The maximum high temperature molar heat capacity at constant volume to be expected for acetylene which is a linear molecule is (a) 9 cal/deg-mole (b) \(12 \mathrm{cal} /\) deg-mole (c) \(19 \mathrm{cal} /\) deg-mole (d) \(14 \mathrm{cal} /\) deg-mole

In the study of thermodynamics and statistical mechanics, 'degrees of freedom' refer to the number of independent ways in which a system can possess energy. For a molecule, these include translational, rotational, and vibrational movements. Translational degrees describe the movement of the molecule as a whole in three-dimensional space, accounting for three degrees of freedom (one for each axis of movement - x, y, and z). Rotational degrees account for the molecule's orientation in space. For linear molecules like acetylene, there are two rotational degrees since rotation along the bond axis does not change the system's energy state. Vibrational degrees are related to the internal vibrations of a molecule's atoms and require a more complex approach for quantification.

Considering the example of acetylene, we can simplify the complex nature of molecular motion into these quantifiable degrees of freedom. At high temperatures, notably, certain degrees disclosed in the problem must be reassessed to accurately represent the molar heat capacity. As seen in the exercise, counting vibrational modes correctly—as two degrees of freedom—is crucial to calculating the correct molar heat capacity.

The equipartition theorem is a fundamental concept in classical thermodynamics that states energy is distributed equally among its degrees of freedom. According to this theorem, for each degree of freedom that contributes to a system's energy, an amount of energy equal to \frac{1}{2}kT\ is granted, where \(k\) is the Boltzmann constant and \(T\) is the temperature. When dealing with molar heat capacities, the ideal gas constant \(R\) is used instead of the Boltzmann constant as it pertains to one mole of substance.

In practical terms, this means that for a diatomic linear molecule like acetylene, each degree of freedom would contribute \(\frac{1}{2} R\) to the molar heat capacity at constant volume. Thus, the problem's solution method hinges on correctly identifying and incorporating the degrees of freedom using the equipartition theorem. This helps explain why a miscalculation in the number of degrees of freedom can lead to a significant error in determining the molar heat capacity.

Vibrational modes of a molecule refer to the patterns of motion that involve periodic motion of the atoms within the molecule with respect to their equilibrium positions. Every molecule can vibrate in different ways, and each distinct pattern represents a vibrational mode. A linear molecule with \(N\) atoms has \(3N-5\) vibrational modes, while a nonlinear molecule has \(3N-6\).

For diatomic molecules or linear polyatomic molecules such as acetylene, vibrations can occur along the bond axis (stretching) or involve changes in the bond angle (bending), although for a diatomic molecule, only the stretching mode is present. It's essential to recognize that at high temperatures each vibrational mode is doubled due to the contribution of both kinetic and potential energy. This distinction is crucial and was underscored in the exercise, where each vibrational degree of freedom at high temperatures contributes as two degrees, highlighting the pivotal role of vibrational modes in calculating the molar heat capacity of a molecule.