Suppose a gas is heated up to a temperature at which all degrees of freedom (translational, rotational, and vibrational) of its molecules are excited. Find the molar heat capacity of such a gas in the isochoric process, as well as the adiabatic exponent `gamma`, if the gas consists of
(a) diatomic ,
(b) linear `N-`atomic ,
( c) network `N-`atomic molecules.

Explanation

(a) A diatomic molecule has `3` translational, `2` rotational and one vibrational degrees of freedom. The corresponding energy per mole is
`(3)/(RT), ("translational") + 2 xx (1)/(2) RT, ("for rotational")`
`+1 xx RT,("for vibrational") = (7)/(2) RT`
Thus, `C_V = (7)/(2) R`, and `gamma = (C_p)/(C_V) = (9)/(7)`
(b) For linear `N- `atomic molecules energy per mole
=` (3 N - (5)/(2)) RT` as before
So, `C_V = (3 N- (5)/(2)) R` and `gamma = (6 N - 3)/(6 N - 5)`
( c) For noncollinear `N-`atomic molecules
`C_V = 3(N - 1) R` as before `(2.68) gamma = (3N - 2)/(3 N - 3) = (N - 2//3)/(N - 1)`.
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