The Equipartition Principle and Molar Heat Capacity

The Equipartition Principle states that in thermodynamic equilibrium, the energy of a molecule is equally distributed among all its degrees of freedom. In other words, each degree of freedom of a molecule has an average energy of kT/2, where k is the Boltzmann constant and T is the temperature.

The molar heat capacity at constant volume, Cv, is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant volume. According to the Equipartition Principle, the molar heat capacity at constant volume for a diatomic molecule, such as C2H2, is given by:

Cv = (f/2)R

where f is the number of degrees of freedom of the molecule and R is the gas constant. For a diatomic molecule, the degrees of freedom are translational (3), rotational (2), and vibrational (1). Therefore, the total number of degrees of freedom for C2H2 is 6, and the predicted high temperature limiting value of the molar heat capacity at constant volume is:

Cv = (6/2)R = 3R

However, this value is only valid at low temperatures. As the temperature increases, the vibrational degrees of freedom become more excited and contribute more to the heat capacity. At high temperatures, the vibrational degrees of freedom become fully excited and the molar heat capacity reaches a limiting value of:

Cv = (7/2)R

This is because the vibrational degrees of freedom are now fully excited and contribute (1/2)kT to the average energy of the molecule, in addition to the (5/2)kT contributed by the translational and rotational degrees of freedom. Therefore, the correct answer is option D, 9.5R.