Polyatomic Gases
Polyatomic gases have a complex structure compared to monatomic and diatomic gases, leading to three translational and three rotational degrees of freedom, along with additional vibrational modes. The law of equipartition of energy applies, meaning each degree of freedom contributes to the overall energy of the system.
Degrees of Freedom
For a mole of polyatomic gas, the total internal energy, U, can be expressed in terms of the translational, rotational, and vibrational contributions:
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Translational energy: Each of the three translational degrees of freedom contributes
\[\frac{3}{2} k_B T \text{ (where } k_B \text{ is Boltzmann's constant and } T \text{ is temperature)}.\]
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Rotational energy: The three rotational degrees of freedom add another
\[\frac{3}{2} k_B T.\]
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Vibrational energy: While the precise number of vibrational modes depends on the gas, each normal mode contributes both potential and kinetic energy, which results in an additional
\[k_B T \text{ per mode (2 contributions)}.\]
The total internal energy thus arrives at:
\[U = (3 + f) R T\]
where \(f\) denotes the number of vibrational modes. Consequently, the specific heat capacities for such gases can be written as:
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Molar specific heat at constant volume,
\[C_v = (3 + f)R\]
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Molar specific heat at constant pressure,
\[C_p = (4 + f)R.\]
The ratio of the specific heats remains consistent with ideal gas theory:
\[\gamma = \frac{C_p}{C_v} = \frac{4 + f}{3 + f}.\]
In summary, the specific heat capacities of polyatomic gases greatly depend on their molecular structure, particularly the vibrational freedom, impacting their thermal behavior and energy storage capabilities.