How molecules take up thermal energy

Vibrational and rotational motions are not possible for monatomic species such as the noble gas elements, so these substances have the lowest heat capacities. Moreover, as you can see in the leftmost column of Table 4, their heat capacities are all the same. This reflects the fact that translational motions are the same for all particles all such motions can be resolved into three directions in space, each contributing one degree of freedom to the molecule and 1/2 R to its heat capacity. (// is the gas constant, 8.314 J K ). 

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N-6 different ways or modes1. For mechanical reasons that we cannot go into here, each vibrational mode contributes R (rather than 1/2 R) to the total heat capacity. 

This result comes from advanced mechanics and will not be proven here. 

The total kinetic energy of a molecule is the sum of those due to the various kinds of motions  

When a monatomic gas absorbs heat, all of the energy ends up in translational motion, and thus goes to increase its temperature. In a polyatomic gas, by contrast, the absorbed energy is partitioned among the other kinds of motions since only the translational motions contribute to the temperature, the temperature rise is smaller, and thus the heat capacity is larger. 

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