT	(13/2)R
H2S	non-linear	3N - 6 = 3	3RT/2 + 3(½RT) + (3)RT = (6)RT	(6)R
CH4	tetrahedral	3N - 6 = 9	3RT/2 + 3(½RT) + (9)RT = (12)RT	(12)R
C4H10	non-linear	3N - 6 = 36	3RT/2 + 3(½RT) + (36)RT = (39)RT	(39)R
C6H6	cyclic	3N - 6 = 30	3RT/2 + 2(½RT) + (30)RT = (65/2)RT	(65/2)R



PROCEDURE:
Peliminary (for practice):
1. Begin the Spartan molecular modeling program and maximize the window.
2. Under the File pull-down menu, click on Open. Ask your instructor in which folder the files for this experiment are located, go to that folder and open the file for the F₂ molecule (F2.spartan). The molecule should appear on your screen.
3. Notice how you can rotate and move the molecule around on the screen with your mouse
4. Under the Display pull-down menu, click on Vibrations. A small window should open listing the frequency (in cm^-1) of the single vibrational mode of this diatomic molecule along with a check-box. Click on the check-box and you will see the animated vibration of this simple molecule. Is this a stretching or bending mode?
5. Under the File pull-down menu, click on Close and answer NO if asked if you wish to store any changes to your file.
6. Repeat Step 2 and bring up the file for the benzene (C₆H₆) molecule.
7. Choose the Vibrations option from the Display pull-down menu and investigate the many vibrational modes for this complex molecule.
8. When finished, Close the file without saving any changes.

The Real Thing (for grade):
You are now ready to proceed with the experiment. For each molecule identified in linked table, use the Open option to bring up a new molecule. Make sure that you have first cleared the screen of the previous molecule with the Close option. Fill in the missing information in the table. The completed table will be your laboratory report: