Predict the contribution to the heat capacity \(C_{V, \mathrm{~m}}\) made by molecular motions for each of the following atoms and molecules: (a) \(\mathrm{HCN}\); (b) \(\mathrm{C}_{2} \mathrm{H}_{6}\); (c) Ar; (d) HBr.

Understanding the concept of degrees of freedom is crucial in physics and chemistry, especially when examining the behavior of gases. Simply put, degrees of freedom are the independent ways in which a system can possess energy. For molecules, these correspond to the various modes of motion they can exhibit. The greater the degrees of freedom, the more ways energy can be distributed within the molecule, directly influencing properties like molar heat capacity.

The contribution of degrees of freedom to the molar heat capacity, denoted as \(C_{V, \mathrm{m}}\), is particularly significant because it allows us to predict how much energy is required to raise the temperature of a substance by one degree. Knowing this helps in understanding thermal properties and behaviors of different substances.

Translational degrees of freedom refer to the motion of the center of mass of a molecule along the three spatial axes - \(x\), \(y\), and \(z\). Essentially, these are movements where the molecule travels from one place to another. Every molecule, regardless of its shape or size, has three translational degrees of freedom, corresponding to its ability to move left or right (\(x\)-axis), up or down (\(y\)-axis), and forward or backward (\(z\)-axis).

In thermodynamics, each translational degree of freedom contributes \(\frac{1}{2}R\) to the molar heat capacity at constant volume, where \(R\) is the ideal gas constant. This forms part of the foundational principles for predicting the heat capacity of gases.

Rotational degrees of freedom describe the molecule's capacity to rotate around its axes. Linear molecules, like \(HBr\), have two rotational degrees of freedom because they can spin around the two axes perpendicular to the line that connects their atoms. Nonlinear molecules, such as \(C_{2}H_{6}\), have three rotational degrees of freedom because they can rotate around all three spatial axes.

These rotational motions, just like translational ones, have an energy component which contributes to the molecule's heat capacity. Rotational degrees of freedom also add \(\frac{1}{2}R\) per degree of freedom to the molar heat capacity at constant volume. This knowledge assists in the detailed calculations of the heat capacities for various molecular structures.

Vibrational degrees of freedom involve the movement of atoms within a molecule as they vibrate about their equilibrium position. For a molecule with \(N\) atoms, the vibrational degrees of freedom are \(3N-5\) for linear molecules and \(3N-6\) for non-linear molecules. These vibrations include stretching and bending movements within the molecule and are more significant at higher temperatures where thay can absorb more energy.

At typical room temperatures, vibrational contributions to heat capacity are generally minimal and often negligible for simplicity. However, at higher temperatures, vibrations become more prominent and can no longer be ignored when computing the molar heat capacity.

The ideal gas constant, denoted as \(R\), is a fundamental constant in chemistry and physics used in the equation of state for an ideal gas. Its value is necessary for the calculation of several thermodynamic properties, including molar heat capacity. \(R\) represents the energy per degree per mole that gases ideally carry, and it is approximately equal to 8.314 J\/mol•K.

This constant is a bridge between the microscopic world of atoms and molecules and the macroscopic world of the gases we can measure and observe. It allows us to use the degrees of freedom discussed to calculate the molar heat capacity of gases by establishing the energy contribution from each translational, rotational, or vibrational motion.