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Question
Calculate the ratio of two specific heats of polyatomic gas molecules.
Solution
- Gases that have molecules containing more than two atoms are termed polyatomic gases.
- Each molecule of the polyatomic gas has 3 translational degrees of freedom. Only linear molecules have 2 degrees of freedom for rotation. All other polyatomic molecules have 3 degrees of freedom for rotation.
- The number of degrees of freedom (f) for the vibrational motion of a polyatomic molecule depends on the geometric structure of the molecule, i.e., the arrangement of atoms in a molecule.
- Each such degree of freedom contributes average energy `2 xx 1/2`kBT from kinetic energy and potential energy terms.
- Therefore for 1 mole of a polyatomic gas, the internal energy is E = `3/2 N_Ak_BT + 3/2 N_Ak_BT + f × 2/2 N_Ak_BT`
= (3 + f) NAkBT - Molar-specificific heats at constant volume and constant pressure are given as
Cv = (3 + f) R and Cp = (4 + f) R
∴ `gamma = C_p/C_v = (4 + f)/(3 + f)`
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