CO2 Molecule Vibrational Modes and Heat Capacity | Thermal Physics Problem

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SUMMARY

The discussion focuses on the vibrational modes of a CO2 molecule and their impact on the constant volume heat capacity. The CO2 molecule exhibits three vibrational modes with frequencies of 2565 cm-1 (asymmetric stretch), 1480 cm-1 (symmetric stretch), and 526 cm-1 (bends). The Equipartition of Energy theorem is applied to determine how energy is distributed among these modes, particularly emphasizing that only translational degrees of freedom contribute to energy at low temperatures. The constant volume heat capacity is defined as the derivative of total energy with respect to temperature (dEtot/dT).

PREREQUISITES
  • Understanding of the Equipartition of Energy theorem
  • Familiarity with vibrational modes and their corresponding wavenumbers
  • Knowledge of heat capacity concepts, specifically constant volume heat capacity
  • Basic principles of thermal physics and molecular behavior
NEXT STEPS
  • Study the relationship between vibrational frequencies and energy levels in diatomic and polyatomic molecules
  • Learn how to sketch heat capacity curves for different gases at varying temperatures
  • Explore the concept of rotational and vibrational degrees of freedom in thermodynamics
  • Investigate the implications of classical versus quantum mechanical treatments of molecular energy
USEFUL FOR

Students and professionals in thermal physics, chemists studying molecular behavior, and anyone interested in the thermodynamic properties of gases, particularly CO2.

kottur
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Homework Statement



Consider a CO2 molecule, which is linear and has vibrational modes with frequency
corresponding to 2565 cm-1 (an asymmetric stretch), 1480 cm-1 (a symmetric stretch)
526 cm-1 (bends). Sketch a curve showing how the constant volume heat capacity of CO2
gas varies with temperature and mark the values of plateaus. (Recall: the spacing between
rotational levels is smaller than the spacing between the vibrational levels).

The Attempt at a Solution



I don't have a clue where to start!
 
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Assume you can treat the CO2 gas classically and try using these main ideas:

1) Equipartition of Energy theorem says that each degree of freedom (for example translational motion along x-axis) which shows up quadratically in the total energy of the molecule contributes an average 1/2kBT to the energy of the molecule.

2) Some degrees of freedom ('modes') require more energy to excite and so they do not contribute to the total energy at lower temperatures. For example, only the three translational degrees of freedom ('x', 'y' and 'z') contribute for the lowest temperatures.

3) The wavenumbers (cm-1) corresponding to each vibrational mode listed are related to the energy of the mode.

4) Constant Vololume heat capacity is defined as dEtot/dT


Bit hand-wavey but I think that's the idea.
 

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