Understanding different variations of the specific heat capacity

Question

The specific heat capacity can be defined as the necessary heat to increase the temperature of one unit of mass of an object made out of a certain material by one unit.

So the specific heat capacity is related to a substance, it's mass and it's temperature increase.

Now, if we consider a polyatomic gas, then for the specific heat capacity one can write (in constant volume):

$C_v=C_v^{tra}+C_v^{vib}+C_v^{rot}$

where:

$C_v^{tra}$ is the translational specific heat capacity.

$C_v^{vib}$ is the vibrational specific heat capacity.

$C_v^{rot}$ is the rotational specific heat capacity.

Let's consider the $C_v^{tra}$ (the same question I have is valid for the rest). How to understand the translational specific heat capacity in constant volume/pressure? And how it's different then the other two?

If I had to guess, I guess the translational specific heat capacity, relates somehow, the increase of temperature with the velocity of the propagation of the particles? I am not sure

Answer 1

The heat capacity tells you how much the internal energy increases when temperature is increased by $dT$. Internal energy has several contributions, e.g., translational, vibrational, rotational and others. When we add an amount $dU$ of energy to a molecule some of that energy will go into translational (molecules will move faster on average), some will go to vibrational (bond will vibrate faster), and so on. Each heat capacity tells us how much of the added energy goes into each different type of molecular motion.

Answer 2

If I had to guess, I guess the translational specific heat capacity, relates somehow, the increase of temperature with the velocity of the propagation of the particles? I am not sure

If you do a classical analogy argument, the translational heat capacity is the increase in kinetic energy with the gas particles moving faster. This is, of course, the standard $1/2mv^{2}$ equation, but the gas particles have different velocities, so it is actually the mean square for the velocity distribution.

You can see this for the Maxwell-Boltzman distribution here.

Perfect ideal gasses have single element particles, so there is no vibration, rotation, or interparticle interaction. Real substances have other energetic phenomena, and so additional contributors to the heat capacity.

The reason way "at constant volume" and "at constant pressure" are different is because at constant pressure, the gas changes volume, so work crosses the system boundary as well. For an ideal gas, the change in temperature is still due to the same change in translational energy, but the amount of heat transfer involved in the process is different due to the work involvement as well. For real substances, the other factors could make the final total internal energy be different, since the states are different after constant volume or constant pressure heat transfer.

Note that we always deal with approximations to idealness. Generally, monatomic gases will fit the ideal behavior closer over more property range. Because O2 for example is diatomic, there must necessarily be additional contributors to the heat capacity than translation (vibration and rotation). At sufficiently low densities, etc., ideal gas theory can include these modified heat capacities. However, the deviation from ideal behavior will generally come faster than monatomic gasses such as He.