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.
(a) A diatomic molecule has 2 translational, 2 rotational and one vibrational degrees of freedom. The corresponding energy per mole is
$$\frac{3}{2}RT$$ (for translational) + 2 $$\times \frac{1}{2}RT$$ (for rotational) + $$1 \times RT$$, (for vibrational) = $$\frac{7}{2} RT$$
Thus, $$C_V = \frac{7}{2}R$$, and $$\gamma = \frac{C_p}{C_V} = \frac{9}{7}$$
(b) For linear N- atomic molecules energy per mole
$$ = \Big( 3N-\frac{5}{2} \Big)RT$$ as before
So, $$C_V = \Big( 3N - \frac{5}{2} \Big)R$$ and $$\gamma = \frac{6N -3}{6N - 5}$$
(c) For noncollinear N- atomic molecules
$$C_V = 3(N-1)R$$ as betore (2.68) $$\gamma = \frac{3N_2}{3N-3} = \frac{N - 2/3}{N-1}$$
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