When the heat conduction model of Eq. (2.54 ) is substituted into Eq. (2.51 ), we produce an equation with two variables and . For example, starting with Eq. (2.56 ) for internal energy, and ignoring mechanical power and heat source terms, gives
For diatomic gases, e.g. nitrogen , molecular motion is translational (left) and rotational (centre) at lower temperature (in the gas state). At higher temperature, vibrational motion is excited along its bond (right). Heat capacity for diatomic gases are then a function of temperature as shown below for .
Rotational motion provides 2 DoFs (rotational energy about the axis is negligible), giving . Beyond 500K, vibrational motion provides 2 additional DoFs (one kinetic, one potential), causing a gradual transition to . At high temperature, molecules dissociate and increases further.
A thermally perfect gas is an ideal gas with . Often can be treated as constant over a range of , e.g. T 600K for . Otherwise, accurate calculations require a suitable model of . An imperfect gas exhibits .