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Vibration
The Law of Equipartition of Energy 1
This law states that the total energy of a system is equally distributed among all accessible degrees of freedom. 2
Each degree of freedom contributes 1/2kT of energy on average, where k is Boltzmann's constant and T is the temperature. 3
This principle ensures that energy is distributed fairly among all possible motions, leading to thermal equilibrium within the system.
Mathematical Formulation: Quantifying Energy per Degree of Freedom
The total energy of a molecule is the sum of its kinetic and potential energy associated with each degree of freedom.
Each degree of freedom receives an average energy of 1/2kT , where k is Boltzmann's constant and T is the temperature.
A diatomic molecule has 6 degrees of freedom, resulting in an average total energy of 3kT.
Limitations and Deviations: When Equipartition Fails
Quantum Effects At low temperatures, quantum mechanics plays a significant role, and the classical equipartition theory breaks down. This means the energy distribution may not be perfectly equal, and the expected heat capacities may deviate from the theoretical values.
Vibrational Modes Vibrational modes can be "frozen out" at low temperatures, as they require a certain amount of energy to be excited. This leads to a decrease in the effective number of degrees of freedom and lower heat capacities than predicted by equipartition.
Complex Molecules For complex molecules, the equipartition theory can become less reliable due to the interplay of multiple vibrational modes and the challenges of accurately calculating their contributions to energy distribution.
