Which gas do you expect to have the higher molar heat capacity, \(\mathrm{NO}\) or \(\mathrm{NO}_{2}\) ? Why?

In physical chemistry, the term 'degrees of freedom' refers to the number of independent ways in which a molecular system can move without breaking any chemical bonds. For gas molecules, these are often divided into three categories: translational, rotational, and vibrational. Translational degrees of freedom involve the motion of the molecule as a whole moving through space. Rotational degrees refer to the molecule spinning or rotating around its axes. Lastly, vibrational degrees of freedom account for the internal motions of atoms within a molecule as they vibrate about their equilibrium positions.

For a linear molecule such as nitric oxide (NO), there are 3 translational and 2 rotational degrees of freedom. This is because a linear molecule can rotate about two axes perpendicular to the axis of the bond. The vibrational degrees of freedom are calculated using the formula 3n - 5, where n is the number of atoms. For NO, with two atoms, this would give 4 vibrational modes. However, each vibrational mode is usually considered as two degrees of freedom since it involves both kinetic and potential energy. Consequently, we typically double the number of vibrational modes to include both energies, resulting in NO having a total of 7 effective degrees of freedom.

In contrast, a non-linear molecule like nitrogen dioxide (NO₂) has 3 translational and 3 rotational degrees of freedom because it can rotate about three different axes. It has 3n - 6 vibrational modes due to its non-linear shape and, after doubling these for kinetic and potential energy consideration, it results in NO₂ having a total of 12 degrees of freedom. More degrees of freedom generally mean that the molecule has more ways to store thermal energy, leading to a higher molar heat capacity.

Vibrational modes are a fundamental concept when understanding molar heat capacity because they directly relate to how molecules absorb and store thermal energy through vibrations. Each atom in a molecule can move in three dimensions, leading to 3n possible vibrations for a molecule with n atoms. However, these include translations and rotations of the whole molecule which are not considered vibrational modes. So, for linear molecules, there are 3n - 5 vibrational modes, and for non-linear molecules, there are 3n - 6. Each vibrational mode involves both stretching and bending movements within the molecule. These movements are quantized, meaning that the energy can only increase or decrease in fixed amounts.

For instance, the linear molecule NO has 5 possible vibrational modes subtracting its translational and rotational motions (4 vibrations after applying the formula). In the case of NO₂, a non-linear molecule, the vibrational modes are 6 (applying the formula 3n - 6). It's pivotal to note that each vibrational mode contributes two degrees of freedom - one for potential energy and one for kinetic energy. These vibrational degrees are particularly important at higher temperatures where molecules have enough energy to engage in these complex motions, heavily influencing the specific heat capacity of the substance.

Translational motion pertains to the movement of a molecule as a whole from one location to another. This type of motion is fundamental to understanding the heat capacity of gases, as it is one of the primary forms of kinetic energy that molecules possess. In a simplistic model, each molecule in a gas moves in random directions, and these movements are described by three degrees of freedom, corresponding to the three spatial dimensions a molecule can traverse: up and down, left and right, and forward and backward.

Gases, due to their low density and high freedom of movement, typically exhibit significant translational motion. When heat is added to a gas, the translational kinetic energy of the gas molecules increases, which is reflected in the temperature of the gas. Having this translational motion as a component of the degrees of freedom is integral because it's one of the ways the gas will absorb and store energy, affecting its molar heat capacity. Both NO and NO₂ possess three translational degrees of freedom, relating to their motion through space.

Rotational motion is concerned with the molecule spinning or turning around its own center of mass. For a molecule, the number of rotational degrees of freedom depends on its geometry. A linear molecule like NO has 2 rotational degrees of freedom because it can rotate about two perpendicular axes that run through its center of mass. The third possible axis of rotation, which lies along the bond axis, does not produce a different configuration and is therefore not counted. In contrast, a non-linear molecule such as NO₂ has 3 rotational degrees of freedom, because it can rotate around three different axes intersecting its center of mass.

These rotations, like translations, are a form of kinetic energy. The energy absorbed in rotational motion contributes less to the internal energy and heat capacity at lower temperatures because quantum effects limit the number of rotational energy levels that can be populated. However, at higher temperatures, rotational motion becomes more significant. The distinction between two and three rotational degrees of freedom among linear and non-linear molecules, respectively, also influences their heat capacities and is a key factor in explaining the difference in molar heat capacity between NO and NO₂.